%!PS-AdobeFont-1.1: CMCSC10 1.0 %%CreationDate: 1991 Aug 18 17:46:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 97 /a put dup 107 /k put dup 109 /m put readonly def /FontBBox{14 -250 1077 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 19 /acute put dup 38 /ampersand put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 123 /endash put dup 124 /emdash put dup 127 /dieresis put readonly def /FontBBox{-163 -250 1146 969}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSL10 %!PS-AdobeFont-1.1: CMSL10 1.0 %%CreationDate: 1991 Aug 20 16:40:20 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSL10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -9.46 def /isFixedPitch false def end readonly def /FontName /CMSL10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 16 /dotlessi put dup 18 /grave put dup 19 /acute put dup 25 /germandbls put dup 39 /quoteright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put dup 122 /z put dup 127 /dieresis put readonly def /FontBBox{-62 -250 1123 750}readonly def currentdict end currentfile eexec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All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 27 /sigma put dup 28 /tau put dup 96 /lscript put readonly def /FontBBox{11 -250 1241 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put readonly def /FontBBox{-20 -250 1193 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: EUFM10 %!PS-AdobeFont-1.1: EUFM10 2.1 %%CreationDate: 1992 Nov 20 17:36:20 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 111 /o put dup 115 /s put readonly def /FontBBox{-26 -224 1055 741}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 65 /A put dup 67 /C put dup 69 /E put dup 72 /H put dup 73 /I put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 84 /T put dup 85 /U put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put readonly def /FontBBox{-53 -251 1139 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 80 /summationtext put dup 88 /summationdisplay put dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 50 /element put dup 54 /negationslash put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSL12 %!PS-AdobeFont-1.1: CMSL12 1.0 %%CreationDate: 1991 Aug 20 16:40:41 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSL12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -9.46 def /isFixedPitch false def end readonly def /FontName /CMSL12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 46 /period put dup 65 /A put dup 79 /O put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-56 -251 1102 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 46 /period put dup 101 /e put dup 103 /g put dup 105 /i put dup 115 /s put readonly def /FontBBox{-36 -251 1103 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 54 /six put dup 56 /eight put dup 61 /equal put dup 100 /d put dup 110 /n put readonly def /FontBBox{-36 -250 1070 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 21 /lambda put dup 22 /mu put dup 27 /sigma put dup 28 /tau put dup 33 /omega put dup 58 /period put dup 59 /comma put dup 61 /slash put dup 73 /I put dup 96 /lscript put dup 100 /d put dup 101 /e put dup 103 /g put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 121 /y put readonly def /FontBBox{-24 -250 1110 750}readonly def currentdict end currentfile eexec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All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /R put dup 84 /T put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 6 /plusminus put dup 7 /minusplus put dup 14 /openbullet put dup 18 /reflexsubset put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 25 /approxequal put dup 33 /arrowright put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 65 /A put dup 66 /B put dup 69 /E put dup 72 /H put dup 77 /M put dup 80 /P put dup 85 /U put dup 94 /logicaland put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 114 /nabla put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec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1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 17 /eta put dup 21 /lambda put dup 22 /mu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 33 /omega put dup 34 /epsilon put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 68 /D put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 88 /X put dup 89 /Y put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 115 /s put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-30 -250 1026 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 60 /less put dup 100 /d put dup 110 /n put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 14 /ffi put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 35 /numbersign put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 18 /grave put dup 22 /macron put dup 25 /germandbls put dup 33 /exclam put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 91 /bracketleft put dup 93 /bracketright put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put readonly def /FontBBox{-34 -251 988 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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66 70 62 70 68 82 57 70 48 35 68 70 59 62 69 65 64 68 6[28 46 46 46 2[46 46 46 46 46 1[28 33 28 5[70 18[46 7[56 11[{}63 90.9091 /CMTI10 rf /Fc 128[45 4[40 48 1[66 48 51 35 36 36 1[51 45 51 76 25 48 1[25 51 45 28 40 51 40 1[45 6[56 68 1[93 68 1[66 51 67 71 62 71 68 83 57 71 47 33 68 71 59 62 69 66 64 68 6[25 11[25 30 25 4[25 13[45 5[45 45 1[25 16[{}56 90.9091 /CMSL10 rf /Fd 159[26 67[29 36 27[{}3 49.8132 /CMMI6 rf /Fe 204[30 30 30 30 48[{}4 49.8132 /CMR6 rf /Ff 140[44 3[49 111[{}2 99.6264 /EUFM10 rf /Fg 134[71 2[71 75 52 53 55 1[75 67 75 112 37 2[37 75 67 1[61 75 60 75 65 12[94 3[92 1[105 4[50 3[88 1[97 13[67 67 67 67 67 2[37 45 45[{}32 119.552 /CMBX12 rf /Fh 143[100 23[144 7[105 46[79 79 12[73 73 8[58 58 6[46 46{}11 99.6264 /CMEX10 rf /Fi 149[20 51[0 3[47 46[35 2[55{}5 66.4176 /CMSY8 rf /Fj 134[51 51 70 51 54 38 38 38 1[54 49 54 81 27 51 1[27 54 1[30 43 54 43 54 49 12[70 4[76 13[73 18[27 1[27 2[38 38 27[54 57 11[{}31 99.6264 /CMSL12 rf /Fk 136[81 2[44 44 46 2[56 62 93 31 59 2[62 56 1[51 62 50 14[78 18[85 14[56 56 2[31 46[{}19 99.6264 /CMBX12 rf /Fl 140[40 9[30 1[45 1[45 54[30 46[{}5 99.6264 /CMTI12 rf /Fm 145[39 9[39 38[55 4[35 1[35 1[35 35 35 35 35 4[55 43[{}11 66.4176 /CMR8 rf /Fn 134[35 8[36 1[43 62 1[37 29 24 1[34 1[33 36 3[29 22[31 11[35 1[20 20 24[44 4[31 40 4[42 41 8[40 45 11[{}22 66.4176 /CMMI8 rf /Fo 165[66 5[66 1[72 82[{}3 99.6264 /MSBM10 rf /Fp 146[74 1[63 9[61 97[{}3 99.6264 /CMCSC10 rf /Fq 141[83 7[28 39 39 50 50 7[66 8[62 4[69 2[120 4[84 2[53 2[65 80 9[0 0 3[66 16[100 7[77 77 2[77 77 1[77 3[50 6[77 77 3[77 28 77{}29 99.6264 /CMSY10 rf /Fr 133[45 48 55 70 47 56 1[46 1[43 49 1[58 85 1[51 40 33 56 47 48 1[51 42 41 51 41 6[57 81 1[57 1[57 60 74 1[63 75 2[66 83 54 43 81 77 63 1[81 3[51 1[76 49 76 27 27 23[46 61 5[55 50 55 43 1[59 57 3[48 3[51 55 62 11[{}56 99.6264 /CMMI12 rf /Fs 134[85 3[90 63 64 66 85 90 81 90 134 45 2[45 90 81 49 74 90 72 90 78 11[124 2[120 4[153 3[60 126 4[117 12[81 81 81 81 81 81 49[{}32 143.462 /CMBX12 rf /Ft 149[25 20[57 84[71{}3 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Fu(d)29 b Ft(\000)g Fu(n)43 b Fv(slo)m(w)h(frequencies.)80 b(Suc)m(h)43 b Fu(d)p Fv({parameter)353 2264 y(families)26 b(of)g Fu(d)p Fv({tori)i(do)d(surviv)m(e)h (Hamiltonian)i(p)s(erturbations)c(as)i(Can)m(tor)h(families)g(of)f Fu(d)p Fv({tori.)353 2377 y(A)k(p)s(erturbation)f(of)h(a)h(sup)s(erin)m (tegrable)f(system)g(that)h(admits)f(a)h(b)s(etter)f(appro)m(ximation)h (b)m(y)353 2490 y(a)37 b(non-degenerate)i(in)m(tegrable)g(p)s (erturbation)d(of)i(the)g(sup)s(erin)m(tegrable)f(system)h(is)f(said)h (to)353 2603 y(remo)m(v)m(e)25 b(the)f(degeneracy)-8 b(.)40 b(In)23 b(the)h(minimal)h(case)g Fu(d)g Fv(=)g Fu(n)7 b Fv(+)g(1)24 b(this)g(can)g(b)s(e)g(ac)m(hiev)m(ed)i(b)m(y)d (means)353 2716 y(of)36 b(a)m(v)m(eraging,)k(but)35 b(the)h(more)g(in)m (tegrals)i(of)e(motion)h(the)f(sup)s(erin)m(tegrable)g(system)g(admits) 353 2829 y(the)30 b(more)h(di\016cult)f(b)s(ecomes)h(the)f(p)s (erturbation)f(analysis.)109 3158 y Fs(1)161 b(In)l(tro)t(duction)109 3377 y Fy(In)m(tegrable)28 b(Hamiltonian)g(systems)i(are)d(in)h Fr(d)f Fq(\025)h Fy(2)g(degrees)g(of)f(freedom)i(the)e(exception)j (rather)d(than)109 3498 y(the)35 b(rule.)53 b(This)36 b(mak)m(es)h(the)f(b)s(eha)m(viour)g(of)f(the)g(dynamics)i(under)g (non-in)m(tegrable)e(p)s(erturbations)109 3618 y(one)i(of)f(the)i(k)m (ey)g(prop)s(erties)g(of)e(an)h(in)m(tegrable)h(system.)58 b(On)37 b(a)g(general)g(lev)m(el)h(one)g(obtains)f(infor-)109 3738 y(mation)c(on)g(motions)h(of)f(an)h(op)s(en)f(subset)i(of)e(all)h (Hamiltonian)g(systems)h([32],)f(and)f(for)g(concretely)109 3859 y(giv)m(en)c(systems)h(one)f(can)f(address)i(the)e(question)i(ho)m (w)f(the)f(dynamics)i(organizes)f(the)f(phases)i(space.)255 3979 y(The)25 b(t)m(ypical)f(b)s(ounded)h(motion)f(in)g(an)f(in)m (tegrable)i(system)g(is)g(quasi-p)s(erio)s(dic,)h(spinning)f(densely) 109 4099 y(around)37 b(in)m(v)-5 b(arian)m(t)39 b(tori.)58 b(This)39 b(mak)m(es)g(the)f(phase)h(space)g(a)e(rami\014ed)i(torus)f (bundle,)i(the)e(regular)109 4220 y(\014bres)f(of)e Fr(n)p Fy({tori)h(forming)f(families)i(that)f(are)g(organized)g(b)m(y)h (families)g(of)e(in)m(v)-5 b(arian)m(t)36 b(\()p Fr(n)25 b Fq(\000)g Fy(1\){tori,)109 4340 y(whic)m(h)40 b(in)f(turn)g(are)g (organized)g(b)m(y)g(bifurcating)g(\()p Fr(n)27 b Fq(\000)g Fy(1\){tori)38 b(and)g(b)m(y)i(\()p Fr(n)27 b Fq(\000)f Fy(2\){tori)38 b(and)h(so)g(on,)109 4461 y(do)m(wn)c(to)g(the)g (equilibria)h(of)f(the)g(Hamiltonian)g(system.)52 b(In)35 b(the)h(case)f Fr(n)d Fy(=)f Fr(d)k Fy(of)f(Lagrangean)g(tori)109 4581 y(one)j(can)h(apply)f Fp(kam)h Fy(theory)g(directly)h(to)e(the)g (in)m(tegrable)i(system,)h(yielding)e(p)s(ersistence)i(of)d(the)109 4701 y(Lagrangean)42 b(tori)g([2,)h(9)o(,)g(18],)i(of)d(elliptic/h)m (yp)s(erb)s(olic)j(lo)m(w)m(er)f(dimensional)g(tori)e([35,)g(10,)h(9)o (])g(and)109 4822 y(of)35 b(in)m(v)-5 b(arian)m(t)37 b(tori)f(that)g(undergo)g(bifurcations)h([28,)f(29,)g(7].)54 b(Suc)m(h)37 b(results)h(are)e(v)-5 b(alid)37 b(under)g(non-)109 4942 y(degeneracy)28 b(conditions)f([30],)h(the)e(b)s(est-kno)m(wn)i(b) s(eing)f(Kolmogoro)m(v's)g(non-degeneracy)h(condition)109 5062 y(whic)m(h)38 b(expresses)h(that)d(the)h(frequency)h(mapping)f (from)f(the)h(families)g(of)f(Lagrangean)g(tori)g(to)g Fo(R)3848 5026 y Fn(d)109 5183 y Fy(is)d(a)f(submersion.)255 5303 y(Sup)s(erin)m(tegrable)46 b(systems)g(fail)e(to)g(satisfy)h(suc)m (h)g(non-degeneracy)h(conditions)f(on)f(a)g(funda-)109 5424 y(men)m(tal)e(lev)m(el)i(and)e(are)g(consequen)m(tly)i(termed)f (prop)s(erly)g(degenerate)g(systems)h(in)e([3].)71 b(F)-8 b(amous)1974 5778 y(1)p eop end %%Page: 2 2 TeXDict begin 2 1 bop 109 208 a Fy(examples)36 b(are)f(the)g(2{b)s(o)s (dy)f(problem,)h(resonan)m(t)h(oscillators,)g(rigid)e(b)s(o)s(dies)h (\014xed)h(at)e(their)h(cen)m(tre)109 329 y(of)29 b(mass)h(and)g(com)m (binations)h(of)e(these;)i(in)f(particular)g(the)g(Kepler)g(appro)m (ximation)g(of)f(the)h Fr(n)p Fy({b)s(o)s(dy)109 449 y(problem,)j(with)g(all)g(planets)g(only)g(in)m(teracting)h(with)f(the) g(sun)g(and)g(not)g(with)g(eac)m(h)g(other.)255 569 y(The)d(additional) f(in)m(tegrals)g(of)g(motion)g(of)f(a)g(sup)s(erin)m(tegrable)j(system) g(simplify)f(the)f(analysis)i(of)109 690 y(the)25 b(\(unp)s(erturb)s (ed\))h(dynamics.)43 b(F)-8 b(or)24 b(instance,)k(in)d(the)g(maximally) h(sup)s(erin)m(tegrable)h(case)f(of)e(2)p Fr(d)7 b Fq(\000)g Fy(1)109 810 y(conserv)m(ed)41 b(quan)m(tities)g(the)e(regular)g (\014bres)h(of)e(the)i(rami\014ed)f(torus)g(bundle)h(are)f(merely)i(p)s (erio)s(dic)109 931 y(orbits)35 b(\(or)f(circles)i(of)e(equilibria\))i (organized)f(b)m(y)h(bifurcating)e(p)s(erio)s(dic)h(orbits)g(and)g(b)m (y)h(\(isolated\))109 1051 y(equilibria.)79 b(Ho)m(w)m(ev)m(er,)50 b(the)44 b(price)h(to)f(pa)m(y)h(is)g(that)f(the)g(p)s(erturbation)h (analysis)g(b)s(ecomes)h(more)109 1171 y(complicated)39 b(and)f(in)m(v)m(olv)m(es)j(the)d(p)s(erturbation)h(as)f(w)m(ell.)61 b(In)38 b(the)h(extreme)h(case)e(of)g(a)g(maximally)109 1292 y(sup)s(erin)m(tegrable)48 b(system)h(one)e(has)h(to)e(study)i (the)g(Hamiltonian)f(system)i(in)e Fr(d)31 b Fq(\000)i Fy(1)46 b(degrees)j(of)109 1412 y(freedom)38 b(that)f(is)h(obtained)g (b)m(y)g(a)m(v)m(eraging)g(the)g(p)s(erturbation)g(along)e(the)i(unp)s (erturb)s(ed)h(p)s(erio)s(dic)109 1532 y(orbits)33 b(and)f(reducing)i (the)f(acquired)h Fr(S)1598 1496 y Fm(1)1637 1532 y Fy({symmetry)-8 b(.)255 1653 y(A)38 b(Hamiltonian)h(system)h(that)f(is)f(closer)i(to)e (the)h(p)s(erturb)s(ed)g(system)h(than)e(the)h(unp)s(erturb)s(ed)109 1773 y(sup)s(erin)m(tegrable)i(system)g(is)e(called)h(an)g(in)m (termediate)g(system)i(if)d(it)g(has)g(additional)h(in)m(tegrals)g(of) 109 1894 y(motion)e(compared)g(to)g(the)g(p)s(erturb)s(ed)h(system)h (but)e(less)h(in)m(tegrals)g(of)f(motion)g(than)g(the)g(unp)s(er-)109 2014 y(turb)s(ed)30 b(system.)44 b(Normal)30 b(forms)f(pro)m(vide)i (the)f(means)h(to)e(obtain)g(in)m(termediate)i(systems,)i(and)d(the)109 2134 y(most)i(ob)m(vious)i(w)m(a)m(y)f(is)g(to)e(normalize)i(b)m(y)g(a) m(v)m(eraging)g(along)f(the)h(unp)s(erturb)s(ed)g(dynamics.)45 b(In)32 b(gen-)109 2255 y(eral)g(suc)m(h)i(an)f(in)m(termediate)h (system)h(inherits)f Fr(n)22 b Fy(+)g(1)32 b(in)m(tegrals)i(of)e (motion)g(in)h(in)m(v)m(olution)h(and)f(the)109 2375 y(p)s(erturbation)d(is)g(said)h(to)e(remo)m(v)m(e)j(the)e(degeneracy)i ([3])e(in)g(case)g(of)g(a)g(\(non-degenerate\))g(in)m(tegrable)109 2496 y(in)m(termediate)k(system.)255 2616 y(In)45 b(the)g(minimally)h (sup)s(erin)m(tegrable)g(case)f Fr(d)j Fy(=)g Fr(n)30 b Fy(+)g(1)44 b(the)h(in)m(termediate)i(system)f(is)f(auto-)109 2736 y(matically)j(in)m(tegrable.)91 b(Bet)m(w)m(een)50 b(these)g(t)m(w)m(o)e(extreme)i(cases)f(there)g(is)f(a)g(whole)h (hierarc)m(h)m(y)h(of)109 2857 y(sup)s(erin)m(tegrable)36 b(systems)g(with)f(2)p Fr(d)22 b Fq(\000)i Fr(n)34 b Fy(indep)s(enden)m(t)i(conserv)m(ed)h(quan)m(tities,)f Fr(n)31 b Fy(=)f(1)p Fr(;)17 b(:)g(:)g(:)e(;)i(d)23 b Fq(\000)g Fy(1.)109 2977 y(Reducing)29 b(the)g(acquired)i Fo(T)1154 2941 y Fn(n)1201 2977 y Fy({symmetry)g(leads)e(to)g Fr(d)14 b Fq(\000)g Fr(n)29 b Fy(degrees)h(of)f(freedom)g(and)g(in)g (general)g(one)109 3097 y(is)j(bac)m(k)h(to)e(studying)i(a)e(general)i (Hamiltonian)f(system)h(\(alb)s(eit)f(with)g(few)m(er)h(degrees)g(of)e (freedom\).)109 3218 y(A)c(general)h(statemen)m(t)h(of)e(quasi-p)s (erio)s(dic)i(p)s(ersistence)h(is)e(pro)m(v)m(en)h(in)e([3])h(under)g (the)g(condition)g(that)109 3338 y(the)h(reduced)h(in)m(termediate)h (system)g(is)e(a)g(non-degenerate)g(in)m(tegrable)h(system)h(and)e(the) g(frequency)109 3459 y(mapping)k(of)f(the)h(sup)s(erin)m(tegrable)h (system)h(on)m(to)d Fo(R)2087 3422 y Fn(n)2167 3459 y Fy(is)h(a)f(submersion)j(as)d(w)m(ell.)255 3579 y(This)41 b(review)g(is)f(organized)g(as)g(follo)m(ws.)65 b(Section)40 b(2)g(con)m(tains)g(the)g(b)s(eha)m(viour)h(of)e(the)h(unp)s(er-)109 3699 y(turb)s(ed)i(system,)j(in)d(particular)g(the)f(geometry)i(imp)s (osed)f(b)m(y)h(the)e(dynamics.)72 b(In)42 b(section)h(3)e(the)109 3820 y(treatmen)m(t)i(of)g(p)s(erturb)s(ed)g(minimally)h(sup)s(erin)m (tegrable)g(systems)i(is)d(exempli\014ed)i(for)d(the)h(Euler)109 3940 y(top)g(and)g(the)h(logarithmic)g(p)s(oten)m(tial.)76 b(Section)45 b(4)e(is)h(dev)m(oted)h(to)e(resonan)m(t)h(oscillators,)j (where)109 4061 y(sup)s(erin)m(tegrabilit)m(y)32 b(ma)m(y)e(or)g(ma)m (y)g(not)g(b)s(e)g(minimal,)h(and)f(more)g(examples)h(of)e(the)i (latter)e(are)h(giv)m(en)109 4181 y(in)i(section)i(5.)43 b(The)34 b(\014nal)f(conclusions)h(constitute)g(section)g(6.)109 4514 y Fs(2)161 b(Unp)t(erturb)t(ed)52 b(dynamics)109 4733 y Fy(Let)26 b(\()p Fq(P)8 b Fr(;)17 b(\033)t Fy(\))27 b(b)s(e)f(a)g(symplectic)j(manifold)e(of)e(dimension)j(dim)18 b Fq(P)36 b Fy(=)27 b(2)p Fr(d)f Fy(and)g(de\014ne)i(\014rst)f (Hamiltonian)109 4853 y(v)m(ector)38 b(\014elds)h(b)m(y)f(means)g(of)f Fr(\033)t Fy(\()p Fr(X)1381 4887 y(H)1474 4853 y(;)17 b(Y)k Fy(\))35 b(=)h(d)p Fr(H)8 b Fy(\()p Fr(Y)21 b Fy(\))37 b(for)g(all)g(v)m(ector)h(\014elds)h Fr(Y)58 b Fy(on)38 b Fq(P)45 b Fy(and)38 b(then)g(the)109 4973 y(P)m(oisson)c(brac)m(k)m (et)g Fq(f)p Fr(f)5 b(;)17 b(g)t Fq(g)26 b Fy(:=)i Fq(\000)p Fr(\033)t Fy(\()p Fr(X)1448 5008 y(f)1512 4973 y(;)17 b(X)1620 4988 y(g)1674 4973 y Fy(\).)255 5105 y(The)28 b(simplest)h(w)m(a)m(y)f(to)e(de\014ne)i(a)f(sup)s(erin)m(tegrable)i (system)g(is)e(to)g(require)h(it)f(to)f(b)s(e)h(an)g(in)m(tegrable)109 5226 y(Hamiltonian)35 b(system,)i(with)e Fr(d)g Fy(in)m(tegrals)g(of)f (motion)h(in)g(in)m(v)m(olution)h Fq(f)p Fr(I)2845 5241 y Fn(k)2888 5226 y Fr(;)17 b(I)2975 5241 y Fn(`)3007 5226 y Fq(g)31 b Fy(=)h(0)i(and)h(admitting)109 5346 y(at)29 b(least)i(one)f(more)g(conserv)m(ed)i(quan)m(tit)m(y)g(indep)s (enden)m(t)g(of)d Fr(I)2399 5361 y Fm(1)2438 5346 y Fr(;)17 b(:)g(:)g(:)f(;)h(I)2700 5361 y Fn(d)2740 5346 y Fy(.)43 b(Since)31 b(the)f(P)m(oisson)i(brac)m(k)m(et)1974 5778 y(2)p eop end %%Page: 3 3 TeXDict begin 3 2 bop 109 208 a Fy(of)39 b(t)m(w)m(o)h(conserv)m(ed)h (quan)m(tities)g(is)f(again)f(an)g(in)m(tegral)h(of)f(motion)g(all)h (these)g(form)f(a)g(Lie)h(algebra,)109 329 y(and)c(as)g(the)h (symplectic)i(structure)f(is)e(non-degenerate)h(suc)m(h)h(a)e(Lie)h (algebra)f(is)h(necessarily)i(non-)109 449 y(comm)m(utativ)m(e)25 b(once)f(the)f(dimension)i(exceeds)h Fr(d)p Fy(.)40 b(One)23 b(therefore)h(also)f(sp)s(eaks)i(of)d(non-comm)m(utativ)m(e)109 569 y(in)m(tegrabililt)m(y)44 b([34)o(].)74 b(The)43 b(follo)m(wing)g(result)g(from)f([23])g(clari\014es)i(the)f(semi-lo)s (cal)g(situation)g(\()p Fl(i.e.)109 690 y Fy(lo)s(cally)33 b(around)f(a)g(maximal)i(torus\).)109 901 y Fk(Theorem)k(2.1)49 b Fj(On)41 b(the)h(subset)h Fq(M)g(\022)h(P)50 b Fj(of)41 b(the)h(phase)g(space)h(with)f(symplectic)i(structure)f Fr(\033)109 1021 y Fj(let)c Fr(f)50 b Fy(:)40 b Fq(M)e(\000)-16 b(!)39 b(B)k(\022)c Fo(R)1075 985 y Fm(2)p Fn(d)p Fi(\000)p Fn(n)1288 1021 y Fj(b)s(e)h(a)f(submersion)i(with)f(compact)g(and)f (connected)i(\014bres)g(\(hence,)109 1141 y(a)35 b(\014bration\).)54 b(Assume)38 b(that)e Fq(f)p Fr(f)1353 1156 y Fn(i)1381 1141 y Fr(;)17 b(f)1473 1156 y Fn(j)1509 1141 y Fq(g)33 b Fy(=)g Fr(P)1764 1156 y Fn(ij)1849 1141 y Fq(\016)25 b Fr(f)i(;)33 b(i;)17 b(j)40 b Fy(=)33 b(1)p Fr(;)17 b(:)g(:)g(:)e(;)i Fy(2)p Fr(d)24 b Fq(\000)h Fr(n)36 b Fj(and)g(that)g(the)g(matrix)h Fr(P)109 1262 y Fj(with)c(en)m(tries)i Fr(P)707 1277 y Fn(ij)796 1262 y Fy(:)28 b Fq(M)h(\000)-17 b(!)29 b Fo(R)k Fj(has)g(rank)g Fy(2\()p Fr(d)22 b Fq(\000)h Fr(n)p Fy(\))33 b Fj(at)g(all)g(p)s(oin)m(ts)h(of)e Fq(B)s Fj(.)46 b(Then)34 b(ev)m(ery)h(\014bre)e(of)g Fr(f)44 b Fj(is)109 1382 y(di\013eomorphic)33 b(to)g Fo(T)904 1346 y Fn(n)983 1382 y Fj(and)g(the)g(\014bration)g Fr(f)43 b Fj(has)33 b(lo)s(cal)f(trivialisations)i(whic)m(h)g(are)f (symplectic.)109 1593 y Fy(T)-8 b(aking)40 b(for)g Fq(M)g Fy(the)g(regular)g(part)g(of)g(the)h(phase)g(space)g(where)g(the)g(in)m (tegrals)g(are)f(indep)s(enden)m(t)109 1714 y(and)k(ha)m(v)m(e)i (compact)f(\(common\))g(lev)m(el)h(sets,)j(ev)m(ery)e(\014bre)e(of)f Fr(f)55 b Fy(has)45 b(a)f(neigh)m(b)s(ourho)s(o)s(d)h Fq(U)55 b Fy(with)109 1834 y(co-ordinates)1003 1954 y(\()p Fr(x;)17 b(y)t(;)g(q)t(;)g(p)p Fy(\))53 b(:)84 b Fq(U)93 b(\000)-16 b(!)82 b Fo(T)2042 1913 y Fn(n)2112 1954 y Fq(\002)22 b Fo(R)2283 1913 y Fn(n)2353 1954 y Fq(\002)g Fo(R)2524 1913 y Fn(d)p Fi(\000)p Fn(n)2684 1954 y Fq(\002)h Fo(R)2856 1913 y Fn(d)p Fi(\000)p Fn(n)3764 1954 y Fy(\(1\))109 2122 y(suc)m(h)34 b(that)e(the)h(lev)m(el)h(sets)g(of)e Fr(f)43 b Fy(coincide)34 b(with)f(the)g(lev)m(el)i(sets)e(of)g(\()p Fr(y)t(;)17 b(q)t(;)g(p)p Fy(\))30 b(and)1122 2420 y Fr(\033)1181 2457 y Ft(jU)1359 2420 y Fy(=)1569 2296 y Fn(n)1518 2326 y Fh(X)1526 2538 y Fn(k)r Fm(=1)1679 2420 y Fy(d)p Fr(x)1788 2435 y Fn(k)1853 2420 y Fq(^)23 b Fy(d)p Fr(y)2044 2435 y Fn(k)2136 2420 y Fy(+)2266 2296 y Fn(d)p Fi(\000)p Fn(n)2261 2326 y Fh(X)2274 2538 y Fn(`)p Fm(=1)2422 2420 y Fy(d)p Fr(q)2519 2435 y Fn(`)2575 2420 y Fq(^)f Fy(d)p Fr(p)2766 2435 y Fn(`)2848 2420 y Fr(:)109 2715 y Fy(These)k(co-ordinates)f(are)g(Nekhoroshev's)i (generalized)f(action)f(angle)g(v)-5 b(ariables)25 b([36].)41 b(In)25 b(particular)109 2836 y(w)m(e)36 b(see)h(that)f(lo)s(cally)g(w) m(e)g(ha)m(v)m(e)h Fr(d)f Fy(comm)m(uting)g(in)m(tegrals)h(of)e(motion) h Fr(y)2806 2851 y Fm(1)2845 2836 y Fr(;)17 b(:)g(:)g(:)e(;)i(y)3111 2851 y Fn(n)3158 2836 y Fr(;)g(p)3251 2851 y Fm(1)3290 2836 y Fr(;)g(:)g(:)g(:)f(;)h(p)3558 2851 y Fn(d)p Fi(\000)p Fn(n)3731 2836 y Fy(and)109 2956 y Fr(d)24 b Fq(\000)h Fr(n)37 b Fy(additional)f(conserv)m(ed)j(quan)m(tities)f Fr(q)1791 2971 y Fm(1)1831 2956 y Fr(;)17 b(:)g(:)g(:)e(;)i(q)2092 2971 y Fn(d)p Fi(\000)p Fn(n)2267 2956 y Fy(that)36 b(are)g(in)h(in)m (v)m(olution)g(with)g(the)g(\014rst)g Fr(n)109 3076 y Fy(in)m(tegrals)c Fr(y)550 3091 y Fm(1)589 3076 y Fr(;)17 b(:)g(:)g(:)f(;)h(y)856 3091 y Fn(n)902 3076 y Fy(.)255 3197 y(On)37 b Fq(M)f Fy(the)h(sup)s(erin)m(tegrable)i(system)g(is)e (equiv)-5 b(arian)m(t)38 b(with)f(resp)s(ect)i(to)d(the)h Fo(T)3305 3161 y Fn(n)3353 3197 y Fy({action)f Fr(x)g Fq(7!)109 3317 y Fr(x)15 b Fy(+)g Fr(\030)34 b Fy(and)29 b(in)g(examples)i(this)f(action)f(often)g(extends)i(to)e(all)g(of)g Fq(P)37 b Fy(\(simple)31 b(coun)m(ter-examples)h(with)109 3438 y Fr(n)c Fy(=)f(1)f(can)h(b)s(e)f(constructed)i(from)e (Hamiltonian)h(systems)h(in)f(one)f(degree)i(of)d(freedom\).)42 b(Reduction)109 3558 y(of)34 b(suc)m(h)i(a)e Fo(T)593 3522 y Fn(n)641 3558 y Fy({action)g(leads)i(to)e Fr(d)23 b Fq(\000)h Fr(n)35 b Fy(degrees)h(of)e(freedom)i(where)f(\()p Fr(q)t(;)17 b(p)p Fy(\))34 b(pro)m(vide)i(canonical)g(co-)109 3678 y(ordinates.)55 b(This)37 b(can)g(in)f(particular)g(b)s(e)h (applied)g(to)f(an)g(in)m(termediate)i(system)g(obtained)e(from)g(a)109 3799 y(giv)m(en)d(p)s(erturbation)g(b)m(y)h(a)m(v)m(eraging)f(o)m(v)m (er)h(the)f(unp)s(erturb)s(ed)g(fast)g(motion.)255 3919 y(The)28 b(p)s(erturbation)g(is)f(said)h(to)f(remo)m(v)m(e)i(the)e (degeneracy)i(\(of)e(the)h(sup)s(erin)m(tegrable)h(or)d(`prop)s(erly) 109 4040 y(degenerate')40 b(unp)s(erturb)s(ed)f(system\))i(if)d(the)h (in)m(termediate)i(system)f(is)f(indeed)h(in)m(tegrable.)63 b(Then)109 4160 y(w)m(e)31 b(ma)m(y)h(c)m(ho)s(ose)g(\()p Fr(q)t(;)17 b(p)p Fy(\))30 b(to)g(b)s(e)h(action)g(angle)g(v)-5 b(ariables)31 b(of)f(the)i(reduced)g(in)m(termediate)g(system)h(and)109 4280 y(the)g(Lagrangean)f(tori)1507 4401 y Fo(T)1573 4360 y Fn(n)1643 4401 y Fq(\002)22 b Fo(T)1808 4360 y Fn(d)p Fi(\000)p Fn(n)1969 4401 y Fq(\002)h(f)p Fr(y)t Fq(g)e(\002)h(f)p Fr(p)p Fq(g)109 4568 y Fy(with)33 b(angular)f(v)-5 b(ariables)33 b(\()p Fr(x;)17 b(q)t Fy(\))33 b(ha)m(v)m(e)h(frequency)g (v)m(ectors)879 4773 y Fr(!)t Fy(\()p Fr(y)t(;)17 b(p)p Fy(\))81 b(=)i(\()p Fr(!)1504 4788 y Fm(1)1543 4773 y Fy(\()p Fr(y)t Fy(\))p Fr(;)17 b(:)g(:)g(:)e(;)i(!)1950 4788 y Fn(n)1996 4773 y Fy(\()p Fr(y)t Fy(\))p Fr(;)g(!)2229 4788 y Fn(n)p Fm(+1)2365 4773 y Fy(\()p Fr(y)t(;)g(p)p Fy(\))p Fr(;)g(:)g(:)g(:)d(!)2820 4788 y Fn(d)2860 4773 y Fy(\()p Fr(y)t(;)j(p)p Fy(\)\))645 b(\(2\))109 4978 y(where)30 b(the)g(last)g Fr(d)16 b Fq(\000)g Fr(n)28 b Fy(comp)s(onen)m(ts)j(|)e(those)h(that)f(do)h(dep)s(end)g(on)f(the)h (v)-5 b(ariables)30 b Fr(p)3316 4993 y Fm(1)3356 4978 y Fr(;)17 b(:)g(:)g(:)f(;)h(p)3624 4993 y Fn(d)p Fi(\000)p Fn(n)3791 4978 y Fy(|)109 5098 y(are)34 b(of)f(the)h(order)g(of)g(the)g (p)s(erturbation)g(\(while)h Fr(!)2020 5113 y Fm(1)2060 5098 y Fr(;)17 b(:)g(:)g(:)e(;)i(!)2339 5113 y Fn(n)2420 5098 y Fy(are)34 b(of)f(order)h(one\).)48 b(Non-degeneracy)109 5219 y(of)32 b(the)h(frequency)h(mapping)1313 5423 y Fr(!)59 b Fy(:)83 b Fq(M)g(\000)-16 b(!)82 b Fo(R)2061 5387 y Fn(d)2129 5423 y Fy(=)28 b Fo(R)2305 5387 y Fn(n)2374 5423 y Fq(\002)23 b Fo(R)2546 5387 y Fn(d)p Fi(\000)p Fn(n)1974 5778 y Fy(3)p eop end %%Page: 4 4 TeXDict begin 4 3 bop 109 208 a Fy(de\014ned)38 b(b)m(y)i(\(2\))d(then) h(yields)g(Can)m(tor)g(families)g(of)f(in)m(v)-5 b(arian)m(t)38 b(Lagrangean)e(tori)h(in)h(the)g(\(original\))109 329 y(p)s(erturb)s(ed)33 b(system,)i(cf.)d([3,)h(4].)255 449 y(The)g(lea)m(v)m(es)i(of)d(the)h(\014bration)f Fr(f)39 b Fy(:)28 b Fq(M)f(\000)-16 b(!)27 b(B)36 b Fy(are)c Fr(n)p Fy({tori)h(on)f(whic)m(h)i(the)f(symplectic)i(structure)109 569 y(v)-5 b(anishes;)34 b(one)f(also)f(sp)s(eaks)i(of)e(an)h (isotropic)g(\014bration)g([23)o(].)44 b(The)33 b(symplectic)j(lea)m(v) m(es)e(of)e(the)h(base)109 690 y(space)28 b Fq(B)j Fy(ha)m(v)m(e)d (dimension)h Fr(d)11 b Fq(\000)g Fr(n)28 b Fy(and)g(constitute)g(the)g (phase)g(space)g(for)f(the)h(reduced)g(in)m(termediate)109 810 y(system.)50 b(In)35 b(case)g(these)h(de\014ne)f(a)f(\014bration)h Fr(s)30 b Fy(:)h Fq(B)j(\000)-16 b(!)30 b(A)k Fy(the)h(comp)s(osition)g Fr(s)23 b Fq(\016)g Fr(f)42 b Fy(:)30 b Fq(M)h(\000)-16 b(!)30 b(A)k Fy(is)109 931 y(the)f(co-isotropic)h(\014bration)f([16,)g (23])g(and)h(the)f(actions)h Fr(y)2277 946 y Fm(1)2316 931 y Fr(;)17 b(:)g(:)g(:)f(;)h(y)2583 946 y Fn(n)2663 931 y Fy(conjugate)33 b(to)g(the)h(toral)f(angles)109 1051 y(are)f(lo)s(cal)h(co-ordinates)f(on)h(the)g(base)g(space)h Fq(A)p Fy(.)255 1171 y(The)j(\014bres)h(of)d(the)i(co-isotropic)g (\014bration)f(are)g(the)h(F)-8 b(ass\022)-49 b(o)36 b(\015o)m(w)m(ers)j([22)o(,)e(23)o(])f(and)h(form)f(them-)109 1292 y(selv)m(es)41 b(\014brations)f Fr(f)876 1307 y Fn(y)957 1292 y Fy(:)f Fr(f)1082 1256 y Fi(\000)p Fm(1)1176 1292 y Fy(\()p Fr(s)1260 1256 y Fi(\000)p Fm(1)1354 1292 y Fy(\()p Fr(y)t Fy(\)\))f Fq(\000)-16 b(!)39 b Fr(s)1804 1256 y Fi(\000)p Fm(1)1898 1292 y Fy(\()p Fr(y)t Fy(\),)h Fr(y)i Fq(2)e(A)p Fy(.)63 b(The)40 b(symplectic)i(leaf)d Fr(s)3387 1256 y Fi(\000)p Fm(1)3482 1292 y Fy(\()p Fr(y)t Fy(\))f(is)i(the)109 1412 y(cen)m(tre)c(of)e(the)i(\015o)m(w)m(er)g (and)f(the)g(isotropic)g Fr(n)p Fy({tori)g Fr(f)2082 1376 y Fi(\000)p Fm(1)2176 1412 y Fy(\()p Fr(z)t Fy(\),)h Fr(z)g Fq(2)c Fr(s)2589 1376 y Fi(\000)p Fm(1)2684 1412 y Fy(\()p Fr(y)t 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Fq(\000)h Fr(n)29 b Fy(=)f(1)k(and)i Fr(@)2802 2581 y Fn(p)2842 2566 y Fr(!)2903 2581 y Fn(d)2943 2566 y Fy(\()p Fr(y)t(;)17 b(p)p Fy(\))27 b Fq(6)p Fy(=)h(0)33 b(is)g(all)g(that)g(is)109 2686 y(needed)25 b(for)f(\(2\))g(to)f(b)s(e)i(non-degenerate)g(\(pro)m (vided)g(that)f(the)h(sup)s(erin)m(tegrable)g(frequency)i(mapping)109 2807 y Fr(y)j Fq(7!)e Fy(\()p Fr(!)414 2822 y Fm(1)453 2807 y Fy(\()p Fr(y)t Fy(\))p Fr(;)17 b(:)g(:)g(:)e(;)i(!)860 2822 y Fn(n)906 2807 y Fy(\()p Fr(y)t Fy(\)\))26 b(is)i (non-degenerate\).)43 b(This)28 b(is)g(a)f(genericit)m(y)i(condition)f (on)g(the)f(p)s(erturba-)109 2927 y(tion)d(whic)m(h)j(can)d(b)s(e)h (explicitly)i(c)m(hec)m(k)m(ed)h(after)c(a)m(v)m(eraging)i(o)m(v)m(er)g (the)f(unp)s(erturb)s(ed)g(fast)g(motion.)41 b(F)-8 b(or)109 3047 y(a)31 b(concretely)i(giv)m(en)g(system)g(the)f(task)g(is)g (therefore)g(to)f(fully)h(analyse)g(the)g(one-degree-of-freedom)109 3168 y(problem)g(parametrised)h(b)m(y)f Fr(n)g Fy(actions)g(|)f(and)h 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00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 %%Trailer %%EndDocument @endspecial 109 2413 a Fy(Figure)49 b(1:)76 b(Sc)m(hematic)51 b(represen)m(tation)g(of)d(the)i(bundle)g(structure)h(of)d(the)i(phase) g(space)g Fq(P)64 b Fy(=)109 2533 y Fr(T)180 2497 y Fi(\003)219 2533 y Fr(S)6 b(O)s Fy(\(3\).)47 b(The)35 b(v)-5 b(alue)35 b Fr(y)1066 2548 y Fm(2)1136 2533 y Fy(=)30 b Fq(j)p Fr(\026)p Fq(j)g Fy(=)g(0)k(is)h(excluded)h(from)e(the)h(picture,)h(in) f(this)f(limit)h(the)g(pro)s(duct)109 2653 y(of)d(the)h(spheres)h(with) f Fr(S)1018 2617 y Fm(1)1090 2653 y Fy(attac)m(hed)h(results)g(in)e(a)h (\014bre)g Fr(S)6 b(O)s Fy(\(3\).)109 2943 y(is)39 b(in)m(v)-5 b(arian)m(t)40 b(under)g(the)f(left)h(action)f(of)f Fr(S)6 b(O)s 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109 2413 a Fy(Figure)34 b(2:)48 b(A)35 b(slice)h Fr(y)912 2428 y Fm(2)983 2413 y Fy(=)31 b Fq(j)p Fr(\026)p Fq(j)j Fy(of)g(the)h(set)h(of)e(critical)h(v)-5 b(alues)36 b(of)e(the)i(energy-momen)m(tum)h(mapping)109 2533 y Fq(E)-24 b(M)26 b Fy(=)i(\()p Fr(y)t(;)546 2508 y Fy(\026)522 2533 y Fr(H)6 b Fy(\))33 b(with)g(axes)g Fr(y)1164 2548 y Fm(1)1231 2533 y Fy(=)28 b Fr(`)1376 2548 y Fm(3)1447 2533 y Fy(and)1662 2508 y(\026)1637 2533 y Fr(H)40 b Fy(for)32 b(the)h(`middle')h(case)g Fr(s)2702 2548 y Fm(3)2763 2533 y Fq(\001)22 b Fy(\()p Fr(\013)q(;)17 b(\014)6 b(;)17 b(\015)5 b Fy(\))26 b Fq(\031)i Fy(\()p Fr(a;)17 b(b;)g(c)p Fy(\).)255 2823 y(The)38 b(phase)h(space)g (structure)f(of)f(the)h(in)m(termediate)i(system)f(can)f(b)s(e)g(read)f (o\013)h(from)f(the)h(slice)109 2944 y Fr(y)157 2959 y Fm(2)223 2944 y Fy(=)28 b Fq(j)p Fr(\026)p Fq(j)j Fy(of)g(the)i(set)f (of)g(singular)g(v)-5 b(alues)33 b(of)e(the)i(energy-momen)m(tum)h (mapping,)f(see)g(\014gure)f(2.)43 b(F)-8 b(or)109 3064 y(v)j(alues)32 b(of)f Fr(y)556 3079 y Fm(1)626 3064 y Fy(near)h Fq(\006)929 3025 y Fm(1)p 929 3041 36 4 v 929 3099 a(3)975 3064 y Fq(j)p Fr(\026)p Fq(j)e Fy(the)i(linear)g(part)f (of)g(the)h(force)f(\014eld)i(has)e(negligeable)i(in\015uence)g(and)f (the)109 3185 y(angular)41 b(momen)m(tum)i(rotates)f(\(=n)m(utates\))h (ab)s(out)e(the)h(direction)g(of)f(the)i(constan)m(t)f(part)g(of)f(the) 109 3305 y(force)d(\014eld.)62 b(Lo)m(w)m(ering)39 b(the)g(v)-5 b(alue)38 b(of)g Fq(j)p Fr(y)1665 3320 y Fm(1)1704 3305 y Fq(j)g Fy(leads)h(to)f(t)m(w)m(o)h(successiv)m(e)j(cen)m(tre-saddle)e (bifurcations)109 3425 y(and)30 b(when)i Fr(y)597 3440 y Fm(1)667 3425 y Fy(passes)g(through)e(0)h(the)g(t)m(w)m(o)g (resulting)h(saddles)g(undergo)f(a)f(connection)i(bifurcation)109 3546 y(\(where)41 b(only)g(the)f(linear)h(part)f(of)g(the)g(force)h (\014eld)g(con)m(tributes,)j(the)c(saddles)i(are)e(connected)i(b)m(y) 109 3666 y(hetero)s(clinic)34 b(orbits\).)255 3787 y(There)f(is)g(no)f (cen)m(tre-saddle)h(bifurcation)g(for)e Fq(j)p Fr(y)2076 3802 y Fm(1)2115 3787 y Fq(j)g Fy(increasing)i(from)2869 3747 y Fm(1)p 2869 3764 V 2869 3821 a(3)2914 3787 y Fq(j)p Fr(\026)p Fq(j)e Fy(if)h(the)h(constan)m(t)f(part)109 3907 y(of)46 b(the)g(force)h(\014eld)g(dominates)g(the)g(linear)g(part) f(while)i(b)s(oth)e(cen)m(tre-saddle)i(bifurcations)f(also)109 4027 y(tak)m(e)d(place)g(for)f Fq(j)p Fr(y)827 4042 y Fm(1)866 4027 y Fq(j)g Fy(increasing)i(from)1655 3988 y Fm(1)p 1655 4004 V 1655 4062 a(3)1701 4027 y Fq(j)p Fr(\026)p Fq(j)d Fy(to)h Fq(j)p Fr(\026)p Fq(j)g Fy(if)g(the)h(constan) m(t)h(part)e(of)g(the)h(force)g(\014eld)h(is)109 4148 y(itself)39 b(dominated)g(b)m(y)g(the)g(linear)g(part.)61 b(Figure)38 b(2)g(depicts)i(the)f(middle)h(case)f(where)h(b)s(oth)e (parts)109 4268 y(of)47 b(the)i(a\016ne)f(force)g(\014eld)h(are)f(of)g (comparable)g(in\015uence,)54 b(then)49 b(only)f(the)h(\014rst)f(cen)m (tre-saddle)109 4388 y(bifurcation)36 b(o)s(ccurs)h(as)g Fq(j)p Fr(y)1113 4403 y Fm(1)1151 4388 y Fq(j)f Fy(comes)i(close)f(to)f Fq(j)p Fr(\026)p Fq(j)p Fy(.)53 b(F)-8 b(or)36 b(\014xed)h(v)-5 b(alues)37 b(of)f Fr(y)j Fy(the)e(Eulerian)g(tori)f(and)109 4509 y(their)h(separatrices)h(th)m(us)f(organize)g(the)g(distribution)g (of)f(3{tori)g(in)m(to)h(1{5)f(families.)56 b(Most)37 b(of)f(the)109 4629 y(in)m(v)-5 b(arian)m(t)32 b(3{tori)f(p)s(ersist)i (under)f(the)g(p)s(erturbation)g(from)f(the)i(in)m(termediate)g(system) h(bac)m(k)e(to)g(the)109 4750 y(original)45 b(rigid)g(b)s(o)s(dy)h(sub) 5 b(ject)47 b(to)e(a)g(w)m(eak)i(a\016ne)f(force)g(\014eld)g([33].)82 b(That)46 b(the)g(necessary)i(non-)109 4870 y(degeneracy)34 b(condition)f(holds)h(true)f(|)f(the)h(slo)m(w)h(frequency)h(v)-5 b(aries)33 b(from)f(torus)h(to)g(torus)g(|)f(is)h(in)109 4990 y(fact)f(enforced)i(b)m(y)f(the)g(w)m(a)m(y)h(the)f(Eulerian)g (tori)g(organize)f(the)h(phase)h(space.)255 5111 y(In)26 b(the)f(in)m(termediate)j(system)f(the)f(connection)g(bifurcation)g (tak)m(es)g(place)h(exactly)f(at)f(the)h(strong)109 5231 y(resonance)32 b Fr(!)613 5246 y Fm(1)653 5231 y Fy(\()p Fr(y)t Fy(\))26 b(=)i(0)j(of)g(the)h(Eulerian)g(tori,)f(see)i(\014gure) f(1.)42 b(Also)32 b(the)g(cen)m(tre-saddle)h(bifurcations)109 5352 y(do)40 b(not)h(p)s(ersist)g(b)m(y)h(general)f(theory)g([29)o(].) 68 b(Indeed,)44 b(the)d(same)g(ratio)2835 5308 y Fn(y)2870 5317 y Fe(1)p 2835 5329 70 4 v 2835 5386 a Fn(y)2870 5395 y Fe(2)2955 5352 y Fy(that)g(parametrises)h(the)1974 5778 y(7)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 109 208 a Fy(lines)35 b(of)f(parab)s(olic)g (Eulerian)h(tori)f(also)h(\014xes)g(the)g(frequency)h(ratio)2729 168 y Fn(!)2773 177 y Fe(1)p 2729 185 79 4 v 2729 243 a Fn(!)2773 252 y Fe(2)2848 208 y Fy(=)2964 168 y Fn(I)2995 177 y Fe(3)3030 168 y Fi(\000)p Fn(I)3116 177 y Fe(1)p 2964 185 186 4 v 3024 243 a Fn(I)3055 252 y Fe(3)3170 164 y Fn(y)3205 173 y Fe(1)p 3170 185 70 4 v 3170 243 a Fn(y)3205 252 y Fe(2)3283 208 y Fy(and)f(the)f(whole)109 329 y(bifurcation)j(scenario)i(ma)m(y)f(fall)f(in)m(to)g(a)h(resonance) g(gap.)58 b(Ho)m(w)m(ev)m(er,)41 b(for)c(most)h(v)-5 b(alues)38 b(of)f(param-)109 449 y(eters)f Fr(s)391 464 y Fm(3)430 449 y Fr(;)17 b(I)517 464 y Fm(1)556 449 y Fr(;)g(I)643 464 y Fm(3)717 449 y Fy(the)35 b(quasi-p)s(erio)s(dic)h (cen)m(tre-saddle)h(bifurcations)f(turn)f(out)f(to)h(b)s(e)g(presen)m (t)i(in)e(the)109 569 y(original)d(dynamics)i([28].)109 856 y Fg(3.2)135 b(The)44 b(logarithmic)j(p)t(oten)l(tial)109 1041 y Fy(The)36 b(motion)g(of)f(a)h(mass)g(p)s(oin)m(t)g(sub)5 b(ject)38 b(to)d(a)g(cen)m(tral)i(force)f(\014eld)g(pro)m(vides)i(for)d (a)g(whole)i(class)g(of)109 1161 y(Hamiltonian)28 b(systems)j(with)e Fr(S)6 b(O)s Fy(\(3\){symmetry;)31 b(here)e(the)g(natural)f(action)h (on)f(the)h(con\014guration)109 1282 y(space)43 b Fo(R)451 1246 y Fm(3)533 1282 y Fy(is)h(lifted)f(to)f(the)h(diagonal)f(action)h (on)g(the)g(phase)g(space)h Fr(T)2861 1246 y Fi(\003)2900 1282 y Fo(R)2972 1246 y Fm(3)3057 1282 y Fy(=)g Fo(R)3249 1246 y Fm(3)3318 1282 y Fq(\002)29 b Fo(R)3496 1246 y Fm(3)3536 1282 y Fy(.)73 b(Again)109 1402 y(the)29 b(comp)s(onen)m(ts)i Fr(\026)866 1417 y Fm(1)905 1402 y Fr(;)17 b(\026)1008 1417 y Fm(2)1046 1402 y Fr(;)g(\026)1149 1417 y Fm(3)1217 1402 y Fy(of)29 b(the)g(angular)g(momen)m(tum)h(are)f(conserv)m(ed)j (quan)m(tities,)f(next)f(to)f(the)109 1522 y(Hamiltonian)k Fr(H)751 1537 y Fm(0)790 1522 y Fy(.)43 b(T)-8 b(o)33 b(\014x)g(though)m(ts)g(w)m(e)h(concen)m(trate)g(on)e(the)h(case)864 1782 y Fr(H)8 b Fy(\()p Fr(\030)d(;)17 b(\021)t Fy(\))82 b(=)1423 1715 y Fr(\021)1475 1678 y Fm(2)1471 1739 y(1)1537 1715 y Fy(+)22 b Fr(\021)1687 1678 y Fm(2)1683 1739 y(2)1748 1715 y Fy(+)g Fr(\021)1898 1678 y Fm(2)1894 1739 y(3)p 1423 1759 514 4 v 1656 1850 a Fy(2)1997 1782 y(+)2132 1715 y(1)p 2132 1759 49 4 v 2132 1850 a(2)2208 1782 y(ln)2306 1701 y Fh(\000)2351 1782 y Fr(R)2426 1741 y Fm(2)2488 1782 y Fy(+)g Fr(\030)2634 1741 y Fm(2)2629 1807 y(1)2695 1782 y Fy(+)g Fr(\030)2841 1741 y Fm(2)2836 1807 y(2)2902 1782 y Fy(+)g Fr(\030)3048 1741 y Fm(2)3043 1807 y(3)3087 1701 y Fh(\001)109 2019 y Fy(of)f(logarithmic)h(p)s(oten)m(tials,)j (where)e(w)m(e)g(scaled)g(the)f(mass)h(to)e(1.)40 b(Note)22 b(that)f(the)i(origin)e(\()p Fr(\030)5 b(;)17 b(\021)t Fy(\))27 b(=)g(\(0)p Fr(;)17 b Fy(0\))109 2139 y(of)29 b(the)i(phase)g(space)h(is)e(an)h(equlibrium)h(for)d Fr(R)g(>)f Fy(0,)i(while)i(in)e(the)h(so-called)g(scale-free)g(case)g Fr(R)e Fy(=)e(0)109 2260 y(the)33 b(p)s(oten)m(tial)g(is)g(not)f (de\014ned)i(at)e(the)h(origin)g Fr(\030)f Fy(=)27 b(0)33 b(of)f(the)h(con\014guration)f(space.)255 2380 y(The)23 b(comp)s(onen)m(ts)g(of)f(the)g(angular)g(momen)m(tum)i(constitute)f (the)f(momen)m(tum)i(mapping)e Fr(J)31 b Fy(of)22 b(the)109 2500 y Fr(S)6 b(O)s Fy(\(3\))27 b(action)i(and)g(the)g(isotrop)m(y)h (group)e Fr(G)1764 2515 y Fn(\026)1840 2500 y Fy(of)g(the)h(co-adjoin)m (t)g(action)g(of)f Fr(S)6 b(O)s Fy(\(3\))27 b(on)i Ff(so)p Fy(\(3\))3605 2464 y Fi(\003)3672 2473 y Fq(\030)3672 2505 y Fy(=)3777 2500 y Fo(R)3849 2464 y Fm(3)109 2621 y Fy(consists)40 b(of)f(the)g(rotations)g(ab)s(out)f(0)g Fq(6)p Fy(=)h Fr(\026)f Fq(2)h Fo(R)1946 2585 y Fm(3)1985 2621 y Fy(.)62 b(Reducing)40 b(the)g Fr(S)6 b(O)s Fy(\(3\){symmetry)39 b(leads)h(to)e(a)109 2741 y(one-degree-of-freedom)32 b(system)j(on)853 2947 y Fr(J)916 2906 y Fi(\000)p Fm(1)1010 2947 y Fy(\()p Fr(\026)p Fy(\))1145 2988 y Fr(=)1194 3021 y(G)1271 3036 y Fn(\026)1409 2919 y Fq(\030)1410 2951 y Fy(=)1569 2866 y Fh(\010)1655 2947 y Fy(\()p Fr(q)t(;)17 b(p)p Fy(\))27 b Fq(2)h Fo(R)2064 2906 y Fm(4)2131 2947 y Fq(j)g Fr(q)2230 2962 y Fm(1)2269 2947 y Fr(p)2318 2962 y Fm(2)2380 2947 y Fq(\000)22 b Fr(q)2522 2962 y Fm(2)2562 2947 y Fr(p)2611 2962 y Fm(1)2678 2947 y Fy(=)28 b Fq(j)p Fr(\026)p Fq(j)2923 2866 y Fh(\011)2982 2988 y Fr(=)3031 3032 y(S)3097 2996 y Fm(1)3764 2947 y Fy(\(5\))109 3222 y(where)43 b(\()p Fr(q)t(;)17 b(p)p Fy(\))42 b Fq(2)h Fr(T)838 3186 y Fi(\003)878 3222 y Fo(R)950 3186 y Fm(2)1031 3222 y Fy(are)e(linear)h(v)-5 b(ariables)43 b(on)e(the)h(plane)g(p)s (erp)s(endicular)h(to)e Fr(\026)h Fy(in)g(whic)m(h)h(the)109 3342 y(motion)36 b(tak)m(es)h(place.)54 b(The)37 b(diagonal)e Fr(S)1656 3306 y Fm(1)1695 3342 y Fy({action)h(on)g(the)g(latter)g(can) g(b)s(e)g(factored)g(out)g(using)h(the)109 3463 y(in)m(v)-5 b(arian)m(ts)893 3617 y Fr(u)55 b Fy(=)1145 3550 y Fr(q)1192 3514 y Fm(2)1188 3575 y(1)1254 3550 y Fy(+)22 b Fr(q)1399 3514 y Fm(2)1395 3575 y(2)p 1145 3594 293 4 v 1267 3686 a Fy(2)1497 3617 y Fr(;)114 b(v)59 b Fy(=)1885 3550 y Fr(p)1934 3514 y Fm(2)1934 3575 y(1)1996 3550 y Fy(+)22 b Fr(p)2143 3514 y Fm(2)2143 3575 y(2)p 1885 3594 297 4 v 2009 3686 a Fy(2)2241 3617 y Fr(;)114 b(w)58 b Fy(=)d Fr(q)2684 3632 y Fm(1)2724 3617 y Fr(p)2773 3632 y Fm(1)2834 3617 y Fy(+)22 b Fr(q)2975 3632 y Fm(2)3015 3617 y Fr(p)3064 3632 y Fm(2)109 3821 y Fy(whic)m(h)47 b(together)f(with)g Fq(j)p Fr(\026)p Fq(j)k Fy(=)g Fr(q)1368 3836 y Fm(1)1407 3821 y Fr(p)1456 3836 y Fm(2)1527 3821 y Fq(\000)32 b Fr(q)1679 3836 y Fm(2)1718 3821 y Fr(p)1767 3836 y Fm(1)1852 3821 y Fy(generate)47 b(the)f(ring)g(of)f Fr(S)2841 3785 y Fm(1)2880 3821 y Fy({in)m(v)-5 b(arian)m(t)46 b(functions)h(on)109 3941 y Fr(T)180 3905 y Fi(\003)219 3941 y Fo(R)291 3905 y Fm(2)358 3941 y Fy(=)27 b Fo(R)533 3905 y Fm(2)595 3941 y Fq(\002)c Fo(R)767 3905 y Fm(2)806 3941 y Fy(.)43 b(This)34 b(turns)f(\(5\))g(in)m(to)773 4147 y Fq(P)842 4163 y Fi(j)p Fn(\026)p Fi(j)1011 4147 y Fy(=)1170 4067 y Fh(\010)1256 4147 y Fy(\()p Fr(u;)17 b(v)t(;)g(w)s Fy(\))25 b Fq(2)j Fo(R)1791 4106 y Fm(3)1858 4147 y Fq(j)g Fr(u)f Fq(\025)h Fy(0)p Fr(;)17 b(v)31 b Fq(\025)d Fy(0)p Fr(;)17 b(S)2531 4163 y Fi(j)p Fn(\026)p Fi(j)2616 4147 y Fy(\()p Fr(u;)g(v)t(;)g(w)s Fy(\))26 b(=)i(0)3166 4067 y Fh(\011)109 4353 y Fy(where)33 b(the)g(syzygy)1227 4527 y Fr(S)1287 4543 y Fi(j)p Fn(\026)p Fi(j)1373 4527 y Fy(\()p Fr(u;)17 b(v)t(;)g(w)s Fy(\))81 b(=)1967 4460 y Fr(w)2040 4424 y Fm(2)p 1967 4504 112 4 v 1998 4596 a Fy(2)2138 4527 y Fq(\000)51 b Fy(2)p Fr(uv)i Fy(+)2606 4460 y Fq(j)p Fr(\026)p Fq(j)2721 4424 y Fm(2)p 2606 4504 154 4 v 2659 4596 a Fy(2)109 4721 y(b)s(et)m(w)m(een)34 b(the)f(in)m(v)-5 b(arian)m(ts)34 b(also)f(allo)m(ws)g(to)f(write)i(do) m(wn)f(the)g(P)m(oisson)h(brac)m(k)m(et)1361 4927 y Fq(f)p Fr(f)5 b(;)17 b(g)t Fq(g)82 b Fy(=)h Fq(hr)p Fr(f)32 b Fq(\002)23 b(r)p Fr(g)31 b Fq(j)c(r)p Fr(S)2511 4942 y Fi(j)p Fn(\026)p Fi(j)2597 4927 y Fq(i)109 5133 y Fy(on)32 b Fo(R)316 5097 y Fm(3)388 5133 y Fy(and)h(th)m(us)g(on)g Fq(P)997 5148 y Fi(j)p Fn(\026)p Fi(j)1083 5133 y Fy(.)43 b(Alternativ)m(ely)-8 b(,)35 b(one)e(can)g(w)m(ork)g(with)g(the)g (e\013ectiv)m(e)i(p)s(oten)m(tial)1237 5405 y Fr(V)1294 5421 y Fi(j)p Fn(\026)p Fi(j)1380 5405 y Fy(\()p Fr(u)p Fy(\))82 b(=)1763 5338 y(1)p 1763 5382 49 4 v 1763 5473 a(2)1839 5405 y(ln\()p Fr(R)2033 5364 y Fm(2)2095 5405 y Fy(+)22 b(2)p Fr(u)p Fy(\))49 b(+)2520 5338 y Fq(j)p Fr(\026)p Fq(j)2635 5302 y Fm(2)p 2520 5382 154 4 v 2545 5473 a Fy(4)p Fr(u)2732 5405 y(:)1974 5778 y Fy(8)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 109 208 a Fy(Action)50 b(angle)f(v)-5 b(ariables)50 b(\()p Fr(x)1228 223 y Fm(1)1268 208 y Fr(;)17 b(y)1360 223 y Fm(1)1399 208 y Fy(\))49 b(of)g(the)h(resulting) h(one-degree-of-freedom)e(system)j(serv)m(e)f(as)109 329 y(starting)34 b(p)s(oin)m(t)h(for)f(the)i(construction)f(of)g (generalized)h(action)f(angle)f(v)-5 b(ariables.)51 b(The)36 b(second)g(an-)109 449 y(gle)31 b Fr(x)314 464 y Fm(2)385 449 y Fy(measures)i(the)f(rotation)f(ab)s(out)f(the)i(angular)f(momen)m (tum)i(v)m(ector,)g(with)f(conjugate)f(action)109 569 y Fr(y)157 584 y Fm(2)223 569 y Fy(=)d Fq(j)p Fr(\026)p Fq(j)p Fy(.)42 b(Restricting)34 b Fr(\026)27 b Fq(2)h Fo(R)1264 533 y Fm(3)1336 569 y Fy(to)k(the)h(sphere)1347 700 y Fh(\010)1433 781 y Fr(\026)27 b Fq(2)h Fo(R)1685 740 y Fm(3)1752 781 y Fq(j)g Fr(\026)1867 740 y Fm(2)1867 805 y(1)1928 781 y Fy(+)22 b Fr(\026)2085 740 y Fm(2)2085 805 y(2)2146 781 y Fy(+)g Fr(\026)2303 740 y Fm(2)2303 805 y(3)2370 781 y Fy(=)28 b Fr(y)2526 740 y Fm(2)2522 805 y(2)2592 700 y Fh(\011)3764 781 y Fy(\(6\))109 992 y(yields)c(a)f(semi-global)g(c)m(hart)h(\()p Fr(x;)17 b(y)t(;)g(\026)p Fy(\),)23 b(while)h(an)m(y)g(c)m(hoice)h(of)d (canonical)i(co-ordinates)f(on)g(\(6\))f(de\014nes)109 1113 y(generalized)34 b(action)e(angle)h(v)-5 b(ariables)33 b(for)f Fr(H)1794 1128 y Fm(0)1834 1113 y Fy(.)43 b(The)34 b(Hamiltonian)410 1378 y Fr(H)8 b Fy(\()p Fr(\030)d(;)17 b(\021)t Fy(\))81 b(=)969 1310 y Fr(\021)1021 1274 y Fm(2)1017 1335 y(1)1083 1310 y Fy(+)22 b Fr(\021)1233 1274 y Fm(2)1229 1335 y(2)1294 1310 y Fy(+)g Fr(\021)1444 1274 y Fm(2)1440 1335 y(3)p 969 1355 514 4 v 1202 1446 a Fy(2)1543 1378 y Fq(\000)50 b Fy(\012\()p Fr(\030)1821 1393 y Fm(1)1860 1378 y Fr(\021)1908 1393 y Fm(2)1970 1378 y Fq(\000)23 b Fr(\030)2113 1393 y Fm(2)2152 1378 y Fr(\021)2200 1393 y Fm(1)2239 1378 y Fy(\))50 b(+)2463 1310 y(1)p 2463 1355 49 4 v 2463 1446 a(2)2538 1378 y(ln)2636 1237 y Fh(\022)2709 1378 y Fr(R)2784 1336 y Fm(2)2846 1378 y Fy(+)2955 1310 y Fr(\030)3003 1274 y Fm(2)2998 1335 y(1)p 2954 1355 90 4 v 2954 1446 a Fr(\032)3004 1412 y Fm(2)3004 1470 y(1)3076 1378 y Fy(+)3185 1310 y Fr(\030)3233 1274 y Fm(2)3228 1335 y(2)p 3184 1355 V 3184 1446 a Fr(\032)3234 1412 y Fm(2)3234 1470 y(2)3306 1378 y Fy(+)3415 1310 y Fr(\030)3463 1274 y Fm(2)3458 1335 y(3)p 3414 1355 V 3414 1446 a Fr(\032)3464 1412 y Fm(2)3464 1470 y(3)3513 1237 y Fh(\023)3764 1378 y Fy(\(7\))109 1641 y(mo)s(dels)41 b(a)g(simple)h(but)g(realistic)g (3{dimensional)g(matter)f(distribution,)j(with)d(re\015ectional)h(sym-) 109 1761 y(metries)d(with)f(resp)s(ect)h(to)f(the)g(principal)h(planes) f(of)g(the)g(system)i(and)d(\(for)h(\012)f Fq(6)p Fy(=)f(0\))h(o)m(v)m (erall)i(bulk)109 1882 y(rotation.)j(The)33 b(scale-factors)f Fr(\032)1320 1897 y Fn(k)1395 1882 y Fy(describing)h(the)f(axial)g (ratios)g(of)f(the)h(equip)s(oten)m(tial)h(surfaces)g(are)109 2002 y(close)j(to)f(eac)m(h)h(other,)h Fl(i.e.)e Fy(the)h (di\013erences)h(b)s(et)m(w)m(een)h(the)d Fr(\032)2393 2017 y Fn(k)2472 2002 y Fy(are)g(small,)i(and)f(the)f(angular)h(v)m (elo)s(c-)109 2122 y(it)m(y)42 b(\012)f(of)g(the)g(rotating)g(galaxy)h (is)f(a)g(small)h(parameter)g(as)f(w)m(ell.)71 b(The)42 b(astroph)m(ysical)h(relev)-5 b(ance)109 2243 y(of)36 b(the)i(mo)s(del)f([5,)g(6])g(is)h(based)g(on)f(its)h(abilit)m(y)g(to)e (describ)s(e)j(in)f(a)e(simple)j(w)m(a)m(y)f(the)g(gross)f(features)109 2363 y(of)c(elliptic)j(galaxies)f(em)m(b)s(edded)h(in)e(a)g(dark)h (matter)f(halo.)48 b(It)35 b(has)f(therefore)h(b)s(een)g(the)g(sub)5 b(ject)36 b(of)109 2483 y(sev)m(eral)e(applications)f(in)g(galactic)g (dynamics)h([24,)f(38)o(,)g(43)o(].)255 2604 y(Reducing)23 b(the)f Fo(T)897 2568 y Fm(2)937 2604 y Fy({symmetry)h(of)f(the)g(in)m (termediate)h(system)h(leads)f(to)e(a)g(one-degree-of-freedom)109 2724 y(problem)38 b(on)f(\(6\).)57 b(Its)38 b(dynamics)h(describ)s(es)g (the)f(slo)m(w)h(\(p)s(erio)s(dic\))e(v)-5 b(ariation)37 b(of)g(the)g(\()p Fr(q)t(;)17 b(p)p Fy(\){plane)109 2845 y(within)28 b(whic)m(h)h(the)e(fast)g(rosetta-orbits)h(of)e Fr(H)1809 2860 y Fm(0)1876 2845 y Fy(tak)m(e)i(place.)42 b(These)29 b(in)m(v)-5 b(arian)m(t)28 b(3{tori)f(are)g(organized)109 2965 y(b)m(y)i(the)g(equilibria)g(of)f(the)h(one-degree-of-freedom)f (problem)h(|)f(the)h(surviving)h(fast)e(2{tori)f(|)h(and)109 3085 y(their)33 b(bifurcations.)255 3206 y(P)m(ersistence)46 b(of)d(in)m(v)-5 b(arian)m(t)43 b(3{tori)g(when)h(p)s(erturbing)f(from) g(the)g(in)m(termediate)i(system)g(bac)m(k)109 3326 y(to)i(the)g (original)g(\(7\))g(relies)i(on)e(separate)h(non-degeneracy)h (conditions)f(on)g(the)f(slo)m(w)i(and)e(fast)109 3446 y(frequencies.)76 b(The)43 b(n)m(umerical)h(results)g(in)f([41])f (indicate)i(that)e(the)h(latter)g(are)g(iso-energetically)109 3567 y(non-degenerate,)27 b(with)f(a)f(frequency)i(ratio)e(that)g (decreases)i(monotonically)f(under)g(v)-5 b(ariation)24 b(of)h Fr(y)3822 3582 y Fm(2)3861 3567 y Fy(.)109 3687 y(The)39 b(3{tori)e(in)h([37])g(are)g(computed)h(for)f(suc)m(h)h(high)g (v)-5 b(alues)39 b(of)e Fr(\032)2599 3702 y Fn(k)2668 3687 y Fq(\000)26 b Fr(\032)2821 3702 y Fn(`)2893 3687 y Fy(\(while)39 b(\012)e(=)g(0\))h(that)g(the)109 3808 y(distinction)26 b(b)s(et)m(w)m(een)i(slo)m(w)e(and)f(fast)g (frequencies)j(disapp)s(ears)e(and)g(lo)m(w)g(order)f(resonances)i (emerge.)109 4139 y Fs(4)161 b(Resonan)l(t)52 b(equilibria)109 4358 y Fy(An)30 b(elliptic)h(equilibrium)h(of)d(a)g(Hamiltonian)i (system)g(has)g(all)e(eigen)m(v)-5 b(alues)32 b(on)e(the)g(imaginary)g (axis)109 4479 y(but)h(no)g(m)m(ultiple)i(eigen)m(v)-5 b(alues.)45 b(In)31 b(particular)h(0)f(is)h(not)f(an)g(eigen)m(v)-5 b(alue,)33 b(the)e(equilibrium)j(p)s(ersists)109 4599 y(and)d(all)g(eigen)m(v)-5 b(alues)33 b(remain)f(on)g(the)f(imaginary)h (axis)g(if)f(the)h(system)h(is)f(sligh)m(tly)g(p)s(erturb)s(ed.)44 b(The)109 4719 y(quadratic)33 b(part)1420 4928 y Fr(H)1509 4886 y Fm(2)1501 4952 y Fn(\013)1550 4928 y Fy(\()p Fr(q)t(;)17 b(p)p Fy(\))82 b(=)2061 4803 y Fn(d)2007 4833 y Fh(X)2020 5045 y Fn(`)p Fm(=1)2168 4928 y Fr(\013)2230 4943 y Fn(`)2273 4860 y Fr(p)2322 4824 y Fm(2)2322 4886 y Fn(`)2383 4860 y Fy(+)22 b Fr(q)2528 4824 y Fm(2)2524 4886 y Fn(`)p 2273 4905 295 4 v 2396 4996 a Fy(2)3764 4928 y(\(8\))109 5183 y(ma)m(y)29 b(ha)m(v)m(e)h(imaginary)f(part)f Fr(\013)1260 5198 y Fn(`)1322 5183 y Fy(of)g(the)h(eigen)m(v)-5 b(alues)30 b(of)e(b)s(oth)h(signs)g(|)f(since)i(the)f(equilibrium)i(do)s(es)109 5303 y(not)k(need)h(to)f(b)s(e)g(a)g(lo)s(cal)g(extrem)m(um)i(of)e(the) h(Hamiltonian)f(it)g(is)h(not)f(p)s(ossible)i(to)d(enforce)i Fr(\013)3666 5318 y Fn(`)3731 5303 y Fr(>)c Fy(0)109 5424 y(for)g(all)g Fr(`)h Fy(|)f(the)h(so-called)g(symplectic)i(sign.) 1974 5778 y(9)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 255 208 a Fy(One)48 b(sp)s(eaks)h(of)e(a)h (resonance)h(b)s(et)m(w)m(een)h(the)e(normal)g(frequencies)i Fr(\013)2981 223 y Fm(1)3020 208 y Fr(;)17 b(:)g(:)g(:)f(;)h(\013)3301 223 y Fn(d)3389 208 y Fy(if)48 b(there)g(are)109 329 y(in)m(tegers)34 b Fr(k)521 344 y Fm(1)560 329 y Fr(;)17 b(:)g(:)g(:)f(;)h(k)830 344 y Fn(d)897 329 y Fq(2)28 b Fo(Z)p Fy(,)33 b(not)g(all)f(zero,)h(suc)m(h)h(that)1390 549 y Fr(k)1441 564 y Fm(1)1480 549 y Fr(\013)1542 564 y Fm(1)1631 549 y Fy(+)50 b Fr(:)17 b(:)g(:)49 b Fy(+)h Fr(k)2098 564 y Fn(d)2138 549 y Fr(\013)2200 564 y Fn(d)2324 549 y Fy(=)82 b(0)49 b Fr(:)1157 b Fy(\(9\))109 769 y(Here)31 b Fr(k)388 784 y Fm(1)427 769 y Fr(;)17 b(:)g(:)g(:)f(;)h(k)697 784 y Fn(d)768 769 y Fy(can)31 b(b)s(e)g(tak)m(en)g(relativ)m(ely)i (prime)f(and)f Fq(j)p Fr(k)s Fq(j)c Fy(=)g Fq(j)p Fr(k)2545 784 y Fm(1)2584 769 y Fq(j)18 b Fy(+)g Fr(:)f(:)g(:)h Fy(+)h Fq(j)p Fr(k)3031 784 y Fn(d)3071 769 y Fq(j)30 b Fy(is)h(called)h(the)f(order)109 889 y(of)c(the)i(resonance.)43 b(Since)29 b(w)m(e)g(excluded)i(the)d(1:0)g(and)g(1:)p Fq(\006)p Fy(1)g(resonances)i(w)m(e)f(ha)m(v)m(e)g Fq(j)p Fr(k)s Fq(j)f(\025)g Fy(3.)41 b(If)28 b(there)109 1009 y(are)33 b(no)g(resonances)i(of)e(order)h Fq(j)p Fr(k)s Fq(j)28 b(\024)h Fr(m)p Fy(,)34 b(then)g(the)g(normal)f(form)g (truncated)i(at)d(order)i Fr(m)g Fy(dep)s(ends)109 1130 y(on)e(\()p Fr(q)t(;)17 b(p)p Fy(\))32 b(only)h(as)g(a)f(function)h(of) f(the)h(in)m(v)-5 b(arian)m(ts)1536 1409 y Fr(I)1579 1424 y Fn(`)1695 1409 y Fy(=)1864 1342 y Fr(p)1913 1305 y Fm(2)1913 1368 y Fn(`)1974 1342 y Fy(+)22 b Fr(q)2119 1305 y Fm(2)2115 1368 y Fn(`)p 1864 1386 295 4 v 1987 1477 a Fy(2)2252 1409 y Fq(\025)83 b Fy(0)109 1660 y(whence)34 b(w)m(e)g(obtain)e(for)g Fr(m)c Fy(=)g(4)k(the)h(\(truncated\))g (Birkho\013)g(normal)g(form)1249 1973 y Fq(H)1333 1988 y Fn(\013;\014)1445 1973 y Fy(\()p Fr(I)8 b Fy(\))83 b(=)g Fr(H)1903 1932 y Fm(2)1895 1998 y Fn(\013)1944 1973 y Fy(\()p Fr(I)8 b Fy(\))49 b(+)2317 1849 y Fn(d)2263 1879 y Fh(X)2246 2091 y Fn(k)r(;`)p Fm(=1)2440 1973 y Fr(\014)2495 1988 y Fn(k)r(`)2576 1906 y Fr(I)2619 1921 y Fn(k)2662 1906 y Fr(I)2705 1921 y Fn(`)p 2576 1951 162 4 v 2633 2042 a Fy(2)109 2297 y(with)42 b(a)g(symmetric)i(matrix)e Fr(\014)49 b Fy(=)44 b(\()p Fr(\014)1558 2312 y Fn(k)r(`)1629 2297 y Fy(\))1667 2312 y Fn(k)r(;`)p Fm(=1)p Fn(;:::)n(;d)1983 2297 y Fy(.)71 b(This)43 b(normal)f(form)g Fq(H)2977 2312 y Fn(\013;\014)3131 2297 y Fy(is)g(\(near)g(the)h(equi-)109 2417 y(librium\))j(an)e(in)m(tegrable)i(appro)m(ximation)g(of)f(the)g (original)g(Hamiltonian)g(and)h(under)g Fl(e.g.)e Fy(Kol-)109 2537 y(mogoro)m(v's)g(non-degeneracy)i(condition)e(det)17 b Fr(\014)53 b Fq(6)p Fy(=)47 b(0)d(a)g(measure-theoretically)i(large)e (part)g(of)f(a)109 2658 y(neigh)m(b)s(ourho)s(o)s(d)32 b(of)g(the)h(equilibrium)i(is)e(foliated)g(b)m(y)g(a)f(Can)m(tor)h (family)g(of)f(in)m(v)-5 b(arian)m(t)34 b Fr(d)p Fy({tori)d([4].)255 2778 y(The)40 b(structure)h(of)e(the)h(rami\014ed)g(torus)g(bundle)h (de\014ned)g(b)m(y)f(the)g(dynamics)h(of)e Fq(H)3450 2793 y Fn(\013;\014)3602 2778 y Fy(is)h(v)m(ery)109 2898 y(transparan)m(t.)h(The)28 b(v)-5 b(alues)27 b Fr(I)1195 2913 y Fn(`)1256 2898 y Fr(>)g Fy(0)g(parametrise)g(the)g(p)s(erio)s (dic)g(orbits)g(in)f(the)h(\()p Fr(q)3111 2913 y Fn(`)3144 2898 y Fr(;)17 b(p)3237 2913 y Fn(`)3270 2898 y Fy(\){plane,)28 b(where)109 3019 y Fr(I)152 3034 y Fm(1)191 3019 y Fr(;)17 b(:)g(:)g(:)f(;)h(I)453 3034 y Fn(`)p Fi(\000)p Fm(1)576 3019 y Fr(;)g(I)663 3034 y Fn(`)p Fm(+1)786 3019 y Fr(;)g(:)g(:)g(:)e (;)i(I)1047 3034 y Fn(d)1124 3019 y Fy(v)-5 b(anish.)55 b(T)-8 b(aking)37 b Fr(n)d Fq(\025)g Fy(2)i(among)g(the)g Fr(I)2623 3034 y Fn(`)2692 3019 y Fy(to)g(b)s(e)g(non-zero)h (parametrises)109 3139 y(the)47 b(v)-5 b(arious)48 b Fr(n)p Fy({parameter)g(families)g(of)f(in)m(v)-5 b(arian)m(t)48 b Fr(n)p Fy({tori,)j(up)d(un)m(til)g(the)g(single)g Fr(d)p Fy({parameter)109 3260 y(family)33 b(of)f(Lagrangean)g(tori.)255 3380 y(Liapuno)m(v's)26 b(Cen)m(tre)f(Theorem)h(yields)g(p)s (ersistence)h(of)d(those)g(\()p Fr(q)2639 3395 y Fn(`)2673 3380 y Fr(;)17 b(p)2766 3395 y Fn(`)2798 3380 y Fy(\){planes)25 b(of)f(p)s(erio)s(dic)h(orbits)109 3500 y(for)31 b(whic)m(h)j(none)f (of)f(the)h(other)f(frequencies)j(is)e(an)g(in)m(teger)g(m)m(ultiple)h (of)e Fr(\013)2912 3515 y Fn(`)2945 3500 y Fy(.)43 b(P)m(ersistence)36 b(of)31 b Fr(n)p Fy({tori)109 3621 y(with)k(2)c Fq(\024)g Fr(n)g Fq(\024)h Fr(d)23 b Fq(\000)h Fy(1,)35 b(parametrised)h(b)m(y)-8 b(,)36 b(sa)m(y)-8 b(,)36 b Fr(I)1983 3636 y Fm(1)2022 3621 y Fr(;)17 b(:)g(:)g(:)f(;)h(I)2284 3636 y Fn(n)2331 3621 y Fy(,)35 b(requires)h(non-degeneracy)g(conditions)109 3741 y(on)28 b(the)i(frequencies)h Fr(!)961 3756 y Fn(`)994 3741 y Fy(\()p Fr(I)8 b Fy(\))27 b(=)h Fr(\013)1314 3756 y Fn(`)1361 3741 y Fy(+)15 b Fr(\014)1507 3756 y Fn(`)p Fm(1)1575 3741 y Fr(I)1618 3756 y Fm(1)1672 3741 y Fy(+)g Fr(:)i(:)g(:)c Fy(+)i Fr(\014)2037 3756 y Fn(`n)2113 3741 y Fr(I)2156 3756 y Fn(n)2231 3741 y Fy(that)29 b(ensure)h(the)g (Diophan)m(tine)f(conditions)689 3984 y Fq(j)e Fr(k)795 3999 y Fm(1)834 3984 y Fr(!)895 3999 y Fm(1)934 3984 y Fy(\()p Fr(I)8 b Fy(\))22 b(+)g Fr(:)17 b(:)g(:)22 b Fy(+)g Fr(k)1467 3999 y Fn(n)1514 3984 y Fr(!)1575 3999 y Fn(n)1621 3984 y Fy(\()p Fr(I)8 b Fy(\))39 b(+)f Fr(k)1952 3999 y Fn(n)p Fm(+1)2089 3984 y Fr(\013)2151 3999 y Fn(n)p Fm(+1)2311 3984 y Fy(+)22 b Fr(:)17 b(:)g(:)k Fy(+)h Fr(k)2694 3999 y Fn(d)2734 3984 y Fr(\013)2796 3999 y Fn(d)2865 3984 y Fq(j)82 b(\025)3194 3917 y Fr(\015)p 3146 3961 153 4 v 3146 4053 a Fq(j)p Fr(k)s Fq(j)3256 4024 y Fn(\034)109 4260 y Fy(to)35 b(hold)h(true)h(for)e(all)h Fr(k)1003 4275 y Fm(1)1042 4260 y Fr(;)17 b(:)g(:)g(:)f(;)h(k)1312 4275 y Fn(n)1392 4260 y Fq(2)34 b Fo(Z)i Fy(\(not)g(all)g(zero\))g(and) g Fr(k)2439 4275 y Fn(n)p Fm(+1)2576 4260 y Fr(;)17 b(:)g(:)g(:)f(;)h (k)2846 4275 y Fn(d)2919 4260 y Fq(2)34 b(f)p Fy(0)p Fr(;)17 b Fq(\006)p Fy(1)p Fr(;)g Fq(\006)p Fy(2)p Fq(g)35 b Fy(on)h(large)109 4380 y(Can)m(tor)d(sets)g(in)g Fr(I)8 b Fy({space)33 b Fo(R)1173 4344 y Fn(n)1220 4380 y Fy(.)109 4669 y Fg(4.1)135 b(In)l(tegrable)47 b(resonances)109 4854 y Fy(The)33 b(resonances)i(\(9\))d(of)g(order)h Fq(j)p Fr(k)s Fq(j)27 b(\024)h Fy(4)k(read)1316 5074 y Fr(\013)1378 5089 y Fn(`)1467 5074 y Fy(=)55 b Fq(\006)p Fy(2)p Fr(\013)1786 5089 y Fn(j)1871 5074 y Fr(;)115 b(\013)2075 5089 y Fn(`)2130 5074 y Fy(+)22 b Fr(\013)2290 5089 y Fn(j)2382 5074 y Fy(=)55 b Fq(\006)p Fr(\013)2652 5089 y Fn(i)3715 5074 y Fy(\(10\))109 5294 y(and)737 5414 y Fr(\013)799 5429 y Fn(`)887 5414 y Fy(=)g Fq(\006)p Fy(3)p Fr(\013)1206 5429 y Fn(j)1292 5414 y Fr(;)114 b(\013)1495 5429 y Fn(`)1584 5414 y Fy(=)55 b Fq(\006)p Fr(\013)1854 5429 y Fn(j)1913 5414 y Fq(\006)23 b Fy(2)p Fr(\013)2124 5429 y Fn(i)2200 5414 y Fr(;)115 b(\013)2404 5429 y Fn(`)2459 5414 y Fy(+)22 b Fr(\013)2619 5429 y Fn(j)2711 5414 y Fy(=)55 b Fq(\006)p Fr(\013)2981 5429 y Fn(i)3032 5414 y Fq(\006)22 b Fr(\013)3193 5429 y Fn(m)3715 5414 y Fy(\(11\))1950 5778 y(10)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 109 208 a Fy(and)43 b(if)h(w)m(e)g(also)g(allo) m(w)f(for)g(m)m(ultiple)i(\(but)f(non-zero\))f(eigen)m(v)-5 b(alues)46 b(on)d(the)h(imaginary)g(axis)g(w)m(e)109 329 y(furthermore)33 b(ha)m(v)m(e)1732 449 y Fr(\013)1794 464 y Fn(`)1882 449 y Fy(=)55 b Fq(\006)p Fr(\013)2152 464 y Fn(j)2238 449 y Fr(:)109 621 y Fy(In)30 b(case)i(of)e(a)g(single) h(lo)m(w)g(order)f(resonance)i(the)f(Hamiltonian)g(\(8\))f(is)g(sup)s (erin)m(tegrable)j(with)e(quasi-)109 741 y(p)s(erio)s(dic)h(motions)h (spinning)g(densely)i(around)d(\()p Fr(d)21 b Fq(\000)h Fy(1\){tori.)42 b(T)-8 b(o)33 b(\014x)g(though)m(ts)g(let)f(the)h (resonance)109 861 y(read)1435 982 y Fr(\013)1497 997 y Fm(1)1591 982 y Fy(=)56 b Fq(\006)p Fr(k)s(\013)1916 997 y Fm(2)2004 982 y Fr(;)115 b(k)30 b Fq(2)e(f)p Fy(2)p Fr(;)17 b Fy(3)p Fq(g)109 1153 y Fy(whence)34 b(the)f(real)g(and)f (imaginary)h(parts)g(of)1378 1405 y Fr(J)92 b Fy(=)1709 1338 y(1)p 1693 1382 82 4 v 1693 1474 a Fr(k)s Fy(!)1784 1405 y(\()p Fr(q)1865 1420 y Fm(1)1927 1405 y Fq(\006)23 b Fy(i)p Fr(p)2103 1420 y Fm(1)2142 1405 y Fy(\)\()p Fr(q)2261 1420 y Fm(2)2323 1405 y Fq(\000)g Fy(i)p Fr(p)2499 1420 y Fm(2)2538 1405 y Fy(\))2576 1364 y Fn(k)109 1657 y Fy(are)32 b(among)h(the)g(generators)g(of)f(the)h Fo(T)1571 1621 y Fn(d)p Fi(\000)p Fm(1)1702 1657 y Fy({symmetric)h(functions,)g (leading)f(to)f(the)h(syzygy)1617 1860 y Fq(j)p Fr(J)9 b Fq(j)1736 1824 y Fm(2)p 1617 1905 158 4 v 1671 1996 a Fy(2)1868 1928 y(=)2037 1860 y(2)2086 1824 y Fn(k)p 2037 1905 92 4 v 2042 1996 a Fr(k)s Fy(!)2138 1928 y Fr(I)2181 1943 y Fm(1)2221 1928 y Fr(I)2272 1886 y Fn(k)2264 1952 y Fm(2)2363 1928 y Fr(:)1325 b Fy(\(12\))109 2166 y(Note)32 b(that)1575 2287 y Fr(K)90 b Fy(:=)83 b Fq(\006)p Fr(k)20 b(I)2125 2302 y Fm(1)2214 2287 y Fy(+)50 b Fr(I)2383 2302 y Fm(2)109 2458 y Fy(has)33 b(v)-5 b(anishing)33 b(P)m(oisson)h(brac)m(k)m(et)g(not)f(only)g(with)g Fr(I)2063 2473 y Fm(1)2102 2458 y Fr(;)17 b(:)g(:)g(:)f(;)h(I)2364 2473 y Fn(d)2404 2458 y Fy(,)33 b(but)g(also)f(with)1441 2672 y Fr(u)55 b Fy(:=)g(Re)17 b Fr(J)58 b(;)114 b(v)59 b Fy(:=)d(Im)17 b Fr(J)109 2885 y Fy(and)32 b(with)1545 3005 y Fr(w)86 b Fy(:=)d Fq(\007)p Fr(k)20 b(I)2078 3020 y Fm(1)2167 3005 y Fy(+)50 b Fr(I)2336 3020 y Fm(2)2424 3005 y Fr(:)109 3177 y Fy(Fixing)32 b Fr(y)461 3192 y Fm(1)528 3177 y Fy(=)c Fr(K)39 b Fy(the)33 b(syzygy)j(\(12\))c(reads) 872 3453 y Fr(S)932 3468 y Fn(y)973 3453 y Fy(\()p Fr(u;)17 b(v)t(;)g(w)s Fy(\))81 b(=)1567 3385 y Fr(u)1623 3349 y Fm(2)1684 3385 y Fy(+)22 b Fr(v)1833 3349 y Fm(2)p 1567 3430 306 4 v 1695 3521 a Fy(2)1932 3453 y Fq(\006)2069 3385 y Fy(\()p Fr(w)j Fq(\000)d Fr(y)2349 3400 y Fm(1)2388 3385 y Fy(\)\()p Fr(w)j Fy(+)d Fr(y)2705 3400 y Fm(1)2744 3385 y Fy(\))2782 3349 y Fn(k)p 2069 3430 756 4 v 2319 3521 a Fy(2)p Fr(k)j Fq(\001)d Fr(k)s Fy(!)2917 3453 y(=)83 b(0)109 3691 y(and)48 b(together)g(with)h(the)f(actions)h Fr(y)1530 3706 y Fn(`)1617 3691 y Fy(:=)54 b Fr(I)1817 3706 y Fn(`)p Fm(+1)1940 3691 y Fy(,)e Fr(`)i Fy(=)g(2)p Fr(;)17 b(:)g(:)g(:)e(;)i(d)33 b Fq(\000)g Fy(1)48 b(and)g(conjugate)g (angles)h Fr(x)3828 3706 y Fn(`)3861 3691 y Fy(,)109 3812 y Fr(`)27 b Fy(=)h(1)p Fr(;)17 b(:)g(:)g(:)e(;)i(d)22 b Fq(\000)h Fy(1)32 b(w)m(e)h(obtain)g(a)f(semi-global)h(c)m(hart)g(\() p Fr(x;)17 b(y)t(;)g(u;)g(v)t(;)g(w)s Fy(\))30 b(when)j(restricting)h (\()p Fr(u;)17 b(v)t(;)g(w)s Fy(\))30 b(to)765 4025 y Fq(P)834 4040 y Fn(y)959 4025 y Fy(:=)1145 3944 y Fh(\010)1231 4025 y Fy(\()p Fr(u;)17 b(v)t(;)g(w)s Fy(\))26 b Fq(2)i Fo(R)1767 3984 y Fm(3)1834 4025 y Fq(j)f(\000)p Fr(y)2014 4040 y Fm(1)2081 4025 y Fq(\024)h Fr(w)i Fq(\024)f Fr(y)2440 4040 y Fm(1)2479 4025 y Fr(;)17 b(S)2583 4040 y Fn(y)2624 4025 y Fy(\()p Fr(u;)g(v)t(;)g(w)s Fy(\))25 b(=)j(0)3173 3944 y Fh(\011)109 4238 y Fy(in)k(the)h(de\014nite)h(case)g Fr(\013)1006 4253 y Fm(1)1073 4238 y Fy(=)27 b Fr(k)s(\013)1292 4253 y Fm(2)1365 4238 y Fy(and)32 b(to)886 4452 y Fq(P)955 4467 y Fn(y)1080 4452 y Fy(:=)1266 4371 y Fh(\010)1352 4452 y Fy(\()p Fr(u;)17 b(v)t(;)g(w)s Fy(\))26 b Fq(2)i Fo(R)1888 4411 y Fm(3)1955 4452 y Fq(j)f Fr(w)j Fq(\025)e(j)p Fr(y)2291 4467 y Fm(1)2330 4452 y Fq(j)p Fr(;)17 b(S)2462 4467 y Fn(y)2503 4452 y Fy(\()p Fr(u;)g(v)t(;)g(w)s Fy(\))25 b(=)j(0)3052 4371 y Fh(\011)109 4665 y Fy(in)j(the)h(inde\014nite)h (case)g Fr(\013)1083 4680 y Fm(1)1150 4665 y Fy(=)27 b Fq(\000)p Fr(k)s(\013)1446 4680 y Fm(2)1518 4665 y Fy(\(recall)32 b Fr(k)e Fq(2)f(f)p Fy(2)p Fr(;)17 b Fy(3)p Fq(g)p Fy(\).)41 b(An)m(y)33 b(c)m(hoice)g(of)e(canonical)h (co-ordinates)109 4785 y(on)g Fq(P)313 4800 y Fn(y)387 4785 y Fy(yields)j(generalized)f(action)e(angle)h(v)-5 b(ariables.)255 4906 y(F)d(or)36 b(a)h(single)i(resonance)f(the)g(in)m (termediate)h(system)g(is)f(giv)m(en)g(b)m(y)g(the)g(truncated)g(Gusta) m(vson)109 5026 y(or)32 b(resonan)m(t)h(\(Birkho\013)7 b(\))33 b(normal)g(form)672 5333 y Fq(H)756 5348 y Fn(\013;\025)867 5333 y Fy(\()p Fr(y)t(;)17 b(u;)g(v)t(;)g(w)s Fy(\))80 b(=)j Fr(H)1635 5292 y Fm(2)1627 5358 y Fn(\013)1676 5333 y Fy(\()p Fr(y)t Fy(\))49 b(+)g Fr(\025w)j Fy(+)e Fr(\015)5 b Fy(\()p Fr(\025)p Fy(\))p Fr(u)49 b Fy(+)2768 5209 y Fn(d)2715 5238 y Fh(X)2703 5451 y Fn(j;`)p Fm(=1)2887 5333 y Fr(\014)2942 5348 y Fn(j)t(`)3007 5333 y Fy(\()p Fr(\025)p Fy(\))3150 5266 y Fr(y)3198 5281 y Fn(j)3234 5266 y Fr(y)3282 5281 y Fn(`)p 3150 5310 165 4 v 3208 5401 a Fy(2)1950 5778 y(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 109 208 a Fy(of)28 b(order)g(4,)h(where)h Fr(\025)e Fy(is)h(the)g(so-called)g(detuning)g(parameter)g(and)f(w)m(e) i(abused)f(notation)f(b)m(y)i(letting)109 329 y Fr(y)157 344 y Fn(d)226 329 y Fy(=)e Fr(w)36 b Fy(in)d(the)h(\014nal)g(sum.)46 b(Note)34 b(that)f(the)g(\(linear\))h(term)g(in)f Fr(v)k Fy(has)d(b)s(een)g(remo)m(v)m(ed)h(b)m(y)f(means)h(of)109 449 y(a)d Fr(\025)p Fy({dep)s(enden)m(t)i(rotation)e(in)h(the)g(\()p Fr(u;)17 b(v)t Fy(\){plane)32 b([19)o(,)h(31)o(].)255 569 y(Reducing)46 b(the)g Fo(T)944 533 y Fn(d)p Fi(\000)p Fm(1)1075 569 y Fy({symmetry)h(of)e(the)h(in)m(termediate)h(system)g (leads)f(to)f(a)g(one-degree-of-)109 690 y(freedom)39 b(problem)h(on)f Fq(P)1083 705 y Fn(y)1163 690 y Fy(that)g(determines)i (ho)m(w)f(the)f(in)m(v)-5 b(arian)m(t)39 b Fr(d)p Fy({tori)f(are)h (organized)h(in)m(to)f Fr(d)p Fy({)109 810 y(parameter)j(families.)73 b(The)43 b(resulting)g(rami\014ed)g(torus)f(bundle)h(has)f Fr(d)29 b Fq(\000)g Fy(1)41 b(fast)h(frequencies)j(\(in)109 931 y(the)e(angles)g Fr(x)645 946 y Fm(1)685 931 y Fr(;)17 b(:)g(:)g(:)f(;)h(x)959 946 y Fn(d)p Fi(\000)p Fm(1)1089 931 y Fy(\))43 b(and)g(one)f(slo)m(w)i(frequency)h(\(for)d Fr(y)2480 946 y Fm(1)2561 931 y Fy(and)h Fr(\025)g Fy(su\016cien)m(tly) i(small\).)74 b(The)109 1051 y(\014nal)29 b(step)g(then)h(is)f(to)f (apply)i Fp(kam)f Fy(theory)g(to)g(obtain)f(Can)m(tor)h(sub-families)h (of)f(in)m(v)-5 b(arian)m(t)29 b(tori)f(that)109 1171 y(surviv)m(e)34 b(the)g(p)s(erturbation)e(bac)m(k)i(to)e(the)h (original)f(Hamiltonian.)109 1458 y Fg(4.2)135 b(Non-in)l(tegrable)47 b(resonances)109 1643 y Fy(F)-8 b(or)40 b(t)m(w)m(o)h(or)g(more)g (resonances)i(\(9\))e(the)g(resonan)m(t)h(Birkho\013)f(or)g(Gusta)m (vson)h(normal)f(form)g(is)g(no)109 1763 y(longer)d(guaran)m(teed)h(to) e(b)s(e)i(in)m(tegrable.)61 b(W)-8 b(riting)38 b Fr(d)26 b Fq(\000)g Fr(n)38 b Fy(for)g(the)h(n)m(um)m(b)s(er)g(of)f(indep)s (enden)m(t)i(reso-)109 1883 y(nances,)34 b(the)f(sup)s(erin)m(tegrable) i(system)g(of)d(coupled)i(harmonic)g(oscillators)g(\(8\))e(de\014nes)j (a)d(rami\014ed)109 2004 y Fo(T)175 1968 y Fn(n)222 2004 y Fy({bundle,)j(the)f(geometry)g(of)f(whic)m(h)i(dep)s(ends)g(on)f(the) g(resonances)h(at)e(hand.)47 b(In)34 b(particular)g(the)109 2124 y(\015o)m(w)e(is)h(p)s(erio)s(dic)f(in)g(the)g(maximal)h(case)g (of)e Fr(d)21 b Fq(\000)g Fy(1)31 b(resonances.)46 b(Let)32 b(us)g(restrict)h(to)f Fr(d)27 b Fy(=)g(3)32 b(coupled)109 2245 y(oscillators)h(whence)h(scaling)g(time)f(allo)m(ws)g(to)g (rewrite)g(\(8\))f(as)1210 2450 y Fr(H)1299 2409 y Fm(2)1291 2475 y Fn(m)1357 2450 y Fy(\()p Fr(q)t(;)17 b(p)p Fy(\))83 b(=)f Fr(m)1899 2465 y Fm(1)1939 2450 y Fr(I)1982 2465 y Fm(1)2071 2450 y Fy(+)50 b Fr(m)2282 2465 y Fm(2)2322 2450 y Fr(I)2365 2465 y Fm(2)2454 2450 y Fy(+)g Fr(m)2665 2465 y Fm(3)2705 2450 y Fr(I)2748 2465 y Fm(3)3715 2450 y Fy(\(13\))109 2655 y(with)33 b(in)m(teger)h(v)m(ector)g Fr(m)28 b Fq(2)g Fo(Z)1220 2619 y Fm(3)1293 2655 y Fy(satisfying)34 b(0)28 b Fr(<)g(m)1994 2670 y Fm(1)2061 2655 y Fr(<)g Fq(j)p Fr(m)2278 2670 y Fm(2)2318 2655 y Fq(j)g Fr(<)f Fq(j)p Fr(m)2590 2670 y Fm(3)2630 2655 y Fq(j)p Fy(.)44 b(The)33 b(cases)i(where)f(b)s(oth)e(reso-)109 2776 y(nances)e(\(9\))f (are)g(of)g(order)h(3)f(are)g(called)h(of)f(gen)m(uine)i(\014rst)e (order)h([39)o(],)g(these)h(are)e(the)h(cases)h(1:)p Fq(\006)p Fy(2:)p Fq(\006)p Fy(3)109 2896 y(and)39 b(1:)p Fq(\006)p Fy(2:)p Fq(\006)p Fy(4.)63 b(Here)40 b(already)g(the)f(cubic) h(normal)f(form)g(is)h(exp)s(ected)h(to)d(b)s(e)i(non-in)m(tegrable,)h (in)109 3016 y(the)33 b(de\014nite)h(cases)f(1:2:3)g(and)f(1:2:4)g (this)h(has)g(b)s(een)h(pro)m(v)m(en)g(in)f([12)o(].)255 3137 y(In)45 b(cases)i(of)d(gen)m(uine)j(second)f(order)f(b)s(oth)g (resonances)i(\(9\))d(are)i(of)e(order)h(at)g(most)g(4.)81 b(F)-8 b(or)109 3257 y Fr(m)36 b Fy(=)h(1:)p Fq(\006)p Fy(3:)p Fq(\006)p Fr(k)s Fy(,)j Fr(k)f Fq(2)e(f)p Fy(5)p Fr(;)17 b Fy(7)p Fr(;)g Fy(9)p Fq(g)36 b Fy(b)s(oth)i(resonances)h(are) f(of)f(the)i(form)e(\(11\))g(and)h(the)g(cubic)h(normal)109 3377 y(form)31 b(is)h(trivial;)h(the)f(fourth)f(order)h(normal)g(form)f (is)i(again)e(exp)s(ected)i(to)f(b)s(e)g(non-in)m(tegrable.)43 b(In)32 b(a)109 3498 y(gen)m(uine)f(second)g(order)f(case)h(where)g (exactly)g(one)g(resonance)g(is)f(of)g(order)g(3,)g Fl(i.e.)f Fy(of)h(the)g(form)g(\(10\))o(,)109 3618 y(the)i(fourth)f(order)g (normal)h(form)f(is)h(still)g(exp)s(ected)i(to)d(b)s(e)g(non-in)m (tegrable,)i(but)f(the)g(cubic)g(normal)109 3739 y(form)g Fl(is)g Fy(in)m(tegrable)i(and)e(ma)m(y)i(serv)m(e)g(as)f(in)m (termediate)h(system.)255 3859 y(If)i(w)m(e)i(allo)m(w)f(for)f(m)m (ultiple)i(eigen)m(v)-5 b(alues)38 b Fr(m)1892 3874 y Fn(`)1960 3859 y Fy(=)c Fq(\006)p Fr(m)2232 3874 y Fn(j)2269 3859 y Fy(,)k(then)f(there)g(are)f(additional)h(cases)h Fr(m)c Fy(=)109 3979 y(1:)p Fq(\006)p Fy(1:)p Fq(\006)p Fy(2)p Fr(;)17 b Fy(1:)p Fq(\006)p Fy(2:)p Fq(\006)p Fy(2)44 b(of)e(gen)m(uine)j(\014rst)f(order)f(and)g Fr(m)k Fy(=)e(1:1:)p Fq(\006)p Fy(1)p Fr(;)17 b Fy(1:)p Fq(\006)p Fy(1:)p Fq(\006)p Fy(3)p Fr(;)g Fy(1:)p Fq(\006)p Fy(3:)p Fq(\006)p Fy(3)44 b(of)f(gen)m(uine)109 4100 y(second)38 b(order)f([39].)56 b(In)38 b(the)f(de\014nite)h(case)g(1:1:2)f(the)g (cubic)h(normal)f(form)g(is)g(kno)m(wn)i(to)d(b)s(e)i(non-)109 4220 y(in)m(tegrable)30 b([20,)g(12)o(])g(while)h(for)e Fr(m)f Fy(=)f Fq(\006)p Fy(1:2:2)j(the)g(cubic)h(normal)e(form)h(ma)m (y)g(serv)m(e)i(as)d(in)m(termediate)109 4340 y(system)f([1,)f(31)o(,)g (25];)h(in)f(the)g(additional)g(cases)h(of)e(gen)m(uine)i(second)g (order)f(the)g(cubic)h(normal)e(form)h(is)109 4461 y(again)h(trivial.) 43 b(In)29 b(inde\014nite)h(cases)h(with)e(double)h(eigen)m(v)-5 b(alues)30 b(of)f(opp)s(osite)g(symplectic)j(sign)d(these)109 4581 y(ma)m(y)38 b(b)s(e)f(in)m(v)m(olv)m(ed)j(in)d(a)g(Krein)h (collision)g(and)g(lea)m(v)m(e)g(the)g(imaginary)g(axis)g(during)f(a)g (sub)s(ordinate)109 4702 y(Hamiltonian)c(Hopf)f(bifurcation.)255 4822 y(T)-8 b(o)24 b(construct)h(generalized)h(action)e(angle)g(v)-5 b(ariables)25 b(for)f(\(13\))f(w)m(e)i(tak)m(e)g Fr(y)31 b Fy(=)c Fr(H)3179 4786 y Fm(2)3171 4847 y Fn(m)3262 4822 y Fy(itself)e(as)f(action,)109 4942 y(the)34 b(conjugate)h(angle)f Fr(x)h Fy(then)g(is)g(the)g(time)g(along)e(the)i(p)s(erio)s(dic)g (\015o)m(w)g(of)f(the)h(resonan)m(t)g(oscillator.)109 5063 y(Abusing)45 b(notation)f(b)m(y)h(also)g(writing)f Fr(y)k Fy(for)43 b(the)i(\(\014xed\))h(v)-5 b(alue)44 b(of)51 b(\(13\))44 b(and)g(dividing)i(out)e(the)109 5183 y Fr(S)175 5147 y Fm(1)214 5183 y Fy({action)32 b(de\014ned)i(b)m(y)g(the)f(p)s(erio)s(dic)g(\015o)m(w)g(w)m(e)h (obtain)e(the)h(reduced)h(phase)g(space)1505 5388 y Fq(P)1574 5403 y Fn(y)1699 5388 y Fy(:=)83 b(\()p Fr(H)2012 5347 y Fm(2)2004 5413 y Fn(m)2070 5388 y Fy(\))2108 5347 y Fi(\000)p Fm(1)2202 5388 y Fy(\()p Fr(y)t Fy(\))2330 5429 y Fr(=)2379 5473 y(S)2445 5437 y Fm(1)1950 5778 y Fy(12)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 109 208 a Fy(and)44 b(an)m(y)i(c)m(hoice)g(\()p Fr(\030)890 223 y Fm(1)929 208 y Fr(;)17 b(\030)1016 223 y Fm(2)1054 208 y Fr(;)g(\021)1146 223 y Fm(1)1186 208 y Fr(;)g(\021)1278 223 y Fm(2)1317 208 y Fy(\))45 b(of)f(canonical)h(co-ordinates)g(on)f Fq(P)2739 223 y Fn(y)2826 208 y Fy(yields)i(generalized)g(action)109 329 y(angle)30 b(v)-5 b(ariables)31 b(\()p Fr(x;)17 b(y)t(;)g(\030)5 b(;)17 b(\021)t Fy(\).)40 b(The)31 b(global)f(structure)i(of)e Fq(P)2322 344 y Fn(y)2393 329 y Fy(dep)s(ends)i(on)e Fr(m)e Fq(2)g Fo(Z)3174 293 y Fm(3)3245 329 y Fy(and)i(can)g Fl(e.g.)g Fy(b)s(e)109 449 y(studied)k(using)f(the)g(in)m(v)-5 b(arian)m(ts)34 b(of)e(the)h Fr(S)1667 413 y Fm(1)1706 449 y Fy({action)f([29,)h(39)o(].)255 569 y(Reduction)23 b(of)e(the)i(\014rst)f(non-trivial)g(normal)g(form)g(leads)h(to)e(a)h (t)m(w)m(o-degree-of-freedom)g(problem)109 690 y(on)30 b Fq(P)311 705 y Fn(y)382 690 y Fy(and)g(since)i(normalization)e (preserv)m(es)j(symmetries)g(a)c(simple)j(w)m(a)m(y)f(to)f(enforce)h (in)m(tegrabilit)m(y)109 810 y(is)41 b(to)g(require)i(the)f(original)f (Hamiltonian)g(to)g(admit)h(the)f(suitable)i(symmetry)-8 b(.)72 b(The)42 b(resonance)109 931 y Fr(m)194 946 y Fn(`)255 931 y Fy(=)29 b Fr(m)445 946 y Fn(j)515 931 y Fy(mak)m(es)34 b(an)f(axial)g Fr(S)1246 894 y Fm(1)1286 931 y Fy({symmetry)i(an)d(ob)m(vious)j(candidate,)f(but)f(all)g(that)g (is)g(needed)i(is)e(to)109 1051 y(prev)m(en)m(t)25 b(certain)e(third)h (or)e(fourth)h(order)g(terms)h(to)f(app)s(ear)g(in)g(the)g(normal)g (form)g(whence)i(in)m(v)-5 b(ariance)109 1171 y(under)36 b(a)g(discrete)h(subgroup)g Fo(Z)1327 1186 y Fn(k)1406 1171 y Fy(of)e Fr(S)1586 1135 y Fm(1)1661 1171 y Fy(su\016ces,)k(with)e Fr(k)f Fy(=)d(2)j(for)f(third)h(order)g(terms,)i Fr(k)e Fy(=)d(3)j(for)109 1292 y(fourth)j(order)g(terms)i(or)e Fr(k)j Fq(\025)e Fy(5)f(just)h(to)f(b)s(e)h(sure.)64 b(Discrete)41 b(symmetries)h(can)e(also)f(pro)m(vide)i(the)109 1412 y(other)35 b(cases)h(with)g(an)f(in)m(tegrable)h(normal)f(form)f ([39],)i(often)f(a)f(rev)m(ersing)j(re\015ection)f Fr(p)3432 1427 y Fn(`)3497 1412 y Fq(7!)31 b(\000)p Fr(p)3754 1427 y Fn(`)3823 1412 y Fy(is)109 1532 y(all)h(that)h(is)g(needed.)109 1865 y Fs(5)161 b(Higher)52 b(sup)t(erin)l(tegrabilit)l(y)109 2084 y Fy(F)-8 b(or)21 b(minimally)j(sup)s(erin)m(tegrable)g (Hamiltonian)e(systems,)27 b(normalization)c(of)f(a)g(giv)m(en)h(p)s (erturbation)109 2205 y(with)30 b(resp)s(ect)h(to)e(the)h(unp)s(erturb) s(ed)g(\015o)m(w)g(alw)m(a)m(ys)h(remo)m(v)m(es)h(the)e(degeneracy)h(|) e(sub)5 b(ject)31 b(to)e(a)h(non-)109 2325 y(degeneracy)h(condition)g (on)f(the)g(in)m(termediate)i(system,)g Fl(i.e.)e Fy(on)g(the)g(p)s (erturbation.)43 b(In)30 b(some)h(cases)109 2445 y(this)42 b(remains)g(true)g(where)h(sup)s(erin)m(tegrabilit)m(y)h(is)e(not)g (minimal,)i(for)d(instance)i(in)f(the)g(presence)109 2566 y(of)47 b(additional)h(symmetry)-8 b(.)92 b(Ho)m(w)m(ev)m(er,)54 b(if)48 b(already)h(the)f(\014rst)h(non-trivial)f(normal)g(form)g(is)g (not)109 2686 y(in)m(tegrable,)i(then)c(alternativ)m(e)h(w)m(a)m(ys)h (to)d(obtain)h(an)g(in)m(tegrable)h(in)m(termediate)g(system)h(b)s (ecome)109 2807 y(imp)s(ortan)m(t.)255 2927 y(F)-8 b(or)36 b(a)g(free)g(rigid)h(b)s(o)s(dy)f(with)h(principal)g(momen)m(ts)h(of)e (inertia)h Fr(I)2717 2942 y Fm(1)2791 2927 y Fy(=)d Fr(I)2944 2942 y Fm(2)3017 2927 y Fy(=)g Fr(I)3170 2942 y Fm(3)3246 2927 y Fy(all)i(motions)h(are)109 3047 y(p)s(erio)s(dic)f(\(as)g(ev)m (ery)h(axis)g(is)f(a)g(main)g(axis)h(of)e(inertia\);)j(recall)e(that)g (w)m(e)h(exclude)h(the)e(zero)g(section)109 3168 y Fr(S)6 b(O)s Fy(\(3\))27 b Fq(\022)i Fr(T)582 3132 y Fi(\003)621 3168 y Fr(S)6 b(O)s Fy(\(3\))32 b(of)h(equilibria.)46 b(This)34 b(is)g(a)f(maximally)h(sup)s(erin)m(tegrable)h(system)g(and)f (a)e(semi-)109 3288 y(global)26 b(c)m(hart)i(\()p Fr(x;)17 b(y)t(;)g(\026;)g(`)p Fy(\))25 b(is)i(giv)m(en)h(b)m(y)g(the)g(angle)f Fr(x)g Fy(measuring)h(the)f(rotation)g(\(=precession\))i(ab)s(out)109 3409 y(the)h(angular)f(momen)m(tum)i(v)m(ector,)g(b)m(y)g(the)f(length) g Fr(y)j Fy(of)c(that)g(v)m(ector)i(and)f(b)m(y)g(restricting)h Fr(\026;)17 b(`)27 b Fq(2)h Fo(R)3849 3372 y Fm(3)109 3529 y Fy(to)k(the)h(symplectic)i(\014bres)673 3749 y Fr(S)739 3708 y Fm(2)733 3774 y Fn(y)801 3749 y Fq(\002)22 b Fr(S)966 3708 y Fm(2)960 3774 y Fn(y)1089 3749 y Fy(=)1247 3668 y Fh(\010)1333 3749 y Fy(\()p Fr(\026;)17 b(`)p Fy(\))27 b Fq(2)h Fo(R)1746 3708 y Fm(6)1813 3749 y Fq(j)g Fr(\026)1928 3708 y Fm(2)1928 3774 y(1)1989 3749 y Fy(+)22 b Fr(\026)2146 3708 y Fm(2)2146 3774 y(2)2207 3749 y Fy(+)g Fr(\026)2364 3708 y Fm(2)2364 3774 y(3)2431 3749 y Fy(=)28 b Fr(`)2576 3708 y Fm(2)2576 3774 y(1)2637 3749 y Fy(+)22 b Fr(`)2776 3708 y Fm(2)2776 3774 y(2)2838 3749 y Fy(+)g Fr(`)2977 3708 y Fm(2)2977 3774 y(3)3044 3749 y Fy(=)27 b Fr(y)3199 3708 y Fm(2)3265 3668 y Fh(\011)109 3969 y Fy(of)35 b(the)h(bi\014bration,)h(see)f(\014gure)h(1.)52 b(Already)36 b(quadratic)h(force)f(\014elds)g(ha)m(v)m(e)h(a)m(v)m (erages)h(that)d(cannot)109 4089 y(b)s(e)h(used)i(as)f(in)m(tegrable)g (in)m(termediate)i(system)f(\(while)g(the)f(linear)g(part)f(of)g(a)g (force)h(\014eld)g(can)g(not)109 4210 y(b)s(e)32 b(`seen')j(b)m(y)e(a)f (dynamically)j(spherically)f(symmetric)h(top\).)255 4330 y(Coupling)41 b Fr(m)f Fy(suc)m(h)h(tops)f(giv)m(es)h(rise)g(to)e(a)h (sup)s(erin)m(tegrable)h(system)h(with)f(tori)e(of)h(dimension)109 4450 y Fr(n)d Fy(=)f Fr(m)i Fy(in)g Fr(d)f Fy(=)f(3)p Fr(m)i Fy(degrees)h(of)f(freedom.)60 b(Replacing)38 b(some)h(\(or)e (all\))h(of)f(them)i(with)f(Euler)h(and)109 4571 y(Lagrange)c(tops)h (allo)m(ws)g(to)f(obtain)h(rami\014ed)g Fo(T)1921 4535 y Fn(n)1969 4571 y Fy({bundles)h(with)f(an)m(y)g Fr(m)d Fq(\024)h Fr(n)f Fq(\024)g Fr(d)f Fy(=)h(3)p Fr(m)p Fy(.)53 b(Recall)109 4691 y(that)37 b(imp)s(osing)g Fr(d)25 b Fq(\000)h Fr(n)37 b Fy(resonances)i(on)f(an)f(elliptic)h(equilibrium)h (ev)m(en)g(allo)m(ws)f(for)f(all)g Fr(n)g Fy(b)s(et)m(w)m(een)109 4812 y(the)c(maximal)g(case)g Fr(n)28 b Fy(=)g(1)k(and)h(the)g(case)g Fr(n)28 b Fy(=)g Fr(d)22 b Fq(\000)g Fy(1)33 b(of)f(minimal)h(sup)s (erin)m(tegrabilit)m(y)-8 b(.)109 5100 y Fg(5.1)135 b(P)l(erturbations) 46 b(that)g(remo)l(v)l(e)g(the)f(degeneracy)109 5285 y Fy(The)34 b(\(spatial\))f(Kepler)i(system)g(is)f(the)f(most)h(famous) g(example)h(of)e(a)g(maximally)h(sup)s(erin)m(tegrable)109 5406 y(system)44 b(in)f(three)g(degrees)h(of)e(freedom.)74 b(Some)44 b(p)s(erturb)s(ed)f(Keplerian)g(systems)i(lik)m(e)f(the)f (lunar)1950 5778 y(13)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 109 208 a Fy(problem)32 b([14)o(,)g(40)o(])g (and)f(in)h(the)g(h)m(ydrogen)g(\(or)f(Rydb)s(erg\))h(atom)f(in)h (crossed)h(\014elds)f([15,)g(21)o(])g(displa)m(y)109 329 y(a)g(helpful)h(time)h(scale)f(phenomenon.)45 b(In)33 b(b)s(oth)f(cases)i(the)f(normal)g(form)1608 516 y(\026)1583 541 y Fr(H)90 b Fy(=)83 b Fr(K)57 b Fy(+)50 b Fr(")17 b(K)23 b(L)109 754 y Fy(obtained)29 b(b)m(y)g(a)m(v)m(eraging)g(along)f (the)h(Keplerian)g(ellipses)h(is)f(the)g(sum)g(of)f(the)h(unp)s(erturb) s(ed)h Fr(H)3628 769 y Fm(0)3694 754 y Fy(=)e Fr(K)109 874 y Fy(and)k(a)h(scalar)g(m)m(ultiple)h(of)e(the)h(pro)s(duct)g Fr(K)7 b(L)p Fy(,)33 b(where)h Fr(L)f Fy(is)g(the)g(third)g(comp)s (onen)m(t)g(of)f(the)h(angular)109 995 y(momen)m(tum.)80 b(This)45 b(sp)s(ecial)g(structure)h(of)1835 969 y(\026)1809 995 y Fr(H)52 b Fy(allo)m(ws)45 b(for)f(a)g(second)h(normalization,)j (no)m(w)d(with)109 1115 y(resp)s(ect)33 b(to)g Fr(L)p Fy(,)g(that)f(results)i(in)1211 1328 y Fq(H)84 b Fy(=)f Fr(K)57 b Fy(+)50 b Fr(")17 b(K)23 b(L)50 b Fy(+)g Fr(")2261 1286 y Fm(2)2300 1328 y Fr(H)2381 1343 y Fm(1)2420 1328 y Fy(\()p Fr(K)r(;)17 b(L;)g(I)8 b Fy(\))109 1540 y(with)27 b(an)f(appropriately)h(c)m(hosen)h(third)e(action)h Fr(I)8 b Fy(.)41 b(F)-8 b(or)25 b(the)i(lunar)f(problem)h(the)g(di\013erence)h (b)s(et)m(w)m(een)109 1661 y(the)33 b(t)m(wice)h(normalized)g Fq(H)g Fy(and)f(the)g(original)g(Hamiltonian)g Fr(H)40 b Fy(is)33 b(of)g(order)g Fr(")3050 1624 y Fm(8)p Fn(=)p Fm(3)3192 1661 y Fy(and)g(taking)g Fq(H)h Fy(as)109 1781 y(in)m(termediate)47 b(system)h(yields)f(a)e(Can)m(tor)h(family)h(of)e (3{tori)g(with)h(three)g(time)h(scales)g(\(and)e(t)m(w)m(o)109 1901 y(time)37 b(scales)h Fr(")e Fy(and)g Fr(")932 1865 y Fm(2)1008 1901 y Fy(of)g(rates)h(of)f(c)m(hange)h(of)f(these)i (freqencies\))h(that)d(p)s(ersist)i(in)m(to)f(the)f(original)109 2022 y(system)e([40].)255 2142 y(P)m(erturbations)23 b(of)f(sup)s(erin)m(tegrable)h(systems)i(alw)m(a)m(ys)e(ha)m(v)m(e)g (three)g(time)f(scales:)40 b(the)22 b(fast)g(motion)109 2263 y(on)35 b(the)h Fr(n)p Fy({tori)f(\(the)g(p)s(etals)h(of)f(the)h (F)-8 b(ass\022)-49 b(o)35 b(\015o)m(w)m(ers\),)j(the)d(slo)m(w)i (motion)e(on)g(the)h(symplectic)i(\014bres)109 2383 y(\(the)31 b(cen)m(tres)h(of)e(the)h(F)-8 b(ass\022)-49 b(o)31 b(\015o)m(w)m (ers\))h(and)f(the)g(v)m(ery)h(\(exp)s(onen)m(tially\))h(slo)m(w)f (motion)e(of)g(the)i(actions)109 2503 y(\(in)26 b(the)g(meado)m(w)h (for)f(the)g(F)-8 b(ass\022)-49 b(o)26 b(\015o)m(w)m(ers\).)43 b(Where)27 b(the)f(dynamics)i(on)e(the)h(symplectic)h(\014bres)f(of)f (the)109 2624 y(reduced)36 b(a)m(v)m(eraged)g(system)h(has)e(as)g(man)m (y)h(time)f(scales)i(as)e(degrees)h(of)e(freedom)i(|)e(slo)m(w,)i(slo)m (w)m(er,)109 2744 y(.)16 b(.)g(.)g(,)30 b(slo)m(w)m(est)h(\(but)e (still)h(faster)f(than)g(exp)s(onen)m(tially)i(slo)m(w\))e(|)g(it)g (should)g(b)s(e)g(p)s(ossible)i(to)d(construct)109 2864 y(an)k(appropriate)h(in)m(termediate)h(system)h([26)o(].)109 3152 y Fg(5.2)135 b(The)44 b(C.Neumann)i(hierarc)l(h)l(y)109 3337 y Fy(The)51 b(Neumann)g(system)g(describ)s(es)h(a)e(particle)h (con\014ned)g(to)e(a)h(sphere)h Fr(S)3052 3301 y Fn(d)3142 3337 y Fy(and)f(mo)m(ving)h(in)f(a)109 3469 y(quadratic)29 b(p)s(oten)m(tial)958 3430 y Fm(1)p 958 3446 36 4 v 958 3504 a(2)1003 3375 y Fh(X)1147 3469 y Fr(b)1188 3484 y Fn(i)1217 3469 y Fr(q)1264 3433 y Fm(2)1260 3494 y Fn(i)1303 3469 y Fy(.)42 b(Only)29 b(the)g(di\013erences)h(b)s(et)m(w)m (een)h(the)e(eigen)m(v)-5 b(alues)30 b Fr(b)3321 3484 y Fn(i)3350 3469 y Fy(,)f Fr(i)f Fy(=)f(0)p Fr(;)17 b(:)g(:)g(:)f(;)h (d)109 3601 y Fy(of)26 b(the)i(linear)g(force)f(\014eld)h(ha)m(v)m(e)h (dynamical)f(consequences)j(and)d(indexing)g(equal)g(eigen)m(v)-5 b(alues)30 b Fr(b)3711 3616 y Fn(\033)3785 3601 y Fy(b)m(y)109 3722 y Fr(I)152 3737 y Fn(\033)199 3722 y Fy(,)i Fr(\033)g Fy(=)27 b(0)p Fr(;)17 b(:)g(:)g(:)f(;)h(`)32 b Fy(with)h Fq(j)p Fr(I)1082 3737 y Fn(\033)1129 3722 y Fq(j)27 b Fy(=)h Fr(m)1373 3737 y Fn(\033)1420 3722 y Fy(,)33 b(the)g (Hamiltonian)f(is)i(in)m(v)-5 b(arian)m(t)33 b(under)g(the)g(symmetry)i (group)1295 3934 y Fr(O)s Fy(\()p Fr(m)1496 3949 y Fm(0)1535 3934 y Fy(\))k Fq(\002)g Fr(O)s Fy(\()p Fr(m)1929 3949 y Fm(1)1968 3934 y Fy(\))g Fq(\002)g Fr(:)17 b(:)g(:)38 b Fq(\002)h Fr(O)s Fy(\()p Fr(m)2631 3949 y Fn(`)2664 3934 y Fy(\))1013 b(\(14\))109 4147 y(\(in)29 b(particular)g Fr(d)15 b Fy(+)g(1)27 b(=)g Fr(m)1124 4162 y Fm(0)1193 4147 y Fy(for)i Fr(`)e Fy(=)h(0)h(where)h(the)f(Neumann)h(system)h(b)s (ecomes)g(the)e(geo)s(desic)h(\015o)m(w)109 4267 y(on)i Fr(S)310 4231 y Fn(d)350 4267 y Fy(\).)44 b(In)m(tegrals)33 b(of)f(motion)h(are)g(the)g(angular)f(momen)m(ta)1306 4480 y Fr(L)1372 4495 y Fn(ij)1516 4480 y Fy(=)83 b Fr(q)1718 4495 y Fn(i)1746 4480 y Fr(p)1795 4495 y Fn(j)1882 4480 y Fq(\000)50 b Fr(q)2052 4495 y Fn(j)2089 4480 y Fr(p)2138 4495 y Fn(i)2215 4480 y Fr(;)114 b(i;)17 b(j)34 b Fq(2)28 b Fr(I)2644 4495 y Fn(\033)109 4693 y Fy(together)k(with)846 4993 y Fr(F)909 5008 y Fn(\033)1039 4993 y Fy(=)1218 4869 y Fn(m)1280 4877 y Fd(\033)1198 4898 y Fh(X)1198 5110 y Fn(i)p Fi(2)p Fn(I)1300 5118 y Fd(\033)1358 4993 y Fr(q)1405 4952 y Fm(2)1401 5018 y Fn(i)1495 4993 y Fy(+)1678 4869 y Fn(`)1620 4898 y Fh(X)1624 5111 y Fn(\034)8 b Fi(6)p Fm(=)p Fn(\033)1805 4869 y(m)1867 4877 y Fd(\033)1785 4898 y Fh(X)1781 5110 y Fn(j)t Fi(2)p Fn(I)1892 5118 y Fd(\033)1976 4869 y Fn(m)2038 4877 y Fd(\034)1955 4898 y Fh(X)1949 5111 y Fn(k)r Fi(2)p Fn(I)2066 5119 y Fd(\034)2214 4917 y Fr(L)2280 4881 y Fm(2)2280 4941 y Fn(ij)p 2131 4970 295 4 v 2131 5061 a Fr(b)2172 5076 y Fn(\033)2241 5061 y Fq(\000)23 b Fr(b)2382 5076 y Fn(\034)2484 4993 y Fr(;)114 b(\033)32 b Fy(=)c(0)p Fr(;)17 b(:)g(:)g(:)e(;)i(`)p Fy(.)109 5303 y(F)-8 b(or)23 b Fr(m)360 5318 y Fn(\033)435 5303 y Fy(=)28 b(1)c(the)h(latter)f(turn)h(in)m(to)g(the)g(Uhlen)m(b)s (ec)m(k)i(in)m(tegrals;)h(the)d(Neumann)h(system)g(is)f(in)m(tegrable) 109 5424 y(ev)m(en)42 b(if)e(all)g(eigen)m(v)-5 b(alues)42 b(are)e(di\013eren)m(t)i(from)e(eac)m(h)h(other.)66 b(F)-8 b(rom)40 b(no)m(w)h(on)f(w)m(e)i(assume)f Fr(m)3633 5439 y Fn(\033)3721 5424 y Fq(\025)g Fy(2)1950 5778 y(14)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 109 208 a Fy(for)33 b(all)h Fr(\033)g Fy(=)c(0)p Fr(;)17 b(:)g(:)g(:)f(;)h(`)34 b Fy(\(see)h([17)o(])f(for)g (the)g(general)h(case\).)49 b(Note)34 b(that)g(the)g(symmetry)i(group)e (\(14\))g(is)109 329 y(comm)m(utativ)m(e)c(if)f Fr(m)858 344 y Fn(\033)933 329 y Fy(=)e(2)i(for)f(all)g Fr(\033)33 b Fy(and)c(almost)g(all)f(quasi-p)s(erio)s(dic)i(motions)f(ha)m(v)m(e)h Fr(d)e Fy(frequencies.)255 449 y(F)-8 b(or)44 b Fr(m)527 464 y Fn(\033)622 449 y Fq(\025)k Fy(3)d(the)f(factor)g Fr(O)s Fy(\()p Fr(m)1512 464 y Fn(\033)1559 449 y Fy(\))g(is)h (non-comm)m(utativ)m(e)h(and)f(enforces)h(sup)s(erin)m(tegrabilit)m(y) -8 b(.)109 569 y(Reducing)36 b(\(14\))c(leads)j(to)e(the)h(Roso)s(c)m (hatius)h(system)g(on)f Fr(S)2355 533 y Fn(`)2421 569 y Fy(and)g(action)f(angle)h(v)-5 b(ariables)35 b(\()p Fr(u;)17 b(v)t Fy(\))28 b Fq(2)109 690 y Fo(T)175 654 y Fn(`)219 690 y Fq(\002)10 b Fo(R)378 654 y Fn(`)438 690 y Fy(of)27 b(this)g(in)m(tegrable)h(system)h(serv)m(e)g(as)e (starting)g(p)s(oin)m(t)g(for)f(the)h(construction)h(of)f(generalized) 109 810 y(action)32 b(angle)h(v)-5 b(ariables.)44 b(The)34 b(additional)e(actions)i(are)e(the)h(total)f(angular)g(momen)m(ta)1202 1145 y Fr(y)1250 1160 y Fn(\033)1380 1145 y Fy(=)1539 974 y Fh( )1663 1020 y Fn(m)1725 1028 y Fd(\033)1643 1050 y Fh(X)1617 1262 y Fn(i;j)t Fi(2)p Fn(I)1772 1270 y Fd(\033)1829 1145 y Fr(L)1895 1104 y Fm(2)1895 1169 y Fn(ij)1956 974 y Fh(!)2045 970 y Fe(1)p 2045 982 31 3 v 2045 1023 a(2)2155 1145 y Fr(;)114 b(\033)32 b Fy(=)27 b(0)p Fr(;)17 b(:)g(:)g(:)f(;)h(`)109 1462 y Fy(with)32 b(conjugate)f(angles)h Fr(x)1117 1477 y Fn(\033)1195 1462 y Fy(measuring)g(the)g(rotation)f(in)g(the)h(plane)g(singled)g (out)f(b)m(y)h Fr(L)3434 1477 y Fn(ij)3495 1462 y Fy(,)g Fr(i;)17 b(j)33 b Fq(2)28 b Fr(I)3841 1477 y Fn(\033)109 1582 y Fy(\(for)h Fr(m)378 1597 y Fn(\033)453 1582 y Fy(=)f(3)h(this)i(is)g(the)g(usual)f(plane)h(p)s(erp)s(endicular)h(to)d (the)i(angular)f(momen)m(tum)h(v)m(ector\).)44 b(The)109 1702 y(symplectic)35 b(lea)m(v)m(es)f(of)f(the)g(bi\014bration)f(are)h (the)g(co-adjoin)m(t)f(orbits)1358 1922 y Fr(G)1435 1937 y Fn(m)1497 1946 y Fe(0)1532 1937 y Fm(2)1621 1922 y Fq(\002)50 b Fr(G)1825 1937 y Fn(m)1887 1946 y Fe(1)1922 1937 y Fm(2)2011 1922 y Fq(\002)h Fr(:)17 b(:)g(:)49 b Fq(\002)h Fr(G)2507 1937 y Fn(m)2569 1949 y Fd(`)2600 1937 y Fm(2)3715 1922 y Fy(\(15\))109 2142 y(of)32 b(pro)s(ducts)h(of)f (Gra\031mannians)1173 2362 y Fr(G)1250 2377 y Fn(m)1312 2385 y Fd(\033)1354 2377 y Fm(2)1477 2362 y Fy(=)83 b Fr(O)s Fy(\()p Fr(m)1837 2377 y Fn(\033)1883 2362 y Fy(\))1921 2403 y Fr(=)1970 2443 y(S)6 b(O)s Fy(\(2\))21 b Fq(\002)h Fr(O)s Fy(\()p Fr(m)2560 2458 y Fn(\033)2629 2443 y Fq(\000)g Fy(2\))109 2651 y(parametrising)35 b(the)f(orien)m(ted)h(planes)h(in)e Fo(R)1764 2615 y Fn(m)1826 2623 y Fd(\033)1872 2651 y Fy(.)48 b(An)m(y)36 b(c)m(hoice)f(of)f(canonical)h(co-ordinates)f(\()p Fr(w)3649 2666 y Fn(k)3691 2651 y Fr(;)17 b(z)3780 2666 y Fn(k)3823 2651 y Fy(\),)109 2771 y Fr(k)42 b Fy(=)e(1)p Fr(;)17 b(:)g(:)g(:)e(;)585 2696 y Fh(P)690 2771 y Fy(\()p Fr(m)813 2786 y Fn(\033)887 2771 y Fq(\000)27 b Fy(2\))39 b(de\014nes)j(generalized)f(action)e(angle)h(v)-5 b(ariables)40 b(\()p Fr(u;)17 b(x;)g(v)t(;)g(y)t(;)g(w)s(;)g(z)t Fy(\))37 b(for)i(the)109 2892 y(Neumann)32 b(system.)45 b(Note)32 b(that)f(the)h(torus)g(dimension)h(is)f Fr(n)c Fy(=)g(2)p Fr(`)19 b Fy(+)h(1)31 b(|)g(ev)m(en)j(dimensions)f Fr(n)f Fy(can)109 3012 y(b)s(e)37 b(ac)m(hiev)m(ed)i(if)e(also)g(allo)m(wing)h (for)f Fr(m)1563 3027 y Fn(\033)1645 3012 y Fy(=)e(1,)j Fl(i.e.)f Fy(eigen)m(v)-5 b(alues)39 b(that)e(are)g(distinct)h(from)f (all)g(other)109 3132 y(eigen)m(v)-5 b(alues)34 b([17].)255 3253 y(Av)m(eraging)49 b(a)f(p)s(erturbation)h(of)f(the)h(Neumann)h (system)g(along)e(the)h(quasi-p)s(erio)s(dic)g Fr(n)p Fy({tori)109 3373 y(in)m(tro)s(duces)28 b(a)f Fo(T)717 3337 y Fn(n)765 3373 y Fy({symmetry)-8 b(,)30 b(the)e(reduction)g(of)f (whic)m(h)i(leads)f(to)f(a)g(Hamiltonian)g(system)i(on)f(\(15\))o(,)109 3511 y(in)44 b Fr(d)30 b Fq(\000)h Fr(n)49 b Fy(=)654 3416 y Fh(X)797 3511 y Fy(\()p Fr(m)920 3526 y Fn(\033)998 3511 y Fq(\000)31 b Fy(2\))44 b(degrees)i(of)e(freedom.)80 b(In)45 b(the)g(minimal)h(case)f Fr(m)3168 3526 y Fn(\033)3263 3511 y Fy(=)j(3)d(\(all)f(other)109 3653 y Fr(m)194 3668 y Fn(\034)265 3653 y Fy(=)27 b(2\))e(the)h(pro)s(duct)f(\(15\))f(has)i (a)f(single)h(non-trivial)f(factor)g Fr(G)2523 3668 y Fn(m)2585 3676 y Fd(\033)2627 3668 y Fm(2)2694 3625 y Fq(\030)2694 3657 y Fy(=)2799 3653 y Fr(S)2865 3617 y Fm(2)2859 3678 y Fn(y)2894 3686 y Fd(\033)2939 3653 y Fy(,)i(the)f(sphere)g(of)f(p)s(ossible)109 3773 y(directions)37 b(of)e(the)h(angular)f(momen)m(tum)i(v)m(ector.)53 b(In)36 b(the)g(maximal)g(case)h Fr(`)32 b Fy(=)h(0,)j Fr(m)3337 3788 y Fm(0)3409 3773 y Fy(=)d Fr(d)24 b Fy(+)g(1)35 b(w)m(e)109 3894 y(ha)m(v)m(e)f(a)e(single)h(Gra\031mannian)f(as)h(w)m (ell.)109 4226 y Fs(6)161 b(Conclusions)109 4445 y Fp(kam)37 b Fy(theory)g(is)g(ab)s(out)f(in)m(tegrable)i(systems)h(|)d(under)h (strong)g(non-resonance)g(conditions)h(their)109 4566 y(quasi-p)s(erio)s(dic)e(solutions)g(p)s(ersist)h(under)f(su\016cien)m (tly)j(small)d(p)s(erturbations.)53 b(This)36 b(do)s(es)g(not)f(di-)109 4686 y(rectly)48 b(apply)f(to)f(sup)s(erin)m(tegrable)j(Hamiltonian)e (systems.)88 b(Here)48 b(it)e(is)i(the)f(rami\014ed)g Fr(d)p Fy({torus)109 4807 y(bundle)32 b(of)e(an)g(in)m(tegrable)i(in)m (termediate)h(system)f(that)f(is)g(Can)m(torised)h(b)m(y)g(the)f(p)s (erturbation,)h(with)109 4927 y(information)g(not)g(only)i(from)e(the)h (unp)s(erturb)s(ed)g(system)i(but)e(also)g(from)f(the)h(p)s (erturbation)g(itself.)255 5047 y(In)j(the)f(minimally)i(sup)s(erin)m (tegrable)g(case)f(of)f Fr(d)23 b Fy(+)h(1)35 b(conserv)m(ed)i(quan)m (tities)g(the)f(p)s(erturbation)109 5168 y(analysis)23 b(consists)i(of)d(three)h(steps:)40 b(compute)23 b(a)f(normal)h(form)f (that)g(serv)m(es)j(as)e(in)m(termediate)h(system,)109 5288 y(study)k(ho)m(w)g(the)g(dynamics)h(of)e(the)h(in)m(termediate)h (system)h(structures)f(the)f(phase)g(space)h(\(organized)109 5409 y(b)m(y)36 b(the)g(fast)f(\()p Fr(d)24 b Fq(\000)h Fy(1\){tori,)35 b(the)h(relativ)m(e)h(equilibria\))g(and)e(c)m(hec)m(k) j(the)e(necessary)i(non-degeneracy)1950 5778 y(15)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 109 208 a Fy(conditions)30 b(concerning)h(b)s (oth)f(the)g(sup)s(erin)m(tegrable)i(system)f(and)f(the)g(reduced)i(in) m(termediate)f(sys-)109 329 y(tem.)255 449 y(Where)38 b(an)f(in)m(tegrable)h(in)m(termediate)h(system)g(can)e(b)s(e)g(found)h (in)f(cases)h(with)g Fr(n)d Fq(\024)h Fr(d)25 b Fq(\000)h Fy(2)36 b(this)109 569 y(three-step-programme)29 b(still)f(applies.)43 b(Ho)m(w)m(ev)m(er,)31 b(while)e(this)g(is)f(automatic)g(in)g(the)g (case)h Fr(n)f Fy(=)f Fr(d)12 b Fq(\000)g Fy(1)109 690 y(of)41 b(minimal)h(sup)s(erin)m(tegrabilit)m(y)-8 b(,)47 b(additional)41 b(prop)s(erties)i(are)e(needed)j(when)e Fr(n)i Fq(\024)f Fr(d)28 b Fq(\000)h Fy(2)42 b(|)f(for)109 810 y(instance)32 b(symmetries)j(of)30 b(the)i(\(p)s(erturb)s(ed\))g (system)i(or)d Fr(d)19 b Fq(\000)i Fr(n)31 b Fy(time)h(scales)h(within) g(the)e(p)s(erturb)s(ed)109 931 y(dynamics)i(\(apart)e(from)g(the)i(m)m (ulti-p)s(erio)s(dic)f(fast)f(angles)i(and)e(the)h(exp)s(onen)m(tially) i(slo)m(w)f(v)-5 b(ariation)109 1051 y(of)44 b(the)h(actions\).)81 b(In)45 b(principle,)50 b(the)45 b(p)s(erturbation)g(problem)h(ma)m(y)g (lead)f(to)f(an)m(y)i(Hamiltonian)109 1171 y(system)34 b(in)f Fr(d)22 b Fq(\000)g Fr(n)33 b Fy(degrees)h(of)e(freedom.)255 1292 y(The)f(resonan)m(t)g(lattices)h(studied)f(in)g([42)o(])g(are)f (maximally)i(sup)s(erin)m(tegrable;)h(reducing)e(the)g(nor-)109 1412 y(mal)38 b(form)g(with)h(resp)s(ect)h(to)e(the)h(p)s(erio)s(dic)g (\015o)m(w)g(leads)g(to)f(a)g(quotien)m(t)i Fr(S)2915 1376 y Fm(2)p Fn(d)p Fi(\000)p Fm(1)3080 1412 y Fr(=S)3195 1376 y Fm(1)3273 1412 y Fy(where)f(the)g Fr(S)3800 1376 y Fm(1)3839 1412 y Fy({)109 1532 y(action)32 b(is)h(free)f(only)h(for)f (the)g(case)h(1:1:1:.)16 b(.)g(.)g(:1.)44 b(In)33 b(the)g(other)f (cases)i(the)e Fr(S)2947 1496 y Fm(1)2986 1532 y Fy({action)g(is)h(lo)s (cally)g(free)109 1653 y(and)38 b(the)h(\(discrete\))h(isotrop)m(y)f (groups)g(lead)g(to)f(singularities)i(of)e(the)h(reduced)h(phase)f (space:)57 b(the)109 1773 y(Hamiltonian)34 b(system)j(in)e Fr(d)23 b Fq(\000)h Fy(1)34 b(degrees)i(of)e(freedom)h(is)g(not)f (de\014ned)i(on)f(a)f(symplectic)j(manifold)109 1894 y(but)29 b(on)g(a)g(P)m(oisson)h(space.)43 b(F)-8 b(or)29 b(inde\014nite)h(lattices)g(the)g(reduced)g(phase)g(space)g(is)g (furthermore)f(not)109 2014 y(compact.)255 2134 y(Ev)m(ery)d(resonance) f(\(9\))e(of)g(an)h(elliptic)g(equilibrium)i(increases)g(the)e (co-dimension)h(of)e(that)g(Hamil-)109 2255 y(tonian)34 b(system)i(within)f(the)g(univ)m(erse)i(of)c(all)i(Hamiltonian)f (systems.)51 b(F)-8 b(or)34 b(instance,)i(t)m(w)m(o)f(param-)109 2375 y(eters)h(are)f(needed)h(to)f(generically)h(encoun)m(ter)h(an)e (equilibrium)i(with)e(quadratic)h(part)f(\(13\))f(of)h(the)109 2496 y(Hamiltonian;)i(b)s(oth)e(parameters)h(detune)g(the)g(frequency)h (ratio.)51 b(Ho)m(w)m(ev)m(er,)38 b(if)d(these)i(are)e(normal)109 2616 y(resonances)41 b(of)f(in)m(v)-5 b(arian)m(t)41 b(tori,)g(then)g(the)f(complete)h(unfolding)g(ma)m(y)f(b)s(e)g(encoun)m (tered)j(within)d(a)109 2736 y(single)33 b(\(generic\))h(Hamiltonian)f (system.)255 2857 y(In)g(this)g(w)m(a)m(y)h(the)f(3{parameter)g (families)g(of)f(in)m(v)-5 b(arian)m(t)33 b(3{tori)f(in)h(six)h (degrees)g(of)e(freedom)h(can)109 2977 y(ha)m(v)m(e)43 b(1{parameter)e(subfamilies)j(with)e(normal)f(b)s(eha)m(viour)i(\(13\)) o(.)71 b(A)41 b(successful)k(application)d(of)109 3097 y Fp(kam)28 b Fy(theory)g(w)m(ould)g(ev)m(en)m(tually)i(lead)d(to)g (1{parameter)h(Can)m(tor)f(families)h(of)f(suc)m(h)i(3{tori,)f (organiz-)109 3218 y(ing)33 b(the)i(Can)m(torised)g(rami\014ed)f(torus) h(bundle)f(of)g(the)g(unfolding)g([29].)47 b(Clearly)-8 b(,)35 b(the)f(dev)m(elopmen)m(t)109 3338 y(of)29 b(suc)m(h)h(a)g (theory)g(has)f(to)g(start)h(with)g(elliptic)g(equilibria)h(of)e (Hamiltonian)h(systems)i(dep)s(ending)e(on)109 3459 y(external)j (parameters.)255 3579 y(P)m(erturbing)28 b(resonan)m(t)g Fr(d)p Fy({tori)e(of)h(a)g(\(non-degenerate\))g(in)m(tegrable)h(system) h(resem)m(bles)h(p)s(erturb-)109 3699 y(ing)h(a)g(sup)s(erin)m (tegrable)i(system.)45 b(Letting)31 b Fr(d)19 b Fq(\000)h Fr(n)31 b Fy(denote)h(the)g(n)m(um)m(b)s(er)g(of)f(indep)s(enden)m(t)i (\(in)m(ternal\))109 3820 y(resonances,)i(normalizing)g(with)f(resp)s (ect)h(to)e(the)h(quasi-p)s(erio)s(dic)g(\015o)m(w)g(with)h Fr(n)e Fy(frequencies)j(yields)109 3940 y(a)43 b(Hamiltonian)h(that)g (admits)g(a)g(lo)s(cal)f Fo(T)1727 3904 y Fn(n)1775 3940 y Fy({action)g(and)h(can)g(b)s(e)g(reduced)h(to)f Fr(d)29 b Fq(\000)i Fr(n)43 b Fy(degrees)j(of)109 4061 y(freedom)34 b(with)g(a)g(cylinder)h Fo(T)1225 4024 y Fn(d)p Fi(\000)p Fn(n)1386 4061 y Fq(\002)24 b Fo(R)1559 4024 y Fn(d)p Fi(\000)p Fn(n)1730 4061 y Fy(as)34 b(phase)h(space.)48 b(F)-8 b(or)33 b Fr(n)c Fy(=)h Fr(d)22 b Fq(\000)i Fy(1)33 b(the)h(resulting)h(family)109 4181 y(of)29 b(in)m(tegrable)j(in)m (termediate)g(systems)g(can)f(in)f(particular)h(b)s(e)f(used)i(to)e (iden)m(tify)h(those)g(\()p Fr(d)17 b Fq(\000)h Fy(1\){tori)109 4301 y(foliating)32 b(the)h(resonan)m(t)g Fr(d)p Fy({tori)f(that)g(p)s (ersist)i(the)f(p)s(erturbation)g([8].)255 4422 y(The)43 b(Can)m(tor)g(families)g(of)f(suc)m(h)i(\()p Fr(d)29 b Fq(\000)g Fy(1\){tori)42 b(organize)g(the)h(phase)h(space)f(and)g(p)s (ersisten)m(tly)109 4542 y(undergo)23 b(bifurcations)g(of)g (co-dimension)h(up)f(to)f Fr(d)r Fq(\000)r Fy(2.)41 b(Similar)23 b(approac)m(hes)h([11)o(,)f(13])g(with)g Fr(n)28 b Fq(\024)g Fr(d)r Fq(\000)r Fy(2)109 4662 y(restrict)c(to)e(elliptic)j(and)e(h)m (yp)s(erb)s(olic)h(tori,)h(the)e(minimal)h(n)m(um)m(b)s(er)g(of)f(whic) m(h)i(is)e(exp)s(ected)i(to)e(coincide)109 4783 y(with)36 b(the)h(minimal)g(n)m(um)m(b)s(er)g Fr(n)25 b Fy(+)f(1)36 b(of)g(critical)g(p)s(oin)m(ts)h(of)f(a)g(smo)s(oth)g(function)g(on)g Fo(T)3393 4747 y Fn(n)3441 4783 y Fy(.)54 b(A)36 b(b)s(etter)109 4903 y(understanding)d(of)f(p)s(erturb)s(ed)h(sup)s(erin)m(tegrable)h (systems)g(w)m(ould)f(en)m(tail)g(a)f(b)s(etter)h(understanding)109 5024 y(of)f(the)h(p)s(erturb)s(ed)g(dynamics)h(near)f(resonan)m(t)h (tori.)109 5244 y Fk(Ac)m(kno)m(wledgemen)m(ts.)64 b Fy(I)39 b(thank)h(Henk)g(Bro)s(er,)h(An)m(tonella)f(Marc)m(hesiello,)j (Giusepp)s(e)d(Pucacco)109 5364 y(and)30 b(F)-8 b(erdinand)30 b(V)-8 b(erh)m(ulst)32 b(for)d(helpful)i(discussions)i(and)d(the)h (referees)g(for)f(their)g(guiding)h(remarks.)1950 5778 y(16)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 109 208 a Fs(References)154 411 y Fv([1])50 b Fc(E.)42 b(v)-5 b(an)43 b(der)f(Aa)h(and)f(F.)h(V)-8 b(erh)m(ulst)43 b(:)108 b Fv(Asymptotic)43 b(in)m(tegrabilit)m(y)i(and) d(p)s(erio)s(dic)g(solutions)h(of)g(a)299 524 y(Hamiltonian)32 b(system)e(in)g(1:2:2-resonance)j(;)e Fb(SIAM)h(J.)g(Math.)h(A)n(nal.)d Fw(15)p Fv(\(5\),)i(p.)e(890)i({)f(911)g(\(1984\))154 712 y([2])50 b Fc(V.I.)29 b(Arnol'd)f(:)67 b Fv(Pro)s(of)28 b(of)h(a)f(theorem)h(of)f(A.)h(N.)g(Kolmogoro)m(v)h(on)e(the)h(in)m(v) -5 b(ariance)29 b(of)f(quasi-p)s(erio)s(dic)299 825 y(motions)g(under)f (small)h(p)s(erturbations)f(of)i(the)f(Hamiltonian)h(;)g Fb(R)n(uss.)i(Math.)f(Surv.)e Fw(18)p Fv(\(5\),)j(p.)c(9)i({)f(36)299 938 y(\(1963\))154 1125 y([3])50 b Fc(V.I.)39 b(Arnol'd)f(:)96 b Fv(Small)39 b(denominators)f(and)g(problems)g(of)h(stabilit)m(y)i(of) d(motion)i(in)e(classical)j(and)299 1238 y(celestial)33 b(mec)m(hanics)e(;)g Fb(R)n(uss.)h(Math.)h(Surv.)d Fw(18)p Fv(\(6\),)j(p.)d(85)h({)g(191)g(\(1963\))154 1426 y([4])50 b Fc(V.I.)39 b(Arnol'd,)i(V.V.)e(Kozlo)m(v)i(and)d(A.I.)h(Neish)m(tadt) h(:)96 b Fv(Mathematical)42 b(Asp)s(ects)d(of)g(Classical)h(and)299 1539 y(Celestial)32 b(Mec)m(hanics)g(;)e(in)h Fb(Dynamic)-5 b(al)34 b(Systems)f(III)d Fv(\(ed.)h(V.I.)g(Arnol'd\))g(Springer)e (\(1988\))154 1726 y([5])50 b Fc(J.)26 b(Binney)h(:)65 b Fv(Resonan)m(t)27 b(excitation)i(of)e(motion)g(p)s(erp)s(endicular)d (to)k(galactic)h(planes)e(;)h Fb(MNRAS)d Fw(196)p Fv(,)299 1839 y(p.)30 b(455)i({)e(467)i(\(1981\))154 2027 y([6])50 b Fc(J.)30 b(Binney)g(and)g(S.)g(T)-8 b(remaine)31 b(:)71 b Fb(Galactic)34 b(Dynamics)d Fv(;)f(Princeton)h(Univ)m(ersit)m(y)g (Press)f(\(2008\))154 2214 y([7])50 b Fc(H.W.)34 b(Bro)s(er,)f(H.)g (Han\031mann)g(and)f(F.O.O)h(W)-8 b(agener)34 b(:)78 b Fb(Quasi-p)-5 b(erio)g(dic)36 b(bifur)-5 b(c)g(ation)36 b(the)-5 b(ory)37 b(:)46 b(the)299 2327 y(ge)-5 b(ometry)34 b(of)f(KAM)c Fv(\(in)h(preparation\))154 2515 y([8])50 b Fc(H.W.)33 b(Bro)s(er,)g(H.)g(Han\031mann)f(and)g(J.)g(Y)-8 b(ou)33 b(:)77 b Fv(On)32 b(the)g(destruction)h(of)f(resonan)m(t)h (Lagrangean)h(tori)299 2628 y(in)e(Hamiltonian)j(Systems)d(;)j(p.)d (317)j({)e(333)h(in)f Fb(R)-5 b(e)g(c)g(ent)36 b(T)-7 b(r)i(ends)36 b(in)f(Dynamic)-5 b(al)36 b(Systems,)h(M)q(\177)-48 b(unchen)299 2741 y(2012)32 b Fv(\(eds.)f(A.)f(Johann)g(et)h(al.\))h (Pro)s(ceedings)e(in)g(Mathematics)j(&)d(Statistics)i Fw(35)p Fv(,)f(Springer)e(\(2013\))154 2928 y([9])50 b Fc(H.W.)39 b(Bro)s(er,)g(G.B.)g(Huitema)g(and)e(M.B.)h(Sevryuk)f(:)93 b Fb(Quasi-Perio)-5 b(dic)39 b(Motions)h(in)f(F)-7 b(amilies)41 b(of)299 3041 y(Dynamic)-5 b(al)34 b(Systems)f(:)42 b(Or)-5 b(der)34 b(amidst)g(Chaos)d Fv(;)g(LNM)g Fw(1645)p Fv(,)g(Springer)e (\(1996\))109 3229 y([10])50 b Fc(H.W.)28 b(Bro)s(er,)g(G.B.)h(Huitema) f(and)f(F.)g(T)-8 b(ak)m(ens)28 b(:)67 b Fv(Unfoldings)27 b(of)g(quasi-p)s(erio)s(dic)g(tori)h(;)h Fb(Mem.)g(AMS)299 3342 y Fw(83)i Fv(#421,)g(p.)f(1)h({)g(82)g(\(1990\))109 3530 y([11])50 b Fc(C.-Q.)35 b(Cheng)g(and)g(S.)g(W)-8 b(ang)37 b(:)86 b Fv(The)35 b(Surviving)f(of)i(Lo)m(w)m(er)g (Dimensional)h(T)-8 b(ori)35 b(from)g(a)h(Resonan)m(t)299 3642 y(T)-8 b(orus)30 b(of)g(Hamiltonian)i(Systems)e(;)g Fb(J.)j(Di\013.)e(Eq.)f Fw(155)p Fv(\(2\),)j(p.)d(311)h({)g(326)h (\(1999\))109 3830 y([12])50 b Fc(O.)33 b(Christo)m(v)g(:)79 b Fv(Non-in)m(tegrabilit)m(y)36 b(of)e(\014rst)e(order)h(resonances)g (of)g(Hamiltonian)i(systems)e(in)g(three)299 3943 y(degrees)e(of)f (freedom)h(;)f Fb(Cel.)j(Me)-5 b(ch.)32 b(&)g(Dyn.)g(Astr.)f Fw(112)p Fv(,)g(p.)f(147)i({)f(167)g(\(2012\))109 4131 y([13])50 b Fc(F.)35 b(Cong,)h(T.)e(K)s(\177)-48 b(upp)s(er,)34 b(Y.)h(Li)g(and)f(J.)g(Y)-8 b(ou)35 b(:)84 b Fv(KAM-T)m(yp)s(e)35 b(Theorem)f(on)h(Resonan)m(t)g(Surfaces)f(for)299 4244 y(Nearly)d(In)m(tegrable)h(Hamiltonian)f(Systems)f(;)h Fb(J.)h(Nonline)-5 b(ar)34 b(Sci.)c Fw(10)p Fv(,)h(p.)f(49)h({)g(68)g (\(2000\))109 4431 y([14])50 b Fc(R.)41 b(Cushman)e(:)102 b Fv(A)41 b(Surv)m(ey)f(of)h(Normalization)i(T)-8 b(ec)m(hniques)41 b(Applied)f(to)i(P)m(erturb)s(ed)d(Keplerian)299 4544 y(Systems)30 b(;)g Fb(Dynamics)k(R)-5 b(ep.,)33 b(new)g(ser.)d Fw(1)p Fv(,)h(p.)f(54)h({)g(112)h(\(1992\))109 4732 y([15])50 b Fc(R.)36 b(Cushman)e(and)i(D.A.)h(Sado)m(vski)-10 b(\023)-35 b(\020)37 b(:)88 b Fv(Mono)s(drom)m(y)36 b(in)f(the)i(h)m(ydrogen)f (atom)h(in)e(crossed)h(\014elds)g(;)299 4845 y Fb(Physic)-5 b(a)33 b(D)d Fw(142)h Fv(p.)f(166)i({)f(196)h(\(1998\))109 5032 y([16])50 b Fc(P)-8 b(.)26 b(Dazord)g(and)f(T.)g(Delzan)m(t)i(:)64 b Fv(Le)25 b(probl)m(\022)-43 b(eme)26 b(g)m(\023)-43 b(en)m(\023)g(eral)29 b(des)c(v)-5 b(ariables)26 b(actions-angles)i(;)f Fb(J.)g(Di\013.)g(Ge)-5 b(o.)299 5145 y Fw(26)p Fv(,)31 b(p.)f(223)i({)e(251)i(\(1987\))1950 5778 y Fy(17)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 109 208 a Fv([17])50 b Fc(H.A.)22 b(Dullin)f(and)f(H.)i(Han\031mann)e(:)57 b Fv(The)21 b(degenerate)h(C.Neumann)e(system)i(I)e(:)i(symmetry)e(reduction)299 321 y(and)k(separation)h(;)i(p.)d(1627)i({)f(1654)h(in)e Fb(Finite)j(dimensional)j(inte)-5 b(gr)g(able)28 b(systems,)h (dynamics,)g(and)f(Lie)299 434 y(the)-5 b(or)g(etic)31 b(metho)-5 b(ds)33 b(in)c(Ge)-5 b(ometry)32 b(and)f(Mathematic)-5 b(al)32 b(Physics,)f(Jena)f(2011)f Fv(\(eds.)f(Andreas)3634 411 y(\024)3623 434 y(Cap)f(et)299 547 y(al.\))k Fb(Cent.)i(Eur.)f(J.)g (Math.)f Fw(10)g Fv(\(2012\))109 735 y([18])50 b Fc(H.S.)30 b(Dumas)g(:)71 b Fb(The)32 b Fa(kam)g Fb(Story:)43 b(A)31 b(friend)5 b(ly)33 b(intr)-5 b(o)g(duction)35 b(to)e(the)g(c)-5 b(ontent,)33 b(history,)h(and)f(signif-)299 848 y(ic)-5 b(anc)g(e)33 b(of)g(classic)-5 b(al)34 b(Kolmo)-5 b(gor)g(ov{A)n (rnold{Moser)38 b(the)-5 b(ory)32 b Fv(;)e(W)-8 b(orld)31 b(Scien)m(ti\014c)h(\(2014\))109 1035 y([19])50 b Fc(J.J.)33 b(Duistermaat)i(:)79 b Fv(Bifurcations)34 b(of)g(p)s(erio)s(dic)e (solutions)i(near)f(equilibrium)g(p)s(oin)m(ts)g(of)g(Hamilto-)299 1148 y(nian)g(systems)g(;)i(p.)e(57)h({)g(105)g(in)f Fb(Bifur)-5 b(c)g(ation)37 b(The)-5 b(ory)37 b(and)f(Applic)-5 b(ations,)38 b(Monte)-5 b(c)g(atini)37 b(1983)e Fv(\(ed.)299 1261 y(L.)30 b(Salv)-5 b(adori\))31 b(LNM)g Fw(1057)p Fv(,)h(Springer)d(\(1984\))109 1449 y([20])50 b Fc(J.J.)27 b(Duistermaat)j(:)67 b Fv(Non-in)m(tegrabilit)m(y)30 b(of)e(the)g(1:1:2-resonance)j(;)d Fb(Er)-5 b(go)g(d.)32 b(Th.)e(&)g(Dynam.)h(Sys.)d Fw(4)p Fv(,)299 1562 y(p.)i(553)i({)e(568)i (\(1984\))109 1749 y([21])50 b Fc(K.)38 b(Efstathiou)h(:)95 b Fb(Metamorphoses)43 b(of)d(Hamiltonian)i(systems)f(with)g(symmetries) e Fv(;)k(LNM)c Fw(1864)p Fv(,)299 1862 y(Springer)29 b(\(2005\))109 2050 y([22])50 b Fc(F.)27 b(F)-8 b(ass\022)-45 b(o)28 b(:)66 b Fv(The)26 b(Euler-P)m(oinsot)i(top)f(:)39 b(A)27 b(non-comm)m(utativ)m(ely)j(in)m(tegrable)e(system)f(without)g (global)299 2163 y(action-angle)33 b(co)s(ordinates)e(;)g Fb(Z.)h(angew.)h(Math.)g(Phys.)d Fw(47)p Fv(,)i(p.)e(953)h({)g(976)h (\(1996\))109 2350 y([23])50 b Fc(F.)36 b(F)-8 b(ass\022)-45 b(o)37 b(:)87 b Fv(Sup)s(erin)m(tegrable)36 b(Hamiltonian)h(systems)f (:)51 b(Geometry)37 b(and)e(P)m(erturbation)h(;)j(p.)c(93)i({)299 2463 y(121)d(in)f Fb(Symmetry)j(and)g(Perturb)-5 b(ation)36 b(The)-5 b(ory,)37 b(Cala)f(Gonone)g(2004)f Fv(\(ed.)f(G.)f(Gaeta\))i Fb(A)-5 b(cta)36 b(Appl.)299 2576 y(Math.)31 b Fw(87)g Fv(\(2005\))109 2764 y([24])50 b Fc(O.)38 b(Gerhard)f(and)h(J.)g (Binney)g(:)94 b Fv(T)-8 b(riaxial)39 b(galaxies)h(con)m(taining)g (massiv)m(e)f(blac)m(k)g(holes)g(or)f(cen)m(tral)299 2877 y(densit)m(y)30 b(cusps)g(;)g Fb(MNRAS)g Fw(216)p Fv(,)h(p.)f(467)i({)f(502)g(\(1985\))109 3064 y([25])50 b Fc(G.)31 b(Haller)g(:)71 b Fb(Chaos)34 b(ne)-5 b(ar)34 b(R)-5 b(esonanc)g(e)32 b Fv(;)f(AMS)f Fw(138)p Fv(,)h(Springer)e (\(1999\))109 3252 y([26])50 b Fc(Y.)28 b(Han,)h(Y.)g(Li)f(and)f(Y.)i (Yi)f(:)68 b Fv(In)m(v)-5 b(arian)m(t)29 b(T)-8 b(ori)28 b(in)g(Hamiltonian)h(Systems)f(with)g(High)g(Order)f(Prop)s(er)299 3365 y(Degeneracy)32 b(;)f Fb(A)n(nn.)h(Inst.)h(H.)f(Poinc)-5 b(ar)n(\023)-44 b(e)31 b Fw(10)p Fv(,)g(p.)f(1419{1436)k(\(2010\))109 3552 y([27])50 b Fc(H.)32 b(Han\031mann)f(:)74 b Fv(Quasi-p)s(erio)s (dic)31 b(Motions)i(of)e(a)h(Rigid)g(Bo)s(dy)f(I)h(|)f(Quadratic)h (Hamiltonians)g(on)299 3665 y(the)e(Sphere)g(with)g(a)g(Distinguished)h (P)m(arameter)h(;)e Fb(R)-5 b(e)g(g.)33 b(&)f(Chaot.)j(Dyn.)30 b Fw(2)p Fv(\(2\),)i(p.)e(41)h({)g(57)g(\(1997\))109 3853 y([28])50 b Fc(H.)37 b(Han\031mann)g(:)92 b Fv(The)37 b(Quasi-P)m(erio)s(dic)h(Cen)m(tre-Saddle)f(Bifurcation)h(;)j Fb(J.)e(Di\013.)f(Eq.)e Fw(142)p Fv(\(2\),)41 b(p.)299 3966 y(305)32 b({)e(370)i(\(1998\))109 4153 y([29])50 b Fc(H.)29 b(Han\031mann)f(:)68 b Fb(L)-5 b(o)g(c)g(al)33 b(and)f(Semi{L)-5 b(o)g(c)g(al)34 b(Bifur)-5 b(c)g(ations)33 b(in)d(Hamiltonian)j(Dynamic)-5 b(al)32 b(Systems)g(|)299 4266 y(R)-5 b(esults)33 b(and)h(Examples)e Fv(;)e(LNM)h Fw(1893)p Fv(,)g(Springer)f(\(2007\))109 4454 y([30])50 b Fc(H.)28 b(Han\031mann)g(:)67 b Fv(Non-degeneracy)29 b(conditions)g(in)e Fa(kam)g Fv(theory)i(;)f Fb(Indag.)j(Math.)e Fw(22)p Fv(,)g(p.)e(241)j({)e(256)299 4567 y(\(2011\))109 4755 y([31])50 b Fc(M.)34 b(Kummer)d(:)80 b Fv(On)32 b(resonan)m(t)i(Hamiltonian)h(systems)e(with)g(\014nitely)g(man)m(y)h (degrees)f(of)h(freedom)f(;)299 4867 y(p.)41 b(19)h({)f(31)h(in)f Fb(L)-5 b(o)g(c)g(al)44 b(and)g(glob)-5 b(al)43 b(metho)-5 b(ds)45 b(of)e(nonline)-5 b(ar)44 b(dynamics,)i(Silver)d(Spring)g(1984) g Fv(\(eds.)299 4980 y(A.V.)31 b(S\023)-45 b(aenz,)31 b(W.W.)h(Zac)m(hary)e(and)g(R.)h(Ca)m(wley\))g(Lect.)h(Notes)f(Ph)m (ys.)f Fw(252)p Fv(,)i(Springer)d(\(1986\))109 5168 y([32])50 b Fc(L.)22 b(Markus)g(and)g(K.R.)g(Mey)m(er)h(:)59 b Fv(Generic)23 b(Hamiltonian)h(dynamical)e(systems)h(are)f(neither)h(in) m(tegrable)299 5281 y(nor)30 b(ergo)s(dic)h(;)f Fb(Mem.)i(AMS)e Fv(#144,)h(p.)g(1)f({)h(52)g(\(1974\))1950 5778 y Fy(18)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 109 208 a Fv([33])50 b Fc(M.)32 b(Mazzo)s(cco)j(:)76 b Fv(KAM)32 b(theorem)g(for)g(generic)h(analytic)h (p)s(erturbations)d(of)h(the)g(Euler)f(system)i(;)g Fb(Z.)299 321 y(angew.)g(Math.)g(Phys.)e Fw(48)p Fv(,)g(p.)f(193)h({)g(219)h (\(1997\))109 509 y([34])50 b Fc(A.S.)44 b(Mishc)m(henk)m(o)i(and)e (A.T.)h(F)-8 b(omenk)m(o)46 b(:)113 b Fv(Generalized)46 b(Liouville)g(metho)s(d)e(of)g(in)m(tegration)j(of)299 622 y(Hamiltonian)32 b(systems)e(;)h Fb(F)-7 b(unct.)32 b(A)n(nal.)h(Appl.)e Fw(12)p Fv(\(2\),)h(p.)e(113)i({)e(121)i(\(1978\)) 109 809 y([35])50 b Fc(J.)29 b(Moser)h(:)70 b Fv(Con)m(v)m(ergen)m(t)31 b(Series)f(Expansion)f(for)g(Quasi-P)m(erio)s(dic)h(Motions)h(;)f Fb(Math.)i(A)n(nn.)d Fw(169)p Fv(\(1\),)299 922 y(p.)h(136)i({)e(176)i (\(1967\))109 1110 y([36])50 b Fc(N.N.)29 b(Nekhoroshev)f(:)67 b Fv(Action-angle)31 b(v)-5 b(ariables)29 b(and)e(their)h (generalizations)j(;)e Fb(T)-7 b(r)i(ans.)31 b(Mosc.)g(Math.)299 1223 y(So)-5 b(c.)31 b Fw(26)p Fv(,)g(p.)f(180)i({)e(198)i(\(1972\))109 1410 y([37])50 b Fc(Y.)39 b(P)m(apaphilipp)s(ou)f(and)h(J.)g(Lask)-5 b(ar)39 b(:)98 b Fv(Global)40 b(dynamics)f(of)g(triaxial)i(galactic)h (mo)s(dels)d(through)299 1523 y(F)-8 b(requency)31 b(map)f(analysis)h (;)f Fb(Astr)-5 b(on.)34 b(&)e(Astr)-5 b(oph.)32 b Fw(329)p Fv(,)g(p.)e(451)h({)g(481)h(\(1998\))109 1711 y([38])50 b Fc(M.)37 b(P)m(ersic,)h(P)-8 b(.)37 b(Salucci)g(and)e(F.)i(Stel)g(:) 88 b Fv(The)36 b(Univ)m(ersal)h(Rotation)h(Curv)m(e)e(of)g(Spiral)g (Galaxies:)54 b(I.)299 1824 y(The)30 b(Dark)h(Matter)g(Connection)g(;)g Fb(MNRAS)e Fw(281)p Fv(,)i(p.)f(27)i({)e(47)i(\(1996\))109 2012 y([39])50 b Fc(J.A.)35 b(Sanders,)f(F.)h(V)-8 b(erh)m(ulst)36 b(and)d(J.)i(Murdo)s(c)m(k)f(:)84 b Fb(A)n(ver)-5 b(aging)36 b(Metho)-5 b(ds)38 b(in)e(Nonline)-5 b(ar)38 b(Dynamic)-5 b(al)299 2124 y(Systems)p Fv(,)31 b(2)715 2091 y Fm(nd)829 2124 y Fv(ed.;)f(AMS)h Fw(59)p Fv(,)g(Springer)e(\(2007\))109 2312 y([40])50 b Fc(B.S.)37 b(Sommer)g(:)91 b Fb(A)39 b(KAM)e(The)-5 b(or)g(em)41 b(for)f(the)f(Sp)-5 b(atial)41 b(Lunar)e(Pr)-5 b(oblem)39 b Fv(;)h(Ph.D.)e(thesis,)h(R)-10 b(WTH)299 2425 y(Aac)m(hen)31 b(\(2003\))109 2613 y([41])50 b Fc(J.)32 b(T)-8 b(ouma)33 b(and)f(S.)h(T)-8 b(remaine)33 b(:)78 b Fv(A)33 b(map)f(for)h(eccen)m(tric)h(orbits)f(in)f (non-axisymmetric)i(p)s(oten)m(tials)g(;)299 2726 y Fb(MNRAS)29 b Fw(292)p Fv(,)j(p.)e(905)h({)g(919)h(\(1997\))109 2913 y([42])50 b Fc(F.)27 b(V)-8 b(erh)m(ulst)28 b(:)66 b Fv(In)m(tegrabilit)m(y)29 b(and)d(mo)s(de-in)m(teraction)j(for)e (Hamiltonian)h(normal)f(forms)g(;)h(to)g(app)s(ear)299 3026 y(in)i Fb(A)-5 b(cta)33 b(Appl.)g(Math.)e Fv(\(2015\))109 3214 y([43])50 b Fc(T.)32 b(de)f(Zeeu)m(w)h(:)76 b Fv(Dynamical)33 b(mo)s(dels)e(for)h(axisymmetric)h(and)e(triaxial)j(galaxies)f(;)g(p.)f (271)h({)f(290)h(in)299 3327 y Fb(Structur)-5 b(e)29 b(and)h(Dynamics)f(of)g(El)5 b(liptic)-5 b(al)30 b(Galaxies,)g(IA)n(U)e (Symp.)h(127)f Fv(\(ed.)e(T.)g(de)g(Zeeu)m(w\))h(D.)f(Reidel)299 3440 y(\(1987\))1950 5778 y Fy(19)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF