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1744 a Fl(1)22 b Fj(\000)h(k)p Fe(v)t Fj(k)2203 1715 y Ff(2)2252 1662 y Fe(v)515 1935 y Fl(b)s(e)35 b(the)h(particle)f (momen)m(tum.)51 b(After)35 b(the)h(impact)e(the)i(pro)5 b(jection)35 b(of)h(the)515 2051 y(momen)m(tum)26 b(to)i(the)g(tangen)m (t)g(plane)f(to)h(the)g(w)m(all)e(remains)h(the)h(same:)41 b Fe(p)3232 2015 y Fd(0)3232 2076 y Fh(\034)3303 2051 y Fl(=)515 2168 y Fe(p)564 2183 y Fh(\034)607 2168 y Fl(,)26 b(while)e(the)i(pro)5 b(jection)25 b(to)f(the)i(normal)d(to)h (the)i(w)m(all)d Fe(p)2635 2131 y Fd(0)2635 2192 y Fh(\027)2703 2168 y Fl(after)i(the)g(impact)515 2284 y(equals)967 2530 y Fe(p)1016 2489 y Fd(0)1016 2555 y Fh(\027)1087 2530 y Fl(=)i(\()p Fj(\000)p Fe(p)1354 2545 y Fh(\027)1398 2530 y Fl(\))1447 2463 y(1)21 b(+)h Fe(V)p 1446 2507 250 4 v 1446 2598 a Fl(1)g Fj(\000)g Fe(V)1727 2530 y Fl(+)1916 2463 y(2)p Fe(V)p 1835 2507 289 4 v 1835 2598 a Fl(1)g Fj(\000)g Fe(V)2084 2569 y Ff(2)2133 2419 y Fc(\020)2193 2440 y(p)p 2293 2440 357 4 v 90 x Fj(k)p Fe(p)p Fj(k)2442 2501 y Ff(2)2503 2530 y Fl(+)g(1)g(+)g Fe(p)2819 2545 y Fh(\027)2862 2530 y Fl(\))p Fe(:)327 b Fl(\(1\))515 2773 y(Here)39 b Fj(k)p Fe(p)p Fj(k)f Fl(=)g Fj(k)p Fe(p)1151 2788 y Fh(\034)1220 2773 y Fl(+)26 b Fe(p)1371 2788 y Fh(\027)1414 2773 y Fj(k)39 b Fl(is)f(the)h(length)f (of)g(the)h(momen)m(tum)f(v)m(ector.)63 b(W)-8 b(e)515 2889 y(also)30 b(supp)s(ose)j(that)f Fe(p)1335 2904 y Fh(\027)1405 2889 y Fj(\024)d Fl(0,)i(whic)m(h)h(means)g(that)f(the)h (particle)f(falls)e(to)j(the)515 3005 y(w)m(all.)515 3204 y(The)25 b(energy)g Fe(E)1088 3168 y Fd(0)1136 3204 y Fl(after)f(the)h(impact)e(dep)s(ends)j(b)s(oth)e(on)g(the)h(energy)h Fe(E)k Fl(b)s(efore)515 3321 y(the)48 b(impact)f(and)i(the)f(tangen)m (tial)f(comp)s(onen)m(t)h(of)g(the)h(momen)m(tum)d(\(or)515 3437 y(v)m(elo)s(cit)m(y\):)1068 3688 y Fe(E)1146 3647 y Fd(0)1197 3688 y Fl(=)28 b Fe(E)1389 3621 y Fl(1)22 b(+)g Fe(V)p 1389 3666 250 4 v 1389 3757 a Fl(1)g Fj(\000)g Fe(V)1670 3688 y Fl(+)1858 3621 y(2)p Fe(V)p 1778 3666 289 4 v 1778 3757 a Fl(1)g Fj(\000)g Fe(V)2027 3728 y Ff(2)2076 3578 y Fc(\020)2136 3602 y Fj(p)p 2219 3602 321 4 v 86 x Fe(E)2297 3660 y Ff(2)2358 3688 y Fj(\000)h Fl(\001)f Fj(\000)h Fe(E)2739 3578 y Fc(\021)2799 3688 y Fe(;)428 b Fl(\(2\))515 3949 y(Here)33 b(\001)28 b(=)g Fj(k)p Fe(p)1057 3964 y Fh(\034)1099 3949 y Fj(k)1149 3913 y Ff(2)1211 3949 y Fl(+)22 b(1.)43 b(Note)33 b(that)f Fe(E)i Fj(\025)2086 3864 y(p)p 2169 3864 82 4 v 85 x Fl(\001)f(for)f(all)e(momen)m(ta)i Fe(p)p Fl(.)515 4148 y(The)39 b(relations)e(\(1\),)j(\(2\))e(app)s(eared)h(\014rst)g(in)76 b([8],)j([10)o(])39 b(and)g(w)m(ere)g(pro)m(v)m(ed)515 4264 y(for)32 b(the)h(one-dimensional)d(case.)44 b(Here)33 b(w)m(e)h(giv)m(e)f(a)f(general)g(pro)s(of.)515 4464 y(F)-8 b(ollo)m(wing)57 b([8],)31 b(w)m(e)h(assume)e(\014rst)h(that)f (the)h(mass)g Fe(M)41 b Fl(of)29 b(the)i(w)m(all)e(is)h(\014nite,)515 4580 y(an)j(it)g(mo)m(v)m(es)i(with)e(in)g(the)h(v)m(ertical)f (direction)g(with)g(the)h(v)m(elo)s(cit)m(y)g Fe(V)21 b Fl(.)47 b(The)515 4696 y(momen)m(tum)30 b(of)h(the)h(w)m(all)f (equals)h Fe(M)10 b(P)k Fl(,)32 b(where)h Fe(P)41 b Fl(=)2631 4657 y Fh(V)p 2539 4673 241 4 v 2539 4683 a Fd(p)p 2598 4683 182 3 v 56 x Ff(1)p Fd(\000)p Fh(V)2745 4720 y Fg(2)2789 4696 y Fl(.)j(The)32 b(tangen-)515 4824 y(tial)j(comp)s(onen)m(t)j(of)f (the)i(w)m(all)d(momen)m(tum)g(equals)i(zero,)i(as)e(the)g(v)m(elo)s (cit)m(y)515 4940 y Fe(V)54 b Fl(is)32 b(directed)h(along)e(the)i (normal)e(to)h(the)h(w)m(all.)1926 5255 y Fn(8)p eop %%Page: 9 9 9 8 bop 515 523 a Fl(Using)26 b(the)g(la)m(ws)h(of)e(conserv)-5 b(ations)27 b(of)f(momen)m(tum)f(and)h(energy)-8 b(,)29 b(w)m(e)e(obtain)515 639 y(the)33 b(follo)m(wing)d(equalities:)1463 838 y Fe(p)1512 853 y Fh(\027)1577 838 y Fl(+)22 b Fe(M)10 b(P)42 b Fl(=)27 b Fe(p)2036 796 y Fd(0)2036 862 y Fh(\027)2102 838 y Fl(+)22 b Fe(M)10 b(P)2381 796 y Fd(0)2404 838 y Fe(;)823 b Fl(\(3\))811 961 y Fc(p)p 910 961 357 4 v 910 1051 a Fj(k)p Fe(p)p Fj(k)1059 1022 y Ff(2)1121 1051 y Fl(+)22 b(1)f(+)h Fe(M)1491 964 y Fj(p)p 1575 964 285 4 v 1575 1051 a Fe(P)1652 1022 y Ff(2)1713 1051 y Fl(+)g(1)28 b(=)1991 961 y Fc(p)p 2091 961 381 4 v 90 x Fj(k)p Fe(p)2190 1022 y Fd(0)2213 1051 y Fj(k)2263 1022 y Ff(2)2324 1051 y Fl(+)22 b(1)g(+)g Fe(M)2695 964 y Fj(p)p 2779 964 304 4 v 2779 1051 a Fe(P)2856 1022 y Fd(0)p Ff(2)2936 1051 y Fl(+)g(1)171 b(\(4\))515 1299 y(Using)32 b(\(4\))g(to)g(\014nd)h Fe(P)1337 1263 y Fd(0)1360 1299 y Fl(,)g(w)m(e)g(obtain:)551 1593 y Fe(M)10 b(P)732 1552 y Fd(0)783 1593 y Fl(=)27 b Fe(M)990 1452 y Fc(\032)1066 1482 y(\020)1135 1438 y(p)p 1235 1438 357 4 v 86 x Fl(1)22 b(+)g Fj(k)p Fe(p)p Fj(k)1553 1495 y Ff(2)1614 1524 y Fj(\000)1714 1438 y Fc(p)p 1813 1438 341 4 v 1813 1524 a Fl(1)g(+)g Fj(k)p Fe(p)2081 1495 y Fd(0)2104 1524 y Fj(kj)2182 1487 y Ff(2)p 1135 1570 1086 4 v 1626 1661 a Fe(M)2253 1593 y Fl(+)2351 1506 y Fj(p)p 2434 1506 285 4 v 87 x Fl(1)g(+)g Fe(P)2680 1564 y Ff(2)2719 1482 y Fc(\021)2779 1499 y Ff(2)2818 1593 y Fj(\000)p Fl(1)2944 1452 y Fc(\033)3029 1447 y Fg(1)p 3029 1459 31 3 v 3029 1500 a(2)3074 1593 y Fl(=)825 1878 y Fe(M)10 b(P)36 b Fl(+)1136 1729 y Fj(p)p 1219 1729 285 4 v 82 x Fl(1)22 b(+)g Fe(P)1465 1782 y Ff(2)p 1136 1855 368 4 v 1282 1947 a Fe(P)1514 1878 y Fl(\()1552 1788 y Fc(p)p 1651 1788 357 4 v 1651 1878 a Fl(1)g(+)g Fj(k)p Fe(p)p Fj(k)1969 1849 y Ff(2)2031 1878 y Fj(\000)2130 1788 y Fc(p)p 2230 1788 381 4 v 90 x Fl(1)g(+)g Fj(k)p Fe(p)2498 1849 y Fd(0)2521 1878 y Fj(k)2571 1849 y Ff(2)2610 1878 y Fl(\))g(+)g Fe(O)2862 1738 y Fc(\022)2973 1811 y Fl(1)p 2945 1855 105 4 v 2945 1947 a Fe(M)3060 1738 y Fc(\023)3150 1878 y Fe(:)515 2221 y Fl(Substituting)27 b(the)i(last)f(equalit)m(y)g(in)f (\(3\))h(and)h(going)e(to)h(the)h(limit)c Fe(M)38 b Fj(!)27 b(1)p Fl(,)515 2347 y(w)m(e)33 b(obtain)d(the)i(equalit)m(y)1498 2262 y Fc(p)p 1598 2262 381 4 v 85 x Fj(k)p Fe(p)1697 2319 y Fd(0)1720 2347 y Fj(k)1770 2319 y Ff(2)1831 2347 y Fl(+)22 b(1)e Fj(\000)2096 2262 y Fc(p)p 2196 2262 357 4 v 85 x Fj(k)p Fe(p)p Fj(k)2345 2319 y Ff(2)2406 2347 y Fl(+)i(1)28 b(=)f Fe(V)22 b Fl(\()p Fe(p)2850 2311 y Fd(0)2850 2372 y Fh(\027)2913 2347 y Fj(\000)f Fe(p)3060 2362 y Fh(\027)3103 2347 y Fl(\))p Fe(;)32 b Fl(that)515 2464 y(leads)g(to)g(the)h(equalities)898 2671 y Fe(p)947 2630 y Fd(0)947 2696 y Fh(\027)991 2620 y Ff(2)1052 2671 y Fj(\000)23 b Fe(p)1201 2630 y Ff(2)1201 2696 y Fh(\027)1272 2671 y Fl(=)k Fe(V)22 b Fl(\()p Fe(p)1541 2630 y Fd(0)1541 2696 y Fh(\027)1606 2671 y Fj(\000)h Fe(p)1755 2686 y Fh(\027)1798 2671 y Fl(\)\()1874 2581 y Fc(p)p 1973 2581 V 1973 2671 a Fl(1)f(+)g Fj(k)p Fe(p)p Fj(k)2291 2643 y Ff(2)2352 2671 y Fl(+)2450 2581 y Fc(p)p 2550 2581 381 4 v 90 x Fl(1)g(+)g Fj(k)p Fe(p)2818 2643 y Fd(0)2841 2671 y Fj(k)2891 2643 y Ff(2)2930 2671 y Fl(\))p Fe(;)1124 2885 y(p)1173 2844 y Fd(0)1173 2910 y Fh(\027)1238 2885 y Fl(+)g Fe(p)1385 2900 y Fh(\027)1456 2885 y Fl(=)27 b Fe(V)22 b Fl(\()1676 2795 y Fc(p)p 1775 2795 357 4 v 1775 2885 a Fl(1)g(+)g Fj(k)p Fe(p)p Fj(k)2093 2856 y Ff(2)2154 2885 y Fl(+)2252 2795 y Fc(p)p 2352 2795 381 4 v 90 x Fl(1)g(+)g Fj(k)p Fe(p)2620 2856 y Fd(0)2643 2885 y Fj(k)2693 2856 y Ff(2)2732 2885 y Fl(\))484 b(\(5\))934 3066 y Fj(\000)p Fe(V)1090 3025 y Ff(2)1129 3066 y Fl(\()p Fe(p)1216 3025 y Fd(0)1216 3091 y Fh(\027)1282 3066 y Fj(\000)22 b Fe(p)1430 3081 y Fh(\027)1473 3066 y Fl(\))28 b(=)g Fj(\000)p Fe(V)22 b Fl(\()1837 2976 y Fc(p)p 1936 2976 V 1936 3066 a Fl(1)g(+)g Fj(k)p Fe(p)2204 3037 y Fd(0)2227 3066 y Fj(k)2277 3037 y Ff(2)2339 3066 y Fj(\000)2438 2976 y Fc(p)p 2538 2976 357 4 v 90 x Fl(1)g(+)g Fj(k)p Fe(p)p Fj(k)2856 3037 y Ff(2)2895 3066 y Fl(\))p Fe(:)294 b Fl(\(6\))515 3315 y(Adding)32 b(\(5\))g(and)h(\(6\),)f(w)m (e)i(obtain)d(the)i(equalit)m(y)1021 3528 y Fe(p)1070 3487 y Fd(0)1070 3553 y Fh(\027)1113 3528 y Fl(\(1)22 b Fj(\000)h Fe(V)1400 3487 y Ff(2)1440 3528 y Fl(\))k(=)h Fj(\000)p Fe(p)1735 3543 y Fh(\027)1779 3528 y Fl(\(1)21 b(+)h Fe(V)2064 3487 y Ff(2)2103 3528 y Fl(\))h(+)f(2)p Fe(V)2389 3438 y Fc(p)p 2488 3438 V 2488 3528 a Fj(k)p Fe(p)p Fj(k)2637 3499 y Ff(2)2699 3528 y Fl(+)g(1)o Fe(;)515 3726 y Fl(from)31 b(whic)m(h)i(the)g(equalit)m(y)f(for)g Fe(p)1761 3690 y Fd(0)1817 3726 y Fl(in)g(\(1\))g(follo)m(ws.)515 3925 y(T)-8 b(o)35 b(pro)m(v)m(e)i(the)f(energy)g(transformation)e(la)m (w,)h(w)m(e)i(do)e(the)h(follo)m(wing)d(tric)m(k.)515 4042 y(Consider)46 b(~)-57 b Fe(p)37 b Fl(=)g Fe(p)1175 4057 y Fh(\027)1218 4042 y Fe(=)p Fl(\001,)i(\001)f(=)e Fj(k)p Fe(p)1744 4057 y Fh(\034)1787 4042 y Fj(k)1837 4006 y Ff(2)1902 4042 y Fl(+)26 b(1.)60 b(The)39 b(transformation)c (form)m(ulas)515 4158 y(\(1\))h(in)h(terms)g(of)45 b(~)-57 b Fe(p)37 b Fl(are)g(exactly)g(the)h(same)f(as)g(for)g(one)g (dimensional)e(case)515 4286 y([8].)61 b(Th)m(us)41 b(the)e("energy") 1572 4260 y(~)1549 4286 y Fe(E)44 b Fl(=)1779 4199 y Fc(p)p 1878 4199 258 4 v 1886 4286 a Fl(~)-57 b Fe(p)1927 4257 y Ff(2)1989 4286 y Fl(+)22 b(1)38 b(is)h(transformed)f(exactly)h (as)g(in)515 4402 y([8])1099 4545 y(~)1076 4570 y Fe(E)1154 4529 y Fd(0)1205 4570 y Fl(=)1331 4545 y(~)1309 4570 y Fe(E)1397 4503 y Fl(1)22 b(+)g Fe(V)p 1397 4548 250 4 v 1397 4639 a Fl(1)f Fj(\000)i Fe(V)1678 4570 y Fl(+)1866 4503 y(2)p Fe(V)p 1786 4548 289 4 v 1786 4639 a Fl(1)f Fj(\000)g Fe(V)2035 4610 y Ff(2)2084 4460 y Fc(\020)2144 4464 y(p)p 2243 4464 V 2266 4545 a Fl(~)2243 4570 y Fe(E)2321 4542 y Ff(2)2383 4570 y Fj(\000)h Fl(1)e Fj(\000)2676 4545 y Fl(~)2653 4570 y Fe(E)2731 4460 y Fc(\021)2791 4570 y Fe(:)515 4774 y Fl(But)j(\001)f(is)h(constan)m(t)g(under)h(the)f (transformation,)f(and)h(one)g(can)g(easily)f(sho)m(w)515 4890 y(that)37 b(the)h("real")e(energy)j Fe(E)j Fl(=)1739 4806 y Fj(p)p 1822 4806 82 4 v 84 x Fl(\001)1926 4865 y(~)1903 4890 y Fe(E)6 b Fl(,)39 b(th)m(us)g(the)f(relation)d(\(2\))i (is)g(pro)m(v)m(ed.)515 5006 y Fb(\003)1926 5255 y Fn(9)p eop %%Page: 10 10 10 9 bop 515 523 a Fl(If)34 b(the)g(particle)f(mo)m(v)m(es)i(up)g(and)f (hits)g(the)g(horizon)m(tal)f(w)m(all,)g(whic)m(h,)i(again,)515 639 y(mo)m(v)m(es)30 b(in)f(the)h(v)m(ertical)f(direction,)g(then)i (the)f(momen)m(tum)e(transformation)515 756 y(will)i(b)s(e)65 b([8])1007 1007 y Fe(p)1056 966 y Fd(0)1056 1032 y Fh(\027)1127 1007 y Fl(=)27 b Fj(\000)p Fe(p)1356 1022 y Fh(\027)1410 940 y Fl(1)22 b Fj(\000)g Fe(V)p 1410 984 250 4 v 1411 1076 a Fl(1)g(+)g Fe(V)1691 1007 y Fl(+)1880 940 y(2)p Fe(V)p 1799 984 289 4 v 1799 1076 a Fl(1)g Fj(\000)g Fe(V)2048 1047 y Ff(2)2097 897 y Fc(\020)2157 917 y(p)p 2257 917 357 4 v 90 x Fj(k)p Fe(p)p Fj(k)2406 978 y Ff(2)2467 1007 y Fl(+)g(1)g Fj(\000)g Fe(p)2784 1022 y Fh(\027)2827 897 y Fc(\021)3254 1007 y Fl(\(7\))515 1244 y(Here)33 b Fe(V)54 b Fl(is)32 b(the)h(v)m(elo)s(cit)m(y)g(of)f(the)h(w)m(all)e (directed)i(up)m(w)m(ards.)515 1444 y(W)-8 b(e)37 b(use)h(the)g (equalities)e(\(1\),)i(\(2\))f(and)g(\(7\))g(to)f(\014nd)i(the)g (relation)d(b)s(et)m(w)m(een)515 1560 y(the)i(particle)e(v)m(elo)s(cit) m(y)h Fe(v)k Fl(b)s(efore)c(the)h(collision)d(with)i(its)g(v)m(elo)s (cit)m(y)g Fe(v)3122 1524 y Fd(0)3181 1560 y Fl(after)515 1676 y(the)d(collision.)515 1958 y Fa(Lemma)k(2.1)42 b Fk(L)-5 b(et)43 b(the)h(p)-5 b(article)43 b(fal)5 b(l)42 b(to)i(the)f(horizontal)g(in\014nitely-he)-5 b(avy)515 2075 y(wal)5 b(l)37 b(with)i(the)f(velo)-5 b(city)38 b Fe(v)t Fk(,)h(dir)-5 b(e)g(cte)g(d)38 b(towar)-5 b(ds)38 b(the)g(wal)5 b(l)38 b(\(i.e.,)g(the)h(normal)515 2191 y(c)-5 b(omp)g(onent)32 b Fe(v)1046 2206 y Fh(\027)1117 2191 y Fe(<)27 b Fl(0)p Fk(\),)33 b(and)g(at)h(the)f(imp)-5 b(act)33 b(the)g(wal)5 b(l)32 b(moves)h(along)f(its)h(nor-)515 2307 y(mal)i(with)g(the)h(velo)-5 b(city)36 b Fe(V)21 b Fk(.)48 b(Then)34 b(after)i(the)g(imp)-5 b(act)35 b(the)h(p)-5 b(article)35 b(velo)-5 b(city)515 2423 y Fe(v)566 2387 y Fd(0)624 2423 y Fk(e)g(quals)934 2680 y Fe(v)985 2639 y Fd(0)981 2705 y Fh(\027)1052 2680 y Fl(=)28 b Fj(\000)1243 2613 y Fe(v)1290 2628 y Fh(\027)1356 2613 y Fj(\000)22 b Fl(2)p Fe(V)44 b Fl(+)22 b Fe(V)1781 2577 y Ff(2)1821 2613 y Fe(v)1868 2628 y Fh(\027)p 1243 2657 668 4 v 1264 2749 a Fl(1)g Fj(\000)h Fl(2)p Fe(V)e(v)1609 2764 y Fh(\027)1674 2749 y Fl(+)h Fe(V)1851 2720 y Ff(2)1921 2680 y Fe(;)117 b(v)2116 2639 y Fd(0)2112 2705 y Fh(\034)2183 2680 y Fl(=)2382 2613 y Fe(v)2429 2628 y Fh(\034)2472 2613 y Fl(\(1)22 b Fj(\000)h Fe(V)2759 2577 y Ff(2)2799 2613 y Fl(\))p 2296 2657 627 4 v 2296 2749 a(1)f Fj(\000)h Fl(2)p Fe(V)e(v)2641 2764 y Fh(\027)2706 2749 y Fl(+)h Fe(V)2883 2720 y Ff(2)2932 2680 y Fe(;)295 b Fl(\(8\))515 2924 y Fe(v)562 2939 y Fh(\034)640 2924 y Fk(is)35 b(the)f(pr)-5 b(oje)g(ction)34 b(of)h(the)g(velo)-5 b(city)35 b(to)g(the)g (horizontal)f(plane.)515 3206 y Fa(Pro)s(of.)59 b Fl(Supp)s(ose)39 b(that)f(b)s(efore)g(the)g(collision)d(the)j(particle)f(has)h(the)g (mo-)515 3323 y(men)m(tum)32 b Fe(p)p Fl(,)g(and)h(after)f(the)h (collision)d(it)h(obtains)h(the)h(momen)m(tum)f Fe(p)3137 3286 y Fd(0)3160 3323 y Fl(.)43 b(As)1578 3565 y Fe(p)28 b Fl(=)1974 3498 y(1)p 1768 3542 460 4 v 1768 3562 a Fc(p)p 1868 3562 361 4 v 85 x Fl(1)22 b Fj(\000)g(k)p Fe(v)t Fj(k)2189 3618 y Ff(2)2238 3565 y Fe(v)t(;)515 3838 y Fl(the)33 b(v)m(elo)s(cit)m(y)f(equals)1593 3997 y Fe(v)g Fl(=)1989 3930 y(1)p 1785 3974 457 4 v 1785 3994 a Fc(p)p 1885 3994 357 4 v 85 x Fl(1)22 b(+)g Fj(k)p Fe(p)p Fj(k)2203 4051 y Ff(2)2252 3997 y Fe(p)515 4237 y Fl(Supp)s(ose)28 b(\014rst)g(that)f(pro)5 b(jection)27 b(of)g(the)h(particle)e(momen)m(tum)g(to)h(the)g(v)m(erti-)515 4353 y(cal)32 b(direction)g(b)s(oth)g(b)s(efore)h(and)h(after)e(the)i (collision)c(is)i(not)h(zero:)44 b Fe(p)3127 4368 y Fh(\027)3199 4353 y Fj(6)p Fl(=)28 b(0,)515 4470 y Fe(p)564 4433 y Fd(0)564 4494 y Fh(\027)635 4470 y Fj(6)p Fl(=)f(0.)42 b(Then)32 b(the)e(normal)e(v)m(elo)s(cit)m(y)i(comp)s(onen)m(ts)h Fe(v)2542 4485 y Fh(\027)2585 4470 y Fl(,)g Fe(v)2694 4433 y Fd(0)2690 4494 y Fh(\027)2763 4470 y Fl(are)f(not)g(zero)s(es.) 515 4586 y(F)-8 b(rom)31 b(\(1\))h(w)m(e)i(get)1248 4775 y Fe(p)1297 4739 y Fd(0)1297 4800 y Fh(\027)p 1248 4820 93 4 v 1248 4911 a Fe(p)1297 4926 y Fh(\027)1378 4843 y Fl(=)27 b Fj(\000)1569 4775 y Fl(1)22 b(+)g Fe(V)p 1568 4820 250 4 v 1568 4911 a Fl(1)g Fj(\000)h Fe(V)1850 4843 y Fj(\000)2040 4775 y Fl(2)p Fe(V)p 1959 4820 289 4 v 1959 4911 a Fl(1)f Fj(\000)h Fe(V)2208 4882 y Ff(2)2258 4732 y Fc(\020)2348 4775 y Fl(1)p 2327 4820 91 4 v 2327 4911 a Fe(v)2374 4926 y Fh(\027)2450 4843 y Fl(+)f(1)2597 4732 y Fc(\021)3254 4843 y Fl(\(9\))1905 5255 y Fn(10)p eop %%Page: 11 11 11 10 bop 515 523 a Fl(F)-8 b(orm)31 b(\(7\))h(w)m(e)i(ha)m(v)m(e)1187 703 y Fe(p)1236 718 y Fh(\027)p 1187 748 93 4 v 1187 839 a Fe(p)1236 810 y Fd(0)1236 864 y Fh(\027)1317 771 y Fl(=)27 b Fj(\000)1507 703 y Fl(1)c Fj(\000)f Fe(V)p 1507 748 250 4 v 1508 839 a Fl(1)g(+)g Fe(V)1789 771 y Fj(\000)1979 703 y Fl(2)p Fe(V)p 1898 748 289 4 v 1898 839 a Fl(1)g Fj(\000)h Fe(V)2147 810 y Ff(2)2197 660 y Fc(\020)2279 771 y Fj(\000)2409 703 y Fl(1)p 2388 748 91 4 v 2388 839 a Fe(v)2439 810 y Fd(0)2435 864 y Fh(\027)2511 771 y Fl(+)f(1)2658 660 y Fc(\021)3206 771 y Fl(\(10\))515 1025 y(W)-8 b(e)33 b(m)m(ultiply)d(equalities)h(\(9\))h(and)h(\(10\):) 696 1273 y(1)28 b(=)f(1)22 b(+)1136 1206 y(2)p Fe(V)p 1055 1250 289 4 v 1055 1341 a Fl(1)g Fj(\000)h Fe(V)1304 1313 y Ff(2)1364 1206 y Fl(1)f(+)g Fe(V)p 1364 1250 250 4 v 1364 1341 a Fl(1)g Fj(\000)g Fe(V)1623 1162 y Fc(\020)1682 1273 y Fl(1)g Fj(\000)1883 1206 y Fl(1)p 1863 1250 91 4 v 1863 1341 a Fe(v)1914 1313 y Fd(0)1910 1366 y Fh(\027)1963 1162 y Fc(\021)2045 1273 y Fj(\000)2235 1206 y Fl(2)p Fe(V)p 2154 1250 289 4 v 2154 1341 a Fl(1)g Fj(\000)h Fe(V)2403 1313 y Ff(2)2463 1206 y Fl(1)f Fj(\000)g Fe(V)p 2463 1250 250 4 v 2464 1341 a Fl(1)f(+)h Fe(V)2722 1162 y Fc(\020)2781 1273 y Fl(1)g(+)2981 1206 y(1)p 2960 1250 91 4 v 2960 1341 a Fe(v)3007 1356 y Fh(\027)3060 1162 y Fc(\021)3120 1273 y Fj(\000)1297 1600 y(\000)1503 1532 y Fl(4)p Fe(V)1631 1496 y Ff(2)p 1384 1577 404 4 v 1384 1668 a Fl(\(1)h Fj(\000)f Fe(V)1672 1639 y Ff(2)1711 1668 y Fl(\))1749 1639 y Ff(2)1798 1489 y Fc(\020)1858 1600 y Fl(1)g Fj(\000)2059 1532 y Fl(1)p 2038 1577 91 4 v 2038 1668 a Fe(v)2089 1639 y Fd(0)2085 1693 y Fh(\027)2139 1489 y Fc(\021\020)2258 1600 y Fl(1)g(+)2457 1532 y(1)p 2437 1577 V 2437 1668 a Fe(v)2484 1683 y Fh(\027)2537 1489 y Fc(\021)515 1821 y Fl(and)32 b(express)j Fe(v)1095 1785 y Fd(0)1091 1846 y Fh(\027)1167 1821 y Fl(through)e Fe(v)1583 1836 y Fh(\027)1658 1821 y Fl(and)g Fe(V)22 b Fl(:)679 2069 y(1)g Fj(\000)880 2001 y Fl(1)p 859 2046 V 859 2137 a Fe(v)910 2108 y Fd(0)906 2162 y Fh(\027)987 2069 y Fl(=)1091 1958 y Fc(\020)1160 2001 y Fl(1)g Fj(\000)h Fe(V)p 1160 2046 250 4 v 1161 2137 a Fl(1)f(+)g Fe(V)1419 1958 y Fc(\020)1479 2069 y Fl(1)g(+)1679 2001 y(1)p 1658 2046 91 4 v 1658 2137 a Fe(v)1705 2152 y Fh(\027)1758 1958 y Fc(\021)q(\021\020)1947 2001 y Fl(1)g(+)g Fe(V)p 1947 2046 250 4 v 1947 2137 a Fl(1)f Fj(\000)i Fe(V)2228 2069 y Fj(\000)2418 2001 y Fl(2)p Fe(V)p 2337 2046 289 4 v 2337 2137 a Fl(1)f Fj(\000)h Fe(V)2586 2108 y Ff(2)2636 1958 y Fc(\020)2695 2069 y Fl(1)f(+)2895 2001 y(1)p 2874 2046 91 4 v 2874 2137 a Fe(v)2921 2152 y Fh(\027)2974 1958 y Fc(\021)q(\021)3093 1981 y Fd(\000)p Ff(1)3188 2069 y Fe(;)1453 2395 y(v)1504 2354 y Fd(0)1500 2420 y Fh(\027)1571 2395 y Fl(=)28 b Fj(\000)1762 2328 y Fe(v)1809 2343 y Fh(\027)1875 2328 y Fj(\000)23 b Fl(2)p Fe(V)43 b Fl(+)22 b Fe(V)2301 2292 y Ff(2)2340 2328 y Fe(v)2387 2343 y Fh(\027)p 1762 2372 668 4 v 1783 2464 a Fe(V)1862 2435 y Ff(2)1923 2464 y Fj(\000)h Fl(2)p Fe(V)e(v)2197 2479 y Fh(\027)2263 2464 y Fl(+)h(1)515 2603 y(As)33 b(the)g(tangen)m(tial)e(comp)s(onen)m(t)h(of)g(the)h(v)m(elo)s(cit)m(y) g(equals)1538 2842 y Fe(v)1585 2857 y Fh(\034)1656 2842 y Fl(=)1974 2774 y(1)p 1770 2819 457 4 v 1770 2839 a Fc(p)p 1869 2839 357 4 v 1869 2924 a Fl(1)22 b(+)g Fj(k)p Fe(p)p Fj(k)2187 2895 y Ff(2)2236 2842 y Fe(p)2285 2857 y Fh(\034)2328 2842 y Fe(;)515 3116 y Fl(and)32 b Fe(p)753 3080 y Fd(0)753 3141 y Fh(\034)824 3116 y Fl(=)c Fe(p)977 3131 y Fh(\034)1020 3116 y Fl(,)k(w)m(e)i(get)f(using)f(\(2\))921 3374 y Fe(v)972 3333 y Fd(0)968 3399 y Fh(\034)1039 3374 y Fl(=)1368 3307 y(1)p 1153 3352 480 4 v 1153 3372 a Fc(p)p 1252 3372 381 4 v 1252 3457 a Fl(1)22 b(+)g Fj(k)p Fe(p)1520 3428 y Fd(0)1543 3457 y Fj(k)1593 3428 y Ff(2)1643 3374 y Fe(p)1692 3389 y Fh(\034)1762 3374 y Fl(=)1888 3307 y Fe(E)p 1876 3352 102 4 v 1876 3443 a(E)1954 3414 y Fd(0)1987 3374 y Fe(v)2034 3389 y Fh(\034)2105 3374 y Fl(=)2388 3307 y(1)g Fj(\000)g Fe(V)2637 3271 y Ff(2)p 2219 3352 627 4 v 2219 3443 a Fl(1)g Fj(\000)g Fl(2)p Fe(V)g(v)2564 3458 y Fh(\027)2629 3443 y Fl(+)g Fe(V)2806 3414 y Ff(2)2855 3374 y Fe(v)2902 3389 y Fh(\034)2945 3374 y Fe(:)234 b Fl(\(11\))515 3732 y(The)39 b(momen)m(ta)f Fe(p)1202 3747 y Fh(\027)1284 3732 y Fl(and)h Fe(p)1529 3696 y Fd(0)1529 3757 y Fh(\027)1611 3732 y Fl(cannot)g(equal)f(zeto)h (sim)m(ultaneously)-8 b(,)39 b(see)h(\(7.)515 3848 y(Supp)s(ose)33 b(that)g Fe(p)1158 3863 y Fh(\027)1229 3848 y Fl(=)27 b(0.)43 b(Then)34 b Fe(v)1753 3863 y Fh(\027)1824 3848 y Fl(=)28 b(0.)43 b(Using)32 b(\(2\),)g(w)m(e)i(get)1161 4101 y Fe(v)1212 4060 y Fd(0)1208 4125 y Fh(\027)1279 4101 y Fl(=)1653 4033 y Fe(p)1702 3997 y Fd(0)1702 4058 y Fh(\027)p 1393 4078 613 4 v 1393 4098 a Fc(p)p 1492 4098 513 4 v 1492 4187 a Fl(1)22 b(+)g Fe(p)1710 4158 y Fd(0)1710 4211 y Fh(\027)1753 4147 y Ff(2)1815 4187 y Fl(+)g Fe(p)1962 4158 y Ff(2)1962 4211 y Fh(\034)2043 4101 y Fl(=)2161 4033 y Fe(p)2210 3997 y Fd(0)2210 4058 y Fh(\027)p 2156 4078 102 4 v 2156 4169 a Fe(E)2234 4140 y Fd(0)2295 4101 y Fl(=)2488 4033 y(2)p Fe(V)p 2409 4078 287 4 v 2409 4169 a Fl(1)g(+)g Fe(V)2656 4140 y Ff(2)2706 4101 y Fe(;)515 4375 y Fl(whic)m(h)33 b(is)f(exactly)h(\(8\))f(for)g Fe(v)1581 4390 y Fh(\027)1652 4375 y Fl(=)c(0.)515 4575 y(Let)d(no)m(w)g Fe(p)926 4538 y Fd(0)926 4599 y Fh(\027)997 4575 y Fl(=)i(0.)41 b(W)-8 b(e)25 b(ha)m(v)m(e)h(to)e(c)m(hec)m(k,)k (that)d(the)g(relation)d(\(8\))j(giv)m(es)g Fe(v)3133 4538 y Fd(0)3129 4599 y Fh(\027)3200 4575 y Fl(=)i(0.)515 4691 y(But)32 b(this)h(follo)m(ws)e(immediately)e(from)j(\(9.)515 4890 y(The)e(expression)g(\(11\))f(is)g(also)f(true)i(for)e(b)s(oth)h (cases,)j(as)d(it)f(is)h(deriv)m(ed)h(using)515 5006 y(the)j(energy)g(relations,)f(rather)g(than)h(the)g(momen)m(ta.)42 b Fb(\003)1905 5255 y Fn(11)p eop %%Page: 12 12 12 11 bop 515 523 a Fa(Lemma)37 b(2.2)42 b Fk(Supp)-5 b(ose)35 b(that)h(the)g(p)-5 b(article)36 b(hits)f(the)h(b)-5 b(oundary)36 b(with)g(some)515 639 y(velo)-5 b(city)37 b Fe(v)t Fk(.)51 b(Then)36 b(after)i(the)f(imp)-5 b(act)36 b(the)i(p)-5 b(article)36 b(velo)-5 b(city)37 b(ve)-5 b(ctor)37 b(length)515 756 y Fj(k)p Fe(v)616 719 y Fd(0)639 756 y Fj(k)d Fk(c)-5 b(an)35 b(b)-5 b(e)34 b(expr)-5 b(esse)g(d)34 b(as)1077 986 y Fl(1)22 b Fj(\000)h(k)p Fe(v)1349 945 y Fd(0)1371 986 y Fj(k)1421 945 y Ff(2)1488 986 y Fl(=)1771 919 y(\(1)f Fj(\000)g Fe(V)2058 883 y Ff(2)2097 919 y Fl(\))2135 883 y Ff(2)p 1602 963 742 4 v 1602 1055 a Fl(\(1)g(+)g Fe(V)1887 1026 y Ff(2)1949 1055 y Fj(\000)g Fl(2)p Fe(V)g(v)2223 1070 y Fh(\027)2266 1055 y Fl(\))2304 1026 y Ff(2)2353 986 y Fl(\(1)g Fj(\000)h(k)p Fe(v)t Fj(k)2713 945 y Ff(2)2752 986 y Fl(\))p Fe(;)389 b Fl(\(12\))515 1208 y Fe(V)56 b Fk(b)-5 b(eing)34 b(the)h("b)-5 b(oundary)35 b(velo)-5 b(city")34 b(at)h(the)g(moment)f(of)h(the)g(imp) -5 b(act.)515 1458 y Fa(Pro)s(of.)43 b Fl(It)33 b(follo)m(ws)e(from)g (\(8\))h(that)1703 1626 y(1)22 b Fj(\000)h(k)p Fe(v)1975 1585 y Fd(0)1998 1626 y Fj(k)2048 1585 y Ff(2)2115 1626 y Fl(=)633 1856 y(=)746 1789 y(\(1)f Fj(\000)h Fl(2)p Fe(V)e(v)1129 1804 y Fh(\027)1194 1789 y Fl(+)h Fe(V)1371 1753 y Ff(2)1410 1789 y Fl(\))1448 1753 y Ff(2)1510 1789 y Fj(\000)g Fl(\()p Fe(v)1694 1804 y Fh(\027)1760 1789 y Fj(\000)h Fl(2)p Fe(V)43 b Fl(+)22 b Fe(V)2186 1753 y Ff(2)2225 1789 y Fe(v)2272 1804 y Fh(\027)2315 1789 y Fl(\))2353 1753 y Ff(2)2415 1789 y Fj(\000)g(k)p Fe(v)2611 1804 y Fh(\034)2655 1789 y Fj(k)2705 1753 y Ff(2)2744 1789 y Fl(\(1)g Fj(\000)g Fe(V)3031 1753 y Ff(2)3070 1789 y Fl(\))3108 1753 y Ff(2)p 746 1833 2402 4 v 1576 1925 a Fl(\(1)g Fj(\000)h Fl(2)p Fe(V)e(v)1959 1940 y Fh(\027)2024 1925 y Fl(+)h Fe(V)2201 1896 y Ff(2)2240 1925 y Fl(\))2278 1896 y Ff(2)3185 1856 y Fl(=)593 2135 y(=)707 2068 y(1)g(+)g(2)p Fe(V)1003 2031 y Ff(2)1042 2068 y Fe(v)1093 2031 y Ff(2)1089 2092 y Fh(\027)1155 2068 y Fl(+)g Fe(V)1331 2031 y Ff(4)1393 2068 y Fj(\000)h Fe(v)1544 2031 y Ff(2)1540 2092 y Fh(\027)1605 2068 y Fj(\000)g Fe(u)1761 2031 y Ff(4)1800 2068 y Fe(v)1851 2031 y Ff(2)1847 2092 y Fh(\027)1912 2068 y Fj(\000)g(k)p Fe(v)t Fj(k)2163 2031 y Ff(2)2224 2068 y Fj(\000)f(k)p Fe(v)2420 2083 y Fh(\034)2464 2068 y Fj(k)2514 2031 y Ff(2)2553 2068 y Fe(V)2631 2031 y Ff(4)2693 2068 y Fl(+)g(2)p Fj(k)p Fe(v)2937 2083 y Fh(\034)2980 2068 y Fj(k)3030 2031 y Ff(2)3069 2068 y Fe(V)3148 2031 y Ff(2)p 707 2112 2481 4 v 1576 2203 a Fl(\(1)g Fj(\000)h Fl(2)p Fe(V)e(v)1959 2218 y Fh(\027)2024 2203 y Fl(+)h Fe(V)2201 2174 y Ff(2)2240 2203 y Fl(\))2278 2174 y Ff(2)3225 2135 y Fl(=)1341 2413 y(=)1455 2346 y(\()p Fe(V)1571 2310 y Ff(4)1633 2346 y Fj(\000)h Fl(2)p Fe(V)1860 2310 y Ff(2)1921 2346 y Fl(+)f(1\)\(1)g Fj(\000)h(k)p Fe(v)t Fj(k)2466 2310 y Ff(2)2504 2346 y Fl(\))p 1455 2391 1088 4 v 1628 2482 a(\(1)f Fj(\000)g Fl(2)p Fe(V)g(v)2011 2497 y Fh(\027)2076 2482 y Fl(+)g Fe(V)2253 2453 y Ff(2)2292 2482 y Fl(\))2330 2453 y Ff(2)515 2613 y Fb(\003)515 2812 y Fl(W)-8 b(e)48 b(no)m(w)h(giv)m(e)f(some)g(basic)g(results)g(from)f(the)i(sp)s(ecial)e (theory)i(of)e(rela-)515 2928 y(tivit)m(y)-8 b(.)81 b(Let)46 b(a)f(particle)g(of)g(mass)g(1)h(mo)m(v)m(e)g(under)g(the)g (in\015uence)g(of)g(the)515 3045 y(\(non-gra)m(vitational\))39 b(p)s(oten)m(tial)i(force)h(\014eld)h Fe(F)14 b Fl(\()p Fe(x)p Fl(\))44 b(=)h Fj(\000)p Fe(g)t(r)s(ad)31 b(U)10 b Fl(\()p Fe(x)p Fl(\),)47 b Fe(x)42 b Fl(is)515 3161 y(the)37 b(spatial)e(co)s(ordinate)h(of)g(the)h(particle.)55 b(W)-8 b(e)37 b(assume)g(that)g(the)g(motion)515 3277 y(o)s(ccurs)31 b(in)f(some)g(compact)h(domain,)e(th)m(us)j(functions)e Fe(U)10 b Fl(\()p Fe(x)p Fl(\))p Fe(;)17 b(F)d Fl(\()p Fe(x)p Fl(\))31 b(are)g(uni-)515 3393 y(formly)k(b)s(ounded.)58 b(The)38 b(dynamics)f(can)g(b)s(e)g(describ)s(ed)h(b)m(y)g(the)f (Lagrange)515 3510 y(equations)c(with)f(the)h(Lagrange)f(function)1283 3686 y Fe(L)p Fl(\()18 b(_)-45 b Fe(x)q(;)17 b(x)p Fl(\))28 b(=)f Fj(\000)1788 3596 y Fc(p)p 1889 3596 366 4 v 1889 3686 a Fl(1)21 b Fj(\000)i(k)17 b Fl(_)-44 b Fe(x)p Fj(k)2214 3658 y Ff(2)2276 3686 y Fj(\000)22 b Fe(U)10 b Fl(\()p Fe(x)p Fl(\))p Fe(;)515 3854 y Fl(see,)29 b(e.g.,)55 b([5],)28 b(Section)f(16.)41 b(Again,)27 b(w)m(e)h(assume)f(that)f(the) i(v)m(elo)s(cit)m(y)e(of)h(ligh)m(t)515 3970 y(equals)33 b Fe(c)27 b Fl(=)h(1.)515 4169 y(The)33 b(equations)g(of)f(motion)f (can)i(b)s(e)f(written)h(as)1498 4317 y Fe(d)p 1481 4362 86 4 v 1481 4453 a(dt)1808 4317 y Fl(_)-44 b Fe(x)p 1586 4362 465 4 v 1586 4382 a Fc(p)p 1686 4382 366 4 v 85 x Fl(1)22 b Fj(\000)h(k)16 b Fl(_)-43 b Fe(x)p Fj(k)2012 4438 y Ff(2)2089 4385 y Fl(=)27 b Fj(\000)2279 4317 y Fe(@)5 b(U)p 2279 4362 134 4 v 2290 4453 a(@)g(x)3206 4385 y Fl(\(13\))515 4714 y(Equations)33 b(\(13\))e(ha)m(v)m(e)j(a)f (\014rst)g(in)m(tegral)1164 4868 y Fe(@)5 b(L)p 1164 4912 124 4 v 1169 5003 a(@)23 b Fl(_)-45 b Fe(x)1314 4935 y Fl(_)h Fe(x)23 b Fj(\000)f Fe(L)28 b Fl(=)1890 4868 y(1)p 1682 4912 465 4 v 1682 4932 a Fc(p)p 1781 4932 366 4 v 1781 5017 a Fl(1)22 b Fj(\000)h(k)17 b Fl(_)-44 b Fe(x)p Fj(k)2107 4989 y Ff(2)2179 4935 y Fl(+)22 b Fe(U)38 b Fl(=)27 b Fe(const;)1905 5255 y Fn(12)p eop %%Page: 13 13 13 12 bop 515 523 a Fl(as)39 b(the)g(Lagrange)f(function)h(do)s(es)g (not)g(dep)s(end)h(explicitly)d(on)h(time)g(\(and)515 639 y(th)m(us)33 b(the)g(energy)h(is)e(preserv)m(ed\).)515 839 y(W)-8 b(e)28 b(denote)h(1)13 b Fj(\000)g(k)j Fl(_)-43 b Fe(x)p Fj(k)1295 802 y Ff(2)1362 839 y Fl(=)27 b Fe(w)s Fl(,)h Fe(w)j(>)c Fl(0.)42 b(Di\013eren)m(tiating)25 b(the)j(energy)h(in)m(tegral,)515 955 y(w)m(e)k(get)1528 1119 y Fj(\000)1726 1052 y Fl(_)-58 b Fe(w)p 1615 1096 232 4 v 1615 1187 a Fl(2)p Fe(w)1737 1159 y Ff(3)p Fh(=)p Ff(2)1879 1119 y Fl(+)1987 1052 y Fe(@)5 b(U)p 1987 1096 134 4 v 1998 1187 a(@)g(x)2147 1119 y Fl(_)-44 b Fe(x)28 b Fl(=)g(0)840 b(\(14\))515 1390 y(Supp)s(ose)39 b(no)m(w)h(that)e(the) h(particle)e(v)m(elo)s(cit)m(y)i(v)-5 b(alue)38 b Fj(k)16 b Fl(_)-43 b Fe(x)p Fj(k)38 b Fl(is)g(close)h(to)f(1,)i(or,)515 1506 y(whic)m(h)35 b(is)f(the)h(same,)g Fe(w)e(<<)e Fl(1.)49 b(As)35 b(b)s(oth)g(the)g(v)m(elo)s(cities)e(and)i(the)g(co)s(ordi-)515 1622 y(nates)i(are)f(b)s(ounded)i(\(as)e(w)m(e)i(supp)s(ose)g(that)e (the)h(motion)d(o)s(ccurs)j(in)f(some)515 1738 y(compact)c(region\),)g (the)h(time)e(deriv)-5 b(ativ)m(e)63 b(_)-58 b Fe(w)35 b Fl(is)d(b)s(ounded)h(b)m(y)1745 1915 y(_)-57 b Fe(w)30 b Fj(\024)e Fe(C)7 b(w)2070 1874 y Ff(3)p Fh(=)p Ff(2)3206 1915 y Fl(\(15\))515 2091 y(for)32 b(some)g(constan)m(t)i Fe(C)7 b Fl(.)515 2290 y(Equations)33 b(\(13\))e(can)i(b)s(e)g (re-written)f(as)1494 2521 y(\177)-55 b Fe(x)22 b Fj(\000)1730 2453 y Fl(_)-58 b Fe(w)p 1675 2498 122 4 v 1675 2589 a Fl(2)p Fe(w)1823 2521 y Fl(_)-44 b Fe(x)28 b Fl(=)g Fj(\000)2070 2444 y(p)p 2153 2444 73 4 v 77 x Fe(w)2235 2453 y(@)5 b(U)p 2235 2498 134 4 v 2246 2589 a(@)g(x)2379 2521 y(;)1460 2755 y Fl(\177)-55 b Fe(x)28 b Fl(=)1641 2665 y Fc(p)p 1740 2665 366 4 v 1740 2755 a Fl(1)22 b Fj(\000)h(k)16 b Fl(_)-43 b Fe(x)p Fj(k)2066 2727 y Ff(2)2105 2755 y Fe(G)p Fl(\()17 b(_)-44 b Fe(x)q(;)17 b(x)p Fl(\))p Fe(;)766 b Fl(\(16\))515 2908 y(where)34 b(the)f(functions)f Fe(G)h Fl(are)f(smo)s(oth)g(and)h(b)s(ounded.)515 3107 y(If)40 b(w)m(e)i(formally)c(substitute)j Fe(w)j Fl(=)d(1)27 b Fj(\000)i(k)16 b Fl(_)-43 b Fe(x)p Fj(k)2197 3071 y Ff(2)2277 3107 y Fl(=)42 b(0)e(in)m(to)g(these)i(equations,)515 3223 y(then)30 b(w)m(e)g(get)36 b(\177)-55 b Fe(x)28 b Fl(=)g(0,)i Fj(k)16 b Fl(_)-43 b Fe(x)p Fj(k)27 b Fl(=)h(1,)i(whic)m (h)g(corresp)s(onds)h(to)e(the)h(motion)d(of)j(the)515 3340 y(ligh)m(t)h(particles)g(in)h(the)h(sp)s(ecial)f(theory)h(of)f (relativit)m(y)-8 b(.)515 3539 y Fa(Remark.)99 b Fl(The)53 b(same)e(considerations)g(w)m(ork)i(also)d(for)h(non-p)s(oten)m(tial) 515 3655 y(forces)40 b(and)g(the)g(electro-magnetic)e(\014elds.)65 b(Indeed,)44 b(our)39 b(estimates)h(w)m(ere)515 3771 y(based)33 b(on)g(the)g(energy)g(relation)1704 3928 y Fe(dE)p 1704 3973 129 4 v 1725 4064 a(dt)1870 3996 y Fl(=)28 b Fe(F)35 b Fj(\001)22 b Fe(v)t(;)515 4226 y Fl(\(where)44 b Fe(E)52 b Fl(=)46 b(1)p Fe(=)1189 4141 y Fc(p)p 1288 4141 361 4 v 1288 4226 a Fl(1)22 b Fj(\000)h(k)p Fe(v)t Fj(k)1610 4197 y Ff(2)1692 4226 y Fl(is)43 b(the)h(particle)e (kinetic)h(energy)h(and)f Fe(F)57 b Fl(is)515 4342 y(the)47 b(force\),)k(whic)m(h)c(is)g(also)f(true)h(in)f(general)h(for)f(non-p)s (oten)m(tial)f(forces.)515 4458 y(The)39 b(electro-magnetic)e(\014eld)h (is)h(de\014ned)h(b)m(y)f(adding)f(some)h(terms)f(to)h(the)515 4575 y(Lagrange)46 b(function)g(that)g(are)h(linear)d(in)i(the)h(v)m (elo)s(cit)m(y)f Fe(v)t Fl(,)k(but)d(one)g(can)515 4691 y(easily)30 b(sho)m(w)h(that)g(the)g(equations)g(of)f(the)h(particle)e (motion)g(are)h(still)e(of)j(the)515 4807 y(form)g(\()i(16\),)f(see,)h (e.g.,)66 b([5],)32 b(Section)h(17.)515 5006 y(No)m(w)g(let)f Fe(K)39 b Fl(b)s(e)33 b(the)g(v)m(elo)s(cit)m(y)g(phase-space)h(of)e (the)h(generalized)f(billiard.)1905 5255 y Fn(13)p eop %%Page: 14 14 14 13 bop 515 523 a Fa(Lemma)37 b(2.3)42 b Fk(The)j(set)g Fj(M)j Fl(=)f Fj(f)p Fl(\()p Fe(x;)17 b(v)t Fl(\))48 b Fj(2)g Fe(K)55 b Fl(:)47 b Fj(k)p Fe(v)t Fj(k)g Fl(=)h(1)p Fj(g)d Fk(is)h(invariant)515 639 y(under)34 b(the)h(dynamics)f(of)h (the)g(gener)-5 b(alize)g(d)33 b(r)-5 b(elativistic)35 b(bil)5 b(liar)-5 b(d.)515 922 y Fa(Pro)s(of)40 b Fl(As)g(the)h (particle)e(v)m(elo)s(cit)m(y)h(equals)g(the)h(v)m(elo)s(cit)m(y)f(of)f (ligh)m(t,)i(an)f(ex-)515 1038 y(ternal)f(force)h(cannot)h(c)m(hange)g (the)f(v)m(elo)s(cit)m(y)g(v)m(ector)i(\(spatial\))c(length.)66 b(It)515 1154 y(follo)m(ws)34 b(from)g(the)i(v)m(elo)s(cit)m(y)f (transformation)e(relations)h(\(12\))h(that)g Fj(j)p Fe(v)3169 1169 y Fh(\027)3212 1154 y Fj(j)3240 1118 y Ff(2)3303 1154 y Fl(+)515 1270 y Fj(k)p Fe(v)612 1285 y Fh(\034)655 1270 y Fj(k)705 1234 y Ff(2)773 1270 y Fl(=)28 b(1)33 b(if)f(and)h(only)f(if)g Fj(j)p Fe(v)1622 1234 y Fd(0)1618 1295 y Fh(\027)1661 1270 y Fj(j)1689 1234 y Ff(2)1751 1270 y Fl(+)22 b Fj(k)p Fe(v)1950 1234 y Fd(0)1946 1295 y Fh(\034)1989 1270 y Fj(k)2039 1234 y Ff(2)2107 1270 y Fl(=)28 b(1,)33 b(whic)m(h)h(pro)m(v)m(es)h(the)e (lemma.)515 1386 y Fb(\003)515 1586 y Fl(When)27 b(a)f(particle)f(mo)m (v)m(es)i(along)e(a)h(segmen)m(t,)i(the)e(motion)f(on)h(the)g(in)m(v)-5 b(arian)m(t)515 1702 y(manifold)21 b Fj(M)j Fl(coincides)h(with)f(the)g (corresp)s(onding)h(classical)e(billiard.)37 b(This)515 1818 y(is)28 b(also)g(true)h(for)f Fe(N)40 b Fl(iden)m(tical)27 b(particles)h(mo)m(ving)f(in)h(a)h(segmen)m(t:)42 b(dynamics)515 1934 y(on)32 b(the)h(in)m(v)-5 b(arian)m(t)31 b(manifold)825 2108 y(^)775 2134 y Fj(M)d Fl(=)f Fj(f)p Fl(\()p Fe(x)1169 2149 y Ff(1)1209 2134 y Fe(;)17 b(:)g(:)g(:)f(;)h(x)1483 2149 y Fh(N)1550 2134 y Fe(;)g(v)1641 2149 y Ff(1)1680 2134 y Fe(;)g(:)g(:)g(:)f(v)1902 2149 y Fh(N)1969 2134 y Fl(\))28 b(:)g Fj(k)p Fe(v)2187 2149 y Ff(1)2226 2134 y Fj(k)g Fl(=)f(1)p Fe(;)17 b(:)g(:)g(:)f(;)h Fj(k)p Fe(v)2772 2149 y Fh(N)2839 2134 y Fj(k)27 b Fl(=)h(1)p Fj(g)515 2333 y Fl(is)k(exactly)h(the)g(motion)d(of)j(a)f(classical)f (billiard)e(in)j(an)g Fe(N)10 b Fl(-simplex.)515 2532 y(As)24 b(so)s(on)g(as)g(the)g(dimension)e(of)i(the)g(domain)e(\005)i (b)s(ecomes)g(greater)g(than)g(one,)515 2648 y(the)i(situation)e(turns) j(out)f(to)f(b)s(e)i(more)e(complicated:)38 b(the)27 b(dynamics)f(on)f Fj(M)515 2765 y Fl(is)h(in)f(some)h(sense)j (dissipativ)m(e,)e(for)f(example,)h(it)e(do)s(es)i(not)f(admit)f(a)h (smo)s(oth)515 2881 y(in)m(v)-5 b(arian)m(t)31 b(measure.)44 b(W)-8 b(e)33 b(giv)m(e)f(a)h(pro)s(of)e(for)h(a)h(v)m(ery)h(simple)d (situation.)515 3163 y Fa(Prop)s(osition)k(2.1)42 b Fk(Supp)-5 b(ose)27 b(that)i(the)f(b)-5 b(oundary)28 b Fl(\000)g Fk(c)-5 b(ontains)28 b(two)g(p)-5 b(ar)g(al)5 b(lel)515 3279 y(se)-5 b(gments)31 b(\(thus)i(ther)-5 b(e)33 b(is)f(a)h(c)-5 b(ontinuum)32 b(of)g(p)-5 b(erio)g(dic)32 b(tr)-5 b(aje)g(ctories)32 b(on)h Fj(M)p Fk(\),)515 3396 y(and)g(supp)-5 b(ose)34 b(that)h(the)f("wal)5 b(l)34 b(velo)-5 b(city")33 b(function)h Fe(V)56 b Fk(do)-5 b(es)34 b(not)g(dep)-5 b(end)33 b(on)515 3512 y(time,)28 b(and)e(is)g(p)-5 b(ositive)26 b(on)h(these)f(se)-5 b(gments.)41 b(Then)26 b(the)h(gener)-5 b(alize)g(d)25 b(bil)5 b(liar)-5 b(d)515 3628 y(on)34 b Fj(M)h Fk(do)-5 b(es)34 b(not)h(admit)f(a)h(smo)-5 b(oth)34 b(invariant)g(me)-5 b(asur)g(e.)515 3910 y Fa(Pro)s(of.)66 b Fl(Consider)41 b(a)f(mapping)1816 3885 y(\026)1796 3910 y Fe(T)55 b Fl(:)41 b(\()p Fe(\015)5 b(;)17 b(v)t Fl(\))40 b Fj(!)g Fl(\()t(\026)-53 b Fe(\015)5 b(;)21 b Fl(\026)-53 b Fe(v)t Fl(\))40 b(consisting)f(of)h(t)m(w)m(o)515 4027 y(successfull)25 b(re\015ections)h(from)d(the)j(b)s(oundary)-8 b(,)26 b(whic)m(h)g(is)e(de\014ned)i(for)f(almost)515 4143 y(an)m(y)45 b(p)s(oin)m(t)f(in)g(\000)30 b Fj(\002)h Fe(S)6 b Fl(,)47 b(where)f Fe(S)54 b Fl(=)48 b Fj(fk)p Fe(v)t Fj(k)g Fl(=)g(1)p Fe(;)17 b(v)2488 4158 y Fh(\027)2579 4143 y Fe(>)48 b Fl(0)p Fj(g)c Fl(is)g(the)h(set)g(of)515 4259 y("admissible)27 b(v)m(elo)s(cities".)41 b(Here,)31 b(as)f(ab)s(o)m(v)m(e,)h(w)m(e)f (denote)h(b)m(y)f Fe(v)2843 4274 y Fh(\027)2916 4259 y Fl(the)g(normal)515 4375 y(comp)s(onen)m(t)37 b(of)f(the)h(particle)f (v)m(elo)s(cit)m(y)g(as)h(it)f(hits)h(the)g(b)s(oundary)-8 b(,)38 b(and)f(b)m(y)515 4491 y Fe(v)562 4506 y Fh(\034)641 4491 y Fl(the)f(tangen)m(tial)e(comp)s(onen)m(t.)53 b(The)36 b(p)s(erio)s(dic)e(tra)5 b(jectories)36 b(corresp)s(ond)515 4608 y(to)c(the)h Fk(stationary)i(p)-5 b(oints)32 b Fl(of)g(the)h (mapping)2247 4583 y(\026)2228 4608 y Fe(T)14 b Fl(.)43 b(Let)33 b Fe(v)2591 4623 y Fh(\034)2662 4608 y Fl(=)28 b Fe(v)j(<<)c Fl(1.)44 b(Then)1500 4865 y Fe(v)1547 4880 y Fh(\027)1619 4865 y Fl(=)27 b(1)22 b Fj(\000)1903 4797 y Fe(v)1954 4761 y Ff(2)p 1903 4842 91 4 v 1923 4933 a Fl(2)2025 4865 y(+)g Fe(O)s Fl(\()p Fe(v)2290 4824 y Ff(4)2328 4865 y Fl(\))p Fe(:)1905 5255 y Fn(14)p eop %%Page: 15 15 15 14 bop 515 523 a Fl(Supp)s(ose)26 b(that)g(the)g(particle)e (re\015ects)j(from)d(these)j(parallel)c(segmen)m(ts)j(of)g(the)515 639 y(b)s(oundary)-8 b(.)44 b(It)32 b(follo)m(ws)f(from)h(\(8\))g(that) 828 814 y(\026)-53 b Fe(v)871 829 y Fh(\027)942 814 y Fl(=)28 b Fe(v)1093 829 y Fh(\027)1158 814 y Fl(+)22 b Fe(O)s Fl(\()p Fe(v)1423 773 y Ff(2)1461 814 y Fl(\))p Fe(;)118 b Fl(\026)-52 b Fe(v)1688 829 y Fh(\034)1759 814 y Fl(=)27 b Fe(\033)t(v)f Fl(+)c Fe(O)s Fl(\()p Fe(v)2259 773 y Ff(2)2297 814 y Fl(\))p Fe(;)121 b Fl(\026)-55 b Fe(x)28 b Fl(=)f Fe(x)c Fl(+)f Fe(O)s Fl(\()p Fe(v)t Fl(\))p Fe(;)515 990 y Fl(where)34 b Fe(x)e Fl(is)h(the)g(co)s (ordinate)e(on)i(one)g(of)f(these)h(segmen)m(ts)h(and)1444 1212 y(0)27 b Fe(<)h(\033)j Fj(\024)1816 1101 y Fc(\020)1885 1144 y Fl(1)22 b Fj(\000)h Fe(V)p 1885 1189 250 4 v 1886 1280 a Fl(1)f(+)g Fe(V)2144 1101 y Fc(\021)2204 1123 y Ff(2)2271 1212 y Fe(<)27 b Fl(1)p Fe(:)515 1424 y Fl(The)j(Jacobian)e (of)h(this)g(mapping)f(in)g(a)h(stationary)f(p)s(oin)m(t)h(is)g(ob)m (viously)g(less)515 1540 y(than)g(1,)h(th)m(us)g(there)h(cannot)e (exist)h(an)f(in)m(v)-5 b(arian)m(t)28 b(metrics)h(\(see,)i(e.g.)72 b([12]\).)515 1656 y Fb(\003)515 1856 y Fl(As)41 b(w)m(e)h(ha)m(v)m(e)g (already)f(men)m(tioned,)i(the)e(main)e(p)s(ostulate)i(of)f(the)i (general)515 1972 y(theory)c(of)g(relativit)m(y)e(is)h(that)h(the)g (gra)m(vitational)d(\014eld)j(is)f(de\014ned)j(b)m(y)f(the)515 2088 y(metric)31 b(of)h(the)h(space-time,)f(i.e.,)h(a)f(quadratic)g (form)1301 2263 y Fe(ds)1398 2222 y Ff(2)1465 2263 y Fl(=)27 b Fe(g)1615 2278 y Fh(ij)1675 2263 y Fe(dx)1781 2222 y Fh(i)1810 2263 y Fe(dx)1916 2222 y Fh(j)1952 2263 y Fe(;)115 b(i;)17 b(j)33 b Fl(=)28 b(0)17 b Fe(:)g(:)g(:)e Fl(3)515 2438 y(of)31 b(signature)h(\(+)p Fe(;)17 b Fj(\000)p Fe(;)g Fj(\000)p Fe(;)g Fj(\000)p Fl(\))33 b(in)e(ev)m(ery)j(p)s(oin)m (t,)e(see,)h(e.g.,)65 b([5],)32 b(Sec.)44 b(87.)f(The)515 2555 y(\014rst)33 b(co)s(ordinate,)f Fe(x)1278 2519 y Ff(0)1345 2555 y Fl(=)c Fe(t)p Fl(,)33 b(will)d(b)s(e)j(re\013ered)g (to)g(as)f(the)h("w)m(orld)g(time".)42 b(W)-8 b(e)515 2671 y(assume)33 b(that)f Fe(g)1110 2686 y Ff(00)1212 2671 y Fe(>)c Fl(0.)515 2870 y(The)41 b(motion)e(of)h(a)g(particle)g (in)g(the)h(gra)m(vitational)c(\014eld)j(is)h(describ)s(ed)g(b)m(y)515 2986 y(the)33 b(geo)s(desics)g(of)g(this)f(metric,)g(th)m(us)i(the)f (length)g(of)f(the)i(particle)d(v)m(elo)s(cit)m(y)515 3103 y(4-v)m(ector)i(is)f(a)g(constan)m(t)i(of)e(motion:)1358 3207 y Fc(\020)1428 3250 y Fe(dx)p 1428 3294 107 4 v 1433 3386 a(ds)1544 3317 y(;)1598 3250 y(dx)p 1598 3294 V 1603 3386 a(ds)1714 3207 y Fc(\021)1801 3317 y Fl(=)27 b Fe(g)1951 3332 y Fh(ij)2022 3250 y Fe(dx)2128 3214 y Fh(i)p 2022 3294 135 4 v 2040 3386 a Fe(ds)2176 3250 y(dx)2282 3214 y Fh(j)p 2176 3294 143 4 v 2199 3386 a Fe(ds)2356 3317 y Fl(=)h(1)p Fe(;)515 3528 y Fl(where)f Fe(s)f Fl(is)g(a)g(natural)f(parameter)g(\(called)g(the)i("particle)d (time"\))h(along)g(the)515 3644 y(geo)s(desic.)515 3843 y(The)41 b(ligh)m(t)d(particles)h(mo)m(v)m(e)h(along)f(so)h(called)f (isotropic)f(geo)s(desics,)k(along)515 3959 y(whic)m(h)29 b(\()839 3920 y Fh(dx)p 838 3936 78 4 v 838 3994 a(d\025)925 3959 y Fe(;)980 3920 y Fh(dx)p 979 3936 V 979 3994 a(d\025)1067 3959 y Fl(\))e(=)h(0.)42 b(Here)29 b Fe(\025)g Fl(is)f(some)h(a\016ne)g (parameter)f(on)h(the)g(isotropic)515 4076 y(geo)s(desic.)70 b(Note)42 b(that)g(the)g(natural)e(parameter)h(is)h(iden)m(tically)d (constan)m(t)515 4192 y(along)31 b(isotropic)g(curv)m(es,)k(as)e Fe(ds)1714 4156 y Ff(2)1780 4192 y Fl(=)28 b Fe(g)1931 4207 y Fh(ij)1991 4192 y Fe(dx)2097 4156 y Fh(i)2125 4192 y Fe(dx)2231 4156 y Fh(j)2296 4192 y Fl(=)f(0.)515 4391 y(The)45 b(time)e(in)m(terv)-5 b(al)43 b(in)g(the)i(general)e (theory)i(of)f(relativit)m(y)e(dep)s(ends)k(not)515 4507 y(only)31 b(on)h(the)h(v)m(elo)s(cit)m(y)f(of)f(the)i(reference)g (frame)e(\(as)i(in)e(the)h(sp)s(ecial)f(theory)515 4624 y(of)37 b(relativit)m(y\),)h(but)g(also)e(on)i(the)g(p)s(osition)e(of)h (the)i(clo)s(c)m(ks)f(in)f(space.)60 b(The)515 4740 y("prop)s(er)32 b(time")f Fe(\034)44 b Fl(in)32 b(a)g(\014xed)i(p)s(oin)m(t)e(in)f (space)j(is)e(de\014ned)i(b)m(y)1608 4962 y Fe(\034)39 b Fl(=)1793 4826 y Fc(Z)1909 4895 y Fj(p)p 1992 4895 122 4 v 67 x Fe(g)2039 4977 y Ff(00)2113 4962 y Fe(dx)2219 4921 y Ff(0)2259 4962 y Fe(:)1905 5255 y Fn(15)p eop %%Page: 16 16 16 15 bop 515 523 a Fl(W)-8 b(e)45 b(shall)e(assume)i(that)g(the)g(gra) m(vitational)d(\014eld)i(is)g(constan)m(t,)49 b(i.e.,)f(the)515 639 y(functions)31 b Fe(g)981 654 y Fh(ij)1073 639 y Fl(do)g(not)h(dep)s(end)g(on)g(the)g(w)m(orld)f(time)f Fe(x)2562 603 y Ff(0)2602 639 y Fl(.)43 b(As)32 b(in)f(the)h(case)g(of) 515 756 y(the)38 b("\015at")e(metric)h Fe(ds)1369 719 y Ff(2)1443 756 y Fl(=)f(\()p Fe(dx)1699 719 y Ff(0)1739 756 y Fl(\))1777 719 y Ff(2)1841 756 y Fj(\000)26 b Fl(\()p Fe(dx)2088 719 y Ff(1)2128 756 y Fl(\))2166 719 y Ff(2)2231 756 y Fj(\000)f Fl(\()p Fe(dx)2477 719 y Ff(2)2517 756 y Fl(\))2555 719 y Ff(2)2620 756 y Fj(\000)h Fl(\()p Fe(dx)2867 719 y Ff(3)2906 756 y Fl(\))2944 719 y Ff(2)2983 756 y Fl(,)39 b(one)f(can)515 872 y(de\014ne)c(the)f(three-dimensional) d(particle)h(v)m(elo)s(cit)m(y)i Fe(v)j Fl(as)1124 1082 y Fe(v)1175 1041 y Fh(\013)1252 1082 y Fl(=)1687 1014 y Fe(dx)1793 978 y Fh(\013)p 1366 1059 798 4 v 1366 1088 a Fj(p)p 1449 1088 122 4 v 62 x Fe(g)1496 1165 y Ff(00)1570 1150 y Fl(\()p Fe(dx)1714 1121 y Ff(0)1776 1150 y Fl(+)22 b Fe(g)1921 1165 y Fh(\013)1970 1150 y Fe(dx)2076 1121 y Fh(\013)2126 1150 y Fl(\))2173 1082 y Fe(;)115 b(\013)28 b Fl(=)g(1)p Fe(;)17 b Fl(2)p Fe(;)g Fl(3)p Fe(;)435 b Fl(\(17\))515 1312 y(see)80 b([5],)41 b(Sec.)65 b(89.)e(Here)40 b(w)m(e)h(denoted)f Fe(g)2120 1327 y Fh(\013)2209 1312 y Fl(=)f Fe(g)2371 1327 y Ff(0)p Fh(\013)2455 1312 y Fe(=g)2551 1327 y Ff(00)2626 1312 y Fl(.)64 b(This)39 b(v)m(elo)s(cit)m(y)g(is)515 1429 y(measured)44 b(in)f(terms)h(of)g (the)g("prop)s(er)g(time",)h(i.e.,)i(the)d(clo)s(c)m(ks)g(are)g(syn-) 515 1545 y(cronized)f(along)f(the)i(tra)5 b(jectory)44 b(of)f(the)h(particle.)75 b(If)43 b(the)h(gra)m(vitational)515 1661 y(\014eld)d(is)g(static,)j(i.e.,)f(the)f(functions)g Fe(g)1996 1676 y Ff(0)p Fh(\013)2123 1661 y Fl(=)h(0)f(and)f Fe(ds)2628 1625 y Ff(2)2711 1661 y Fl(=)h Fe(g)2876 1676 y Ff(00)2951 1661 y Fe(dx)3057 1625 y Ff(0)3125 1661 y Fj(\000)29 b Fe(dl)3313 1625 y Ff(2)3352 1661 y Fl(,)515 1777 y(where)38 b Fj(\000)p Fe(dl)960 1741 y Ff(2)1034 1777 y Fl(=)d Fe(g)1192 1792 y Fh(ab)1263 1777 y Fe(dx)1369 1741 y Fh(a)1411 1777 y Fe(dx)1517 1741 y Fh(b)1551 1777 y Fl(,)j Fe(a;)17 b(b)35 b Fl(=)f(1)p Fe(;)17 b Fl(2)p Fe(;)g Fl(3,)37 b(then)g(this)f(expression)i(for)f(the)515 1894 y(v)m(elo)s(cit)m(y)32 b(giv)m(es)h Fe(v)f Fl(=)27 b Fe(dl)r(=d\034)11 b Fl(.)515 2093 y(The)33 b(four-dimensional)d(in)m (terv)-5 b(al)31 b Fe(ds)1905 2057 y Ff(2)1976 2093 y Fl(can)i(b)s(e)g(expressed)i(as)1233 2281 y Fe(ds)1330 2240 y Ff(2)1396 2281 y Fl(=)28 b Fe(g)1547 2296 y Ff(00)1621 2281 y Fl(\()p Fe(dx)1765 2240 y Ff(0)1827 2281 y Fl(+)22 b Fe(g)1972 2296 y Fh(\013)2021 2281 y Fe(dx)2127 2240 y Fh(\013)2177 2281 y Fl(\))2215 2240 y Ff(2)2254 2170 y Fc(\020)2314 2281 y Fl(1)g Fj(\000)g Fe(v)2535 2240 y Ff(2)2574 2170 y Fc(\021)2634 2281 y Fe(;)515 2482 y Fl(where)34 b Fe(v)848 2446 y Ff(2)914 2482 y Fl(=)28 b Fj(\000)p Fe(g)1142 2497 y Fh(ab)1214 2482 y Fe(v)1265 2446 y Fh(a)1306 2482 y Fe(v)1357 2446 y Fh(b)1391 2482 y Fl(,)33 b Fe(a;)17 b(b)28 b Fl(=)g(1)p Fe(;)17 b Fl(2)p Fe(;)g Fl(3.)515 2681 y(The)33 b(particle)f(energy)h(giv)m(en)g(b)m(y) 1653 2897 y Fj(E)1706 2912 y Ff(0)1773 2897 y Fl(=)1956 2762 y Fj(p)p 2039 2762 V 63 x Fe(g)2086 2840 y Ff(00)p 1887 2874 344 4 v 1887 2894 a Fj(p)p 1970 2894 261 4 v 82 x Fl(1)22 b Fj(\000)g Fe(v)2191 2947 y Ff(2)3206 2897 y Fl(\(18\))515 3113 y(is)32 b(a)g(constan)m(t)i(of)e(motion.)515 3312 y(The)g(de\014nition)e(of)h(the)h(generalized)f(billiard)d (re\015ection)j(from)f(the)i(b)s(ound-)515 3428 y(ary)42 b(remains)e(almost)g(the)i(same:)62 b(the)42 b(normal)e(should)h(b)s(e) h(tak)m(en)h(in)e(the)515 3545 y(metric)32 b Fe(g)866 3560 y Fh(ij)927 3545 y Fl(,)i(and)g(to)f(obtain)g(the)h(new)g(v)m(elo) s(cit)m(y)g(one)g(should)g(mak)m(e)f(a)h(co)s(or-)515 3661 y(dinate)k(transformation)f(in)h(the)h(neigh)m(b)s(ourho)s(o)s(d)f (of)h(the)g(collision)d(p)s(oin)m(t,)515 3777 y(suc)m(h)e(that)e(in)g (this)g(p)s(oin)m(t)g(the)h(metric)e(b)s(ecomes)i("\015at":)1109 3939 y Fe(ds)1206 3898 y Ff(2)1273 3939 y Fl(=)27 b(\()p Fe(dx)1520 3898 y Ff(0)1560 3939 y Fl(\))1598 3898 y Ff(2)1659 3939 y Fj(\000)c Fl(\()p Fe(dx)1903 3898 y Ff(1)1942 3939 y Fl(\))1980 3898 y Ff(2)2042 3939 y Fj(\000)f Fl(\()p Fe(dx)2285 3898 y Ff(2)2325 3939 y Fl(\))2363 3898 y Ff(2)2424 3939 y Fj(\000)h Fl(\()p Fe(dx)2668 3898 y Ff(3)2708 3939 y Fl(\))2746 3898 y Ff(2)515 4101 y Fl(and)32 b(use)i(form)m(ulas)d(\(8\).)515 4476 y Fm(3)134 b(The)45 b(main)g(theorem)515 4774 y Fl(Here)35 b(w)m(e)g(form)m(ulate) e(and)h(pro)m(v)m(e)h(a)f(general)g(theorem,)g(whic)m(h)h(is)f(the)h (main)515 4890 y(to)s(ol)i(in)h(the)h(pro)s(ofs)f(of)h(the)g(un)m(b)s (ounded)h(energy)g(grouth)e(for)h(generalized)515 5006 y(relativistic)30 b(billiards.)1905 5255 y Fn(16)p eop %%Page: 17 17 17 16 bop 515 523 a Fl(Let)45 b Fe(M)55 b Fl(b)s(e)45 b(a)f(smo)s(oth)g Fe(n)p Fl(-dimensional)e(manifold,)j(whic)m(h)g(is)f (\(lo)s(cally\))e(a)515 639 y(direct)c(pro)s(duct)g(of)f(a)h(circle)f Fe(S)1697 603 y Ff(1)1775 639 y Fl(and)h(an)g Fe(n)26 b Fj(\000)g Fl(1-dimensional)35 b(disk)j Fe(D)3215 603 y Fh(n)p Fd(\000)p Ff(1)3352 639 y Fl(.)515 756 y(F)-8 b(or)37 b(example)h(one)h(can)f(consider)h(a)f(smo)s(oth)g(\014b)s(er)h (bundle)f Fe(\031)k Fl(:)37 b Fe(M)49 b Fj(!)37 b Fe(B)5 b Fl(,)515 872 y Fe(D)599 836 y Fh(n)p Fd(\000)p Ff(1)763 872 y Fj(2)28 b Fe(B)5 b Fl(,)29 b(and)f(the)g(\014b)s(ers)g(are)g(the) f(one-dimensional)e(circles.)42 b(Let)28 b Fe(t;)17 b(w)s(;)g(x)515 988 y Fl(b)s(e)41 b(the)h(lo)s(cal)d(co)s(ordinates)i(on)g Fe(M)10 b Fl(,)45 b(where)d Fe(t)100 b Fl(\(mo)s(d)32 b(1\))43 b Fj(2)g Fe(S)2907 952 y Ff(1)2946 988 y Fl(,)g Fe(w)i Fj(2)f Fi(R)3306 952 y Ff(1)3352 988 y Fl(,)515 1104 y Fe(x)28 b Fj(2)g Fi(R)758 1068 y Fh(n)p Fd(\000)p Ff(2)901 1104 y Fl(.)515 1303 y(Consider)39 b(a)f(smo)s(oth)g(mapping)e Fe(T)52 b Fl(:)38 b Fe(t;)17 b(w)s(;)g(x)37 b Fj(!)h Fe(t)2407 1267 y Fd(0)2430 1303 y Fe(;)17 b(w)2547 1267 y Fd(0)2569 1303 y Fe(;)g(x)2668 1267 y Fd(0)2730 1303 y Fl(of)38 b(the)h(compact)515 1420 y(domain)d Fe(S)933 1384 y Ff(1)998 1420 y Fj(\002)26 b Fe(D)1185 1384 y Fh(n)p Fd(\000)p Ff(1)1360 1420 y Fl(to)37 b(itself.)58 b(Supp)s(ose)39 b(that)f(the)g(mapping)f(lea)m(v)m(es)i(the)515 1536 y(manifold)30 b Fe(w)g Fl(=)e(0)33 b(in)m(v)-5 b(arian)m(t,)31 b(and)i(on)g(this)g(manifold)d(the)j(restriction)f(of)g Fe(T)515 1652 y Fl(to)g(the)h(\014b)s(ers)g Fe(S)1131 1616 y Ff(1)1203 1652 y Fl(is)f(the)h(rotation)e(of)h(the)h(circle:) 1167 1825 y Fe(T)41 b Fl(:)28 b(\()p Fe(t;)17 b Fl(0)p Fe(;)g(x)p Fl(\))28 b Fj(!)f Fl(\()p Fe(t)22 b Fl(+)g Fe(l)102 b Fl(\(mo)s(d)32 b(1\))p Fe(;)17 b Fl(0)p Fe(;)g(x)2639 1784 y Fd(0)2662 1825 y Fl(\))p Fe(:)515 1998 y Fl(W)-8 b(e)38 b(also)f(assume)h(that)f(in)g(the)h(neigh)m(b)s(ourho)s(o)s(d)f (of)g(the)h(manifold)d Fe(w)k Fl(=)d(0)515 2115 y(the)d(mapping)e Fe(T)46 b Fl(can)33 b(b)s(e)g(writen)f(as)924 2288 y Fe(w)997 2246 y Fd(0)1048 2288 y Fl(=)27 b Fe(A)p Fl(\()p Fe(t)p Fl(\))p Fe(w)e Fl(+)d Fe(B)1602 2303 y Ff(1)1642 2288 y Fl(\()p Fe(t;)17 b(w)s(;)g(x)p Fl(\))p Fe(;)113 b Fj(j)p Fe(B)2211 2303 y Ff(1)2250 2288 y Fj(j)28 b(\024)g Fe(c)2453 2303 y Ff(1)2492 2288 y Fj(j)p Fe(w)s Fj(j)2621 2246 y Ff(1+)p Fh(\013)2760 2288 y Fe(;)1188 2429 y(t)1223 2388 y Fd(0)1274 2429 y Fl(=)f Fe(t)c Fl(+)f Fe(l)i Fl(+)e Fe(B)1758 2444 y Ff(2)1798 2429 y Fl(\()p Fe(t;)17 b(w)s(;)g(x)p Fl(\))p Fe(;)113 b Fj(j)p Fe(B)2367 2444 y Ff(2)2407 2429 y Fj(j)27 b(\024)h Fe(c)2609 2444 y Ff(2)2649 2429 y Fj(j)p Fe(w)s Fj(j)p Fe(;)401 b Fl(\(19\))515 2602 y(where)34 b(the)f(constan)m(ts)h Fe(c)1439 2617 y Ff(1)1478 2602 y Fl(,)f Fe(c)1580 2617 y Ff(2)1652 2602 y Fl(and)g Fe(\013)h Fl(are)f(p)s(ositiv)m(e.)43 b(All)31 b(the)i(functions)g(are) 515 2718 y(1-p)s(erio)s(dic)d(in)i Fe(t)p Fl(.)515 2917 y(W)-8 b(e)27 b(denote)h(b)m(y)g Fe(T)1187 2881 y Fh(n)1233 2917 y Fl(\()p Fe(w)s Fl(\),)g Fe(T)1508 2881 y Fh(n)1555 2917 y Fl(\()p Fe(t)p Fl(\))f(and)g Fe(T)1948 2881 y Fh(n)1994 2917 y Fl(\()p Fe(x)p Fl(\))h(the)f Fe(w)s Fl(-,)g Fe(t)p Fj(\000)h Fl(and)f Fe(x)p Fj(\000)p Fl(co)s(ordinate)515 3034 y(of)32 b(the)h Fe(n)852 2997 y Fh(th)955 3034 y Fl(p)s(o)m(w)m(er)g(of)g(the)g(mapping)e Fe(T)14 b Fl(\()p Fe(t;)j(w)s(;)g(x)p Fl(\))26 b Fj(!)i Fl(\()p Fe(t)2545 2997 y Fd(0)2568 3034 y Fe(;)17 b(w)2685 2997 y Fd(0)2707 3034 y Fe(;)g(x)2806 2997 y Fd(0)2830 3034 y Fl(\).)515 3290 y Fa(Theorem)37 b(3.1)42 b Fk(L)-5 b(et)1454 3371 y Fc(Z)1553 3398 y Ff(1)1509 3597 y(0)1609 3507 y Fl(ln)16 b Fe(A)p Fl(\()p Fe(t)p Fl(\))p Fe(dt)28 b Fl(=)g Fj(\000)p Fe(\016)k(<)27 b Fl(0)p Fe(:)515 3754 y Fk(Then)35 b(for)h(any)1120 3728 y Fl(~)1115 3754 y Fe(\016)e(>)d Fl(0)p Fk(,)1418 3728 y Fl(~)1414 3754 y Fe(\016)j(<)c(\016)40 b Fk(ther)-5 b(e)36 b(exists)g(a)g(c)-5 b(onstant)36 b Fe(N)41 b Fj(2)30 b Fi(N)52 b Fk(such)36 b(that)515 3871 y(for)g(any)h Fe(l)d Fj(6)p Fl(=)1041 3827 y Fh(p)p 1041 3848 36 4 v 1042 3905 a(q)1086 3871 y Fk(,)k Fe(p;)17 b(q)35 b Fj(2)c Fi(N)9 b Fk(,)43 b Fe(q)36 b Fj(\024)31 b Fe(N)48 b Fk(ther)-5 b(e)37 b(is)f(a)h(c)-5 b(onstant)36 b Fe(!)f(>)c Fl(0)37 b Fk(such)g(that)515 4001 y(for)d(any)h Fe(w)s Fk(,)f Fj(j)p Fe(w)s Fj(j)27 b(\024)h Fe(!)t Fk(,)34 b(the)h(estimate)1527 4194 y Fj(j)p Fe(T)1626 4153 y Fh(n)1672 4194 y Fl(\()p Fe(w)s Fl(\))p Fj(j)27 b(\024)h Fe(C)7 b(e)2103 4153 y Fd(\000)2161 4136 y Ff(~)2158 4153 y Fh(\016)s(n)2239 4194 y Fj(j)p Fe(w)s Fj(j)515 4367 y Fk(holds)34 b(for)g(some)h(c)-5 b(onstant)34 b Fe(C)42 b Fk(for)35 b(any)f Fe(n)28 b Fj(2)g Fi(N)9 b Fk(.)515 4623 y Fa(Pro)s(of.)90 b Fl(Consider)49 b(the)f("linearized")e (mapping)2502 4598 y(~)2483 4623 y Fe(T)68 b Fl(:)54 b(\()p Fe(w)s(;)17 b(t)p Fl(\))54 b Fj(!)g Fl(\()20 b(~)-69 b Fe(w)r(;)3280 4605 y Fl(~)3279 4623 y Fe(t)p Fl(\),)515 4740 y(where)1476 4856 y(~)g Fe(w)30 b Fl(=)e Fe(A)p Fl(\()p Fe(t)p Fl(\))p Fe(w)s(;)2059 4837 y Fl(~)2058 4856 y Fe(t)f Fl(=)h Fe(t)22 b Fl(+)g Fe(l)r(;)515 5006 y Fl(whic)m(h)33 b(is)f(a)g(sk)m(ew)j(pro)s(duct)e(of)f(a)g(circle)g (rotation)f(an)h(a)g(linear)f(mapping.)1905 5255 y Fn(17)p eop %%Page: 18 18 18 17 bop 515 535 a Fa(Lemma)37 b(3.1)42 b Fk(F)-7 b(or)25 b(any)1452 509 y Fl(^)1447 535 y Fe(\016)32 b(>)c Fl(0)p Fk(,)1736 509 y Fl(^)1732 535 y Fe(\016)k(<)27 b(\016)j Fk(ther)-5 b(e)26 b(exists)g(such)f(c)-5 b(onstant)26 b Fe(N)38 b Fj(2)28 b Fi(N)9 b Fk(,)515 652 y(that)35 b(for)g(any)f Fe(l)c Fj(6)p Fl(=)1229 608 y Fh(p)p 1229 629 36 4 v 1230 686 a(q)1274 652 y Fk(,)35 b Fe(p;)17 b(q)31 b Fj(2)d Fi(N)9 b Fk(,)41 b Fe(q)32 b Fj(\024)c Fe(N)10 b Fk(,)35 b(ther)-5 b(e)35 b(exists)f Fe(m)28 b Fj(2)g Fi(N)50 b Fk(such)35 b(that)1546 886 y Fj(j)1593 861 y Fl(~)1574 886 y Fe(T)1645 845 y Fh(m)1711 886 y Fl(\()p Fe(w)s Fl(\))p Fj(j)27 b(\024)h Fe(e)2065 845 y Fd(\000)2123 827 y Ff(^)2120 845 y Fh(\016)r(m)2220 886 y Fj(j)p Fe(w)s Fj(j)515 1076 y Fk(for)34 b(any)h Fe(w)s Fk(.)515 1349 y Fa(Pro)s(of)44 b(of)g(Lemma)88 b(3.1)38 b Fl(If)g Fe(l)j Fl(is)d(an)g(irrational)d(n)m(um)m(b)s(er,)40 b(then)f(Lemma)515 1465 y(3.1)c(follo)m(ws)g(from)g(the)h(corresp)s (onding)g(result)g(in)72 b([8])36 b(\(Chapter)g(1,)h(Sec.3,)515 1581 y(Lemma)43 b(3\):)66 b(for)44 b(an)g(irrational)d(rotation)i(n)m (um)m(b)s(er)h(the)h(rotation)e(of)h(the)515 1697 y(circle)28 b Fe(t)g Fj(!)f Fe(t)15 b Fl(+)g Fe(l)101 b Fl(\(mo)s(d)33 b(1\))28 b(is)h(a)f(uniquely)h(ergo)s(dic)f(mapping,)h(and)g(b)m(y)h (the)515 1814 y(ergo)s(dic)25 b(theorem,)i(the)g(sum)f(\(in)f(the)i (logarithmic)22 b(co)s(ordinates\))k(con)m(v)m(erges)515 1930 y(uniformly)33 b(to)j(this)f Fe(\016)t Fl(.)53 b(Note)36 b(that)g(in)f(this)g(is)g(a)h(stronger)g(result)g(than)g(the)515 2046 y(one)d(of)f(the)h(lemma)d(3.1:)43 b(for)32 b(an)m(y)h(su\016cien) m(tly)h(large)d Fe(m)d Fj(2)g Fi(N)1429 2236 y Fj(j)1476 2211 y Fl(~)1457 2236 y Fe(T)1528 2195 y Fh(m)1594 2236 y Fl(\()p Fe(w)s Fl(\))p Fj(j)f(\024)h Fe(e)1948 2195 y Fd(\000)p Fh(\016)r(m)p Ff(+)p Fh(o)p Ff(\()p Fh(m)p Ff(\))2310 2236 y Fj(j)p Fe(w)s Fj(j)p Fe(;)515 2426 y Fl(where)34 b(the)f(function)f Fe(o)p Fl(\()p Fe(m)p Fl(\))p Fe(=m)27 b Fj(!)h Fl(0,)k(as)h Fe(m)28 b Fj(!)f(1)p Fl(.)515 2625 y(Supp)s(ose)33 b(no)m(w)h(that)e Fe(l)e Fl(=)1484 2581 y Fh(p)p 1484 2602 V 1485 2660 a(q)1530 2625 y Fl(.)515 2833 y(Consider)j(the)g(sum)1566 2894 y Fh(q)r Fd(\000)p Ff(1)1556 2928 y Fc(X)1564 3140 y Fh(k)r Ff(=0)1717 3023 y Fl(ln)15 b Fe(A)p Fl(\()p Fe(t)23 b Fl(+)2091 2955 y Fe(k)s(p)p 2091 3000 104 4 v 2119 3091 a(q)2204 3023 y Fl(\))2252 2955 y(1)p 2252 3000 49 4 v 2253 3091 a Fe(q)2311 3023 y(:)515 3272 y Fl(It)32 b(is)g(easy)i(to)e(see)i(that)e(it)g(is)g(exactly)h(an)g(in)m(tegral)d (sum)1590 3426 y Fh(q)r Fd(\000)p Ff(1)1581 3460 y Fc(X)1588 3672 y Fh(k)r Ff(=0)1741 3555 y Fl(ln)16 b Fe(A)p Fl(\()p Fe(t)22 b Fl(+)2115 3487 y Fe(k)p 2115 3532 55 4 v 2119 3623 a(q)2179 3555 y Fl(\))2227 3487 y(1)p 2227 3532 49 4 v 2228 3623 a Fe(q)2286 3555 y(;)515 3861 y Fl(of)40 b(the)h(in)m(tegral)1173 3781 y Fc(R)1239 3807 y Ff(1)1220 3896 y(0)1296 3861 y Fl(ln)15 b Fe(A)p Fl(\()p Fe(t)p Fl(\))p Fe(dt)p Fl(,)43 b(as)e(all)e(the)i Fe(q)j Fl(p)s(oin)m(ts)d Fe(t)27 b Fl(+)h Fe(pk)s(=q)103 b Fl(\(mo)s(d)32 b(1\))515 3977 y(are)d(di\013eren)m(t)h(on)g(the)g(cycle)g Fe(S)1658 3941 y Ff(1)1727 3977 y Fl(\()p Fe(k)h Fl(=)c(0)17 b Fe(:)g(:)g(:)f(q)j Fj(\000)d Fl(1\).)43 b(Th)m(us)31 b(c)m(ho)s(osing)f Fe(q)j Fl(large)515 4093 y(enough,)g(w)m(e)g(can)g (appro)m(ximate)f(the)h(in)m(tegral)1557 4211 y Fc(Z)1657 4238 y Ff(1)1613 4437 y(0)1713 4347 y Fl(ln)16 b Fe(A)p Fl(\()p Fe(t)p Fl(\))p Fe(dt)28 b Fl(=)f Fj(\000)p Fe(\016)515 4614 y Fl(with)j(an)m(y)h(giv)m(en)g(precision,)g(i.e.)42 b(for)30 b(an)m(y)h(0)d Fe(<)2284 4588 y Fl(^)2279 4614 y Fe(\016)k(<)c(\016)34 b Fl(there)d(is)g(suc)m(h)h Fe(N)38 b Fj(2)28 b Fi(N)515 4731 y Fl(that)1451 4792 y Fh(q)r Fd(\000)p Ff(1)1441 4826 y Fc(X)1449 5038 y Fh(k)r Ff(=0)1602 4920 y Fl(ln)15 b Fe(A)p Fl(\()p Fe(t)23 b Fl(+)1976 4853 y Fe(k)s(p)p 1976 4897 104 4 v 2004 4989 a(q)2089 4920 y Fl(\))2137 4853 y(1)p 2137 4897 49 4 v 2138 4989 a Fe(q)2223 4920 y Fj(\024)28 b(\000)2410 4894 y Fl(^)2405 4920 y Fe(\016)1905 5255 y Fn(18)p eop %%Page: 19 19 19 18 bop 515 523 a Fl(for)32 b(an)m(y)h Fe(t)28 b Fj(2)g Fe(S)1071 487 y Ff(1)1110 523 y Fl(,)33 b(an)m(y)g Fe(q)f(>)27 b(N)10 b Fl(.)515 722 y(No)m(w)33 b(for)f(a)g(rational)e Fe(l)g Fl(=)1503 678 y Fh(p)p 1503 699 36 4 v 1504 757 a(q)1548 722 y Fl(,)j Fe(q)e(>)d(N)10 b Fl(,)33 b(tak)m(e)g Fe(m)28 b Fl(=)g Fe(q)t Fl(.)43 b(Then)34 b(for)e(an)m(y)h Fe(w)1532 966 y Fj(j)1579 940 y Fl(~)1560 966 y Fe(T)1631 925 y Fh(m)1697 966 y Fl(\()p Fe(w)s Fl(\))p Fj(j)27 b(\024)h Fe(e)2051 925 y Fd(\000)2109 907 y Ff(^)2106 925 y Fh(\016)s(m)2207 966 y Fj(j)p Fe(w)s Fj(j)p Fe(:)515 1165 y Fb(\003)515 1364 y Fl(Consider)33 b(no)m(w)g(our)f(original)e (mapping)h Fe(T)14 b Fl(.)515 1646 y Fa(Lemma)37 b(3.2)42 b Fk(F)-7 b(or)33 b(any)i Fe(n)28 b Fj(2)g Fi(N)50 b Fk(and)34 b(for)h(any)g Fj(j)p Fe(w)s Fj(j)26 b(\024)i Fl(1)553 1846 y Fe(T)624 1805 y Fh(n)671 1846 y Fl(\()p Fe(w)s Fl(\))f(=)g Fe(A)p Fl(\()p Fe(t)c Fl(+)f(\()p Fe(n)g Fj(\000)h Fl(1\))p Fe(l)r Fl(\))p Fe(A)p Fl(\()p Fe(t)f Fl(+)g(\()p Fe(n)g Fj(\000)h Fl(2\))p Fe(l)r Fl(\))f Fj(\001)g Fe(:)17 b(:)g(:)k Fj(\001)h Fe(A)p Fl(\()p Fe(t)p Fl(\))p Fe(w)j Fl(+)d Fe(B)2940 1861 y Fh(n)2987 1846 y Fl(\()p Fe(t;)17 b(w)s(;)g(x)p Fl(\))p Fe(;)1423 2045 y Fj(j)p Fe(B)1525 2060 y Fh(n)1572 2045 y Fl(\()p Fe(t;)g(w)s(;)g(x)p Fl(\))p Fj(j)27 b(\024)h Fe(C)2129 2060 y Fh(n)2176 2045 y Fj(j)p Fe(w)s Fj(j)2305 2004 y Ff(1+)p Fh(\013)2443 2045 y Fe(;)849 2211 y(T)920 2170 y Fh(n)967 2211 y Fl(\()p Fe(t)p Fl(\))f(=)h Fe(t)22 b Fl(+)g Fe(nl)j Fl(+)d Fe(B)1653 2170 y Fh(t)1648 2236 y(n)1695 2211 y Fl(\()p Fe(t;)17 b(w)s(;)g(x)p Fl(\))p Fe(;)116 b Fj(j)p Fe(B)2272 2170 y Fh(t)2267 2236 y(n)2313 2211 y Fl(\()p Fe(t;)17 b(w)s(;)g(x)p Fl(\))p Fj(j)27 b(\024)h Fe(C)2870 2226 y Fh(n)2917 2211 y Fj(j)p Fe(w)s Fj(j)515 2377 y Fk(for)34 b(some)g(c)-5 b(onstant)35 b Fe(C)1378 2392 y Fh(n)1425 2377 y Fk(.)515 2659 y Fa(Pro)s(of)42 b(of)h(Lemma)f(3.2)37 b Fl(Pro)s(of)g(b)m(y)h(induction.)55 b(The)38 b(statemen)m(t)g(is)f(true)515 2776 y(for)32 b Fe(n)c Fl(=)f(1.)43 b(Supp)s(ose)34 b(that)e(for)g Fe(n)c Fl(=)g Fe(k)880 2975 y(T)951 2934 y Fh(k)993 2975 y Fl(\()p Fe(w)s Fl(\))f(=)h Fe(A)p Fl(\()p Fe(t)22 b Fl(+)g(\()p Fe(k)j Fj(\000)e Fl(1\))p Fe(l)r Fl(\))f Fj(\001)g Fe(:)17 b(:)g(:)k Fj(\001)h Fe(A)p Fl(\()p Fe(t)p Fl(\))p Fe(w)j Fl(+)d Fe(B)2618 2990 y Fh(k)2661 2975 y Fl(\()p Fe(t;)17 b(w)s(;)g(x)p Fl(\))p Fe(;)1353 3174 y(T)1424 3133 y Fh(k)1467 3174 y Fl(\()p Fe(t)p Fl(\))28 b(=)f Fe(t)c Fl(+)f Fe(k)s(l)i Fl(+)e Fe(B)2149 3133 y Fh(t)2144 3199 y(k)2187 3174 y Fl(\()p Fe(t;)17 b(w)s(;)g(x)p Fl(\))p Fe(;)515 3340 y Fl(and)32 b(the)h(corresp)s (onding)g(estimates)f(hold.)43 b(Then)515 3539 y Fe(T)586 3498 y Fh(k)r Ff(+1)718 3539 y Fl(\()p Fe(w)s Fl(\))27 b(=)h Fe(T)14 b Fl(\()p Fe(T)1178 3498 y Fh(k)1220 3539 y Fl(\()p Fe(w)s Fl(\)\))27 b(=)g Fe(A)p Fl(\()p Fe(T)1719 3498 y Fh(k)1762 3539 y Fl(\()p Fe(t)p Fl(\)\))p Fe(T)1982 3498 y Fh(k)2024 3539 y Fl(\()p Fe(w)s Fl(\))10 b(+)g Fe(B)2343 3554 y Ff(1)2381 3539 y Fl(\()p Fe(T)2490 3498 y Fh(k)2533 3539 y Fl(\()p Fe(t)p Fl(\))p Fe(;)17 b(T)2759 3498 y Fh(k)2801 3539 y Fl(\()p Fe(w)s Fl(\))p Fe(;)g(T)3065 3498 y Fh(k)3106 3539 y Fl(\()p Fe(x)p Fl(\)\))28 b(=)515 3739 y(=)f Fe(A)p Fl(\()p Fe(t)19 b Fl(+)e Fe(k)s(l)k Fl(+)d Fe(B)1153 3697 y Fh(t)1148 3763 y(k)1190 3739 y Fl(\()p Fe(t;)f(w)s(;)g(x)p Fl(\)\)\()p Fe(A)p Fl(\()p Fe(t)h Fl(+)g(\()p Fe(k)j Fj(\000)d Fl(1\))p Fe(l)r Fl(\))g Fj(\001)g Fe(:)f(:)g(:)g Fj(\001)h Fe(A)p Fl(\()p Fe(t)p Fl(\))p Fe(w)i Fl(+)e Fe(B)2896 3754 y Fh(k)2939 3739 y Fl(\()p Fe(t;)f(w)s(;)g(x)p Fl(\)\)+)1353 3905 y(+)p Fe(B)1503 3920 y Ff(1)1543 3905 y Fl(\()p Fe(T)1652 3864 y Fh(k)1694 3905 y Fl(\()p Fe(t)p Fl(\))p Fe(;)g(T)1920 3864 y Fh(k)1962 3905 y Fl(\()p Fe(w)s Fl(\))p Fe(;)g(T)2226 3864 y Fh(k)2268 3905 y Fl(\()p Fe(x)p Fl(\)\))28 b(=)529 4124 y(=)g Fe(A)p Fl(\()p Fe(t)22 b Fl(+)g Fe(k)s(l)r Fl(\))p Fe(A)p Fl(\()p Fe(t)h Fl(+)f(\()p Fe(k)j Fj(\000)e Fl(1\))p Fe(l)r Fl(\))f Fj(\001)g Fe(:)17 b(:)g(:)k Fj(\001)h Fe(A)p Fl(\()p Fe(t)p Fl(\))p Fe(w)j Fl(+)2304 4057 y Fe(dA)p 2304 4101 124 4 v 2323 4193 a(dt)2438 4124 y Fj(j)2466 4139 y Fh(t)p Ff(=)p Fh(\030)2584 4124 y Fe(B)2663 4083 y Fh(t)2658 4149 y(k)2700 4124 y Fl(\()p Fe(t;)17 b(w)s(;)g(x)p Fl(\))p Fe(T)3098 4083 y Fh(k)3140 4124 y Fl(\()p Fe(w)s Fl(\)+)915 4338 y(+)p Fe(A)p Fl(\()p Fe(t)22 b Fl(+)g Fe(k)s(l)r Fl(\))p Fe(B)1454 4353 y Fh(k)1497 4338 y Fl(\()p Fe(t;)17 b(w)s(;)g(x)p Fl(\))22 b(+)g Fe(B)2018 4353 y Ff(1)2057 4338 y Fl(\()p Fe(T)2166 4297 y Fh(k)2209 4338 y Fl(\()p Fe(t)p Fl(\))p Fe(;)17 b(T)2435 4297 y Fh(k)2477 4338 y Fl(\()p Fe(w)s Fl(\))p Fe(;)g(T)2741 4297 y Fh(k)2782 4338 y Fl(\()p Fe(x)p Fl(\)\))p Fe(:)515 4504 y Fl(W)-8 b(e)33 b(no)m(w)g(denote)1080 4732 y Fe(B)1154 4747 y Fh(k)r Ff(+1)1286 4732 y Fl(\()p Fe(t;)17 b(w)s(;)g(x)p Fl(\))27 b(=)1754 4665 y Fe(dA)p 1754 4709 V 1773 4800 a(dt)1888 4732 y Fj(j)1916 4747 y Fh(t)p Ff(=)p Fh(\030)2034 4732 y Fe(B)2113 4691 y Fh(t)2108 4757 y(k)2150 4732 y Fl(\()p Fe(t;)17 b(w)s(;)g(x)p Fl(\))p Fe(T)2548 4691 y Fh(k)2590 4732 y Fl(\()p Fe(w)s Fl(\)+)929 4979 y(+)p Fe(A)p Fl(\()p Fe(t)22 b Fl(+)g Fe(k)s(l)r Fl(\))p Fe(B)1468 4994 y Fh(k)1511 4979 y Fl(\()p Fe(t;)17 b(w)s(;)g(x)p Fl(\))k(+)h Fe(B)2031 4994 y Ff(1)2071 4979 y Fl(\()p Fe(T)2180 4938 y Fh(k)2222 4979 y Fl(\()p Fe(t)p Fl(\))p Fe(;)17 b(T)2448 4938 y Fh(k)2491 4979 y Fl(\()p Fe(w)s Fl(\))p Fe(;)g(T)2755 4938 y Fh(k)2796 4979 y Fl(\()p Fe(x)p Fl(\)\))1905 5255 y Fn(19)p eop %%Page: 20 20 20 19 bop 515 525 a Fl(As)683 486 y Fh(d)p 671 502 62 4 v 671 560 a(dt)742 525 y Fe(A)35 b Fl(is)f(b)s(ounded)h(\()p Fe(t)c Fj(2)g Fe(S)1617 489 y Ff(1)1656 525 y Fl(\))j(and)h Fj(j)p Fe(w)s Fj(j)p Fl(,)f Fj(k)p Fe(x)p Fj(k)g Fl(are)g(b)s(ounded,)i (there)f(exists)515 659 y(a)d(constan)m(t)1012 634 y(~)989 659 y Fe(C)1059 674 y Fh(k)r Ff(+1)1225 659 y Fl(suc)m(h)i(that)1338 859 y Fj(j)p Fe(B)1440 874 y Fh(k)r Ff(+1)1572 859 y Fl(\()p Fe(t;)17 b(w)s(;)g(x)p Fl(\))p Fj(j)27 b(\024)2081 833 y Fl(~)2059 859 y Fe(C)2129 874 y Fh(k)r Ff(+1)2262 859 y Fj(j)p Fe(w)s Fj(j)2391 817 y Ff(1+)p Fh(\013)2529 859 y Fe(:)515 1058 y Fl(The)33 b(same)g(w)m(a)m(y)g(w)m(e)h(sho)m(w)g (that)1417 1283 y Fj(j)p Fe(B)1524 1242 y Fh(t)1519 1308 y(k)r Ff(+1)1652 1283 y Fl(\()p Fe(t;)17 b(w)s(;)g(x)p Fl(\))p Fj(j)27 b(\024)2153 1235 y Fl(~)2161 1258 y(~)2139 1283 y Fe(C)2216 1298 y Fh(k)r Ff(+1)2349 1283 y Fj(j)p Fe(w)s Fj(j)515 1520 y Fl(and)32 b(tak)m(e)i Fe(C)986 1535 y Fh(k)r Ff(+1)1146 1520 y Fl(=)28 b Fe(max)p Fj(f)1514 1495 y Fl(~)1491 1520 y Fe(C)1561 1535 y Fh(k)r Ff(+1)1694 1520 y Fe(;)1752 1472 y Fl(~)1760 1495 y(~)1738 1520 y Fe(C)1815 1535 y Fh(k)r Ff(+1)1948 1520 y Fj(g)p Fl(.)43 b Fb(\003)515 1732 y Fl(W)-8 b(e)34 b(no)m(w)h(tak)m(e)1107 1706 y(^)1102 1732 y Fe(\016)g Fl(=)30 b(\()p Fe(\016)d Fl(+)1498 1706 y(~)1493 1732 y Fe(\016)t Fl(\))p Fe(=)p Fl(2.)48 b(By)35 b(Lemmas)e(1)h(and)h(2,)f(there)h(are)f Fe(m)d Fj(2)g Fi(N)515 1848 y Fl(and)h Fe(C)774 1863 y Fh(m)873 1848 y Fl(suc)m(h)i(that)1270 2059 y Fj(j)p Fe(T)1369 2018 y Fh(m)1435 2059 y Fl(\()p Fe(w)s Fl(\))p Fj(j)27 b(\024)h Fe(e)1789 2018 y Fd(\000)1847 2001 y Ff(^)1844 2018 y Fh(\016)s(m)1945 2059 y Fj(j)p Fe(w)s Fj(j)20 b Fl(+)i Fe(C)2262 2074 y Fh(m)2329 2059 y Fj(j)p Fe(w)s Fj(j)2458 2018 y Ff(1+)p Fh(\013)2596 2059 y Fe(:)515 2259 y Fl(W)-8 b(e)34 b(c)m(ho)s(ose)42 b(~)-57 b Fe(!)32 b(>)d Fl(0)k(suc)m(h)i(that)e(for)g Fj(j)p Fe(w)s Fj(j)28 b(\024)37 b Fl(~)-56 b Fe(!)36 b Fl(the)e(second)h(term)e(is)g(so)g (small)515 2375 y(that)1532 2491 y Fj(j)p Fe(T)1631 2450 y Fh(m)1697 2491 y Fl(\()p Fe(w)s Fl(\))p Fj(j)27 b(\024)h Fe(e)2051 2450 y Fd(\000)2109 2433 y Ff(~)2106 2450 y Fh(\016)s(m)2207 2491 y Fj(j)p Fe(w)s Fj(j)p Fe(:)515 2740 y Fl(The)39 b(follo)m(wing)c(lemma)h(is)h(ob)m(vious)i(\(for)e (example,)i(it)e(can)h(b)s(e)h(pro)m(v)m(ed)g(b)m(y)515 2856 y(induction\):)515 3139 y Fa(Lemma)e(3.3)42 b Fk(F)-7 b(or)42 b(any)i Fe(k)j Fj(2)d Fi(N)9 b Fk(,)51 b(any)44 b Fe(w)s Fk(,)h Fj(j)p Fe(w)s Fj(j)d(\024)j Fl(1)e Fk(ther)-5 b(e)44 b(is)f(a)g(c)-5 b(onstant)515 3255 y Fe(C)592 3219 y Fd(\003)585 3281 y Fh(k)659 3255 y Fe(>)27 b Fl(0)35 b Fk(such)g(that)g Fj(j)p Fe(T)1366 3219 y Fh(j)1402 3255 y Fl(\()p Fe(w)s Fl(\))p Fj(j)26 b(\024)j Fe(C)1788 3219 y Fd(\003)1781 3281 y Fh(k)1827 3255 y Fj(j)p Fe(w)s Fj(j)34 b Fk(for)g(any)h Fe(j)f Fl(=)27 b(0)p Fe(;)17 b(:)g(:)g(:)f(;)h(k)515 3537 y Fl(No)m(w)49 b(tak)m(e)h Fe(!)59 b(<)c Fl(1)49 b(suc)m(h)h(that)f Fj(j)p Fe(T)1893 3501 y Fh(j)1929 3537 y Fl(\()p Fe(w)s Fl(\))p Fj(j)54 b(\024)64 b Fl(~)-57 b Fe(!)52 b Fl(for)d Fe(j)61 b Fj(\024)56 b Fe(m)p Fl(,)d Fj(j)p Fe(w)s Fj(j)i(\024)h Fe(!)t Fl(.)515 3653 y(By)40 b(Lemma)f(3,)j(w)m(e)g(can)e(\014nd)h(a)f(constan)m(t)h Fe(C)48 b(>)40 b Fl(0)g(suc)m(h)i(that)e Fj(j)p Fe(T)3049 3617 y Fh(j)3085 3653 y Fl(\()p Fe(w)s Fl(\))p Fj(j)g(\024)515 3785 y Fe(C)7 b(e)637 3749 y Fd(\000)695 3731 y Ff(~)692 3749 y Fh(\016)r(j)762 3785 y Fj(j)p Fe(w)s Fj(j)31 b Fl(for)h Fe(j)i Fl(=)28 b(0)p Fe(;)17 b(:)g(:)g(:)e(;)i(m)p Fl(,)33 b Fj(j)p Fe(w)s Fj(j)26 b(\024)i Fl(1.)515 3984 y(Consider)33 b Fj(j)p Fe(T)1021 3948 y Fh(mn)p Ff(+)p Fh(j)1217 3984 y Fl(\()p Fe(w)s Fl(\))p Fj(j)p Fl(,)e Fj(j)p Fe(w)s Fj(j)c(\024)h Fe(!)t Fl(.)43 b(As)33 b Fj(j)p Fe(T)2091 3948 y Fh(j)2127 3984 y Fl(\()p Fe(w)s Fl(\))p Fj(j)26 b(\024)36 b Fl(~)-56 b Fe(!)s Fl(,)1070 4204 y Fj(j)p Fe(T)1169 4162 y Fh(m)p Ff(+)p Fh(j)1322 4204 y Fl(\()p Fe(w)s Fl(\))p Fj(j)26 b Fl(=)i Fj(j)p Fe(T)1728 4162 y Fh(m)1794 4204 y Fl(\()p Fe(T)1903 4162 y Fh(j)1939 4204 y Fl(\()p Fe(w)s Fl(\)\))p Fj(j)f(\024)h Fe(e)2331 4162 y Fd(\000)2389 4145 y Ff(~)2386 4162 y Fh(\016)s(m)2486 4204 y Fj(j)p Fe(T)2585 4162 y Fh(j)2621 4204 y Fl(\()p Fe(w)s Fl(\))p Fj(j)p Fe(:)515 4403 y Fl(But)k(this)h(is)f(again)f(less)i(than)40 b(~)-57 b Fe(!)t Fl(,)32 b(th)m(us)896 4617 y Fj(j)p Fe(T)995 4576 y Ff(2)p Fh(m)p Ff(+)p Fh(j)1183 4617 y Fl(\()p Fe(w)s Fl(\))p Fj(j)27 b Fl(=)g Fj(j)p Fe(T)1589 4576 y Fh(m)1655 4617 y Fl(\()p Fe(T)1764 4576 y Fh(m)p Ff(+)p Fh(j)1918 4617 y Fl(\()p Fe(w)s Fl(\)\))p Fj(j)f(\024)i Fe(e)2309 4576 y Fd(\000)2367 4558 y Ff(~)2364 4576 y Fh(\016)s(m)2465 4617 y Fj(j)p Fe(T)2564 4576 y Fh(m)p Ff(+)p Fh(j)2717 4617 y Fl(\()p Fe(w)s Fl(\))p Fj(j)f(\024)1607 4836 y(\024)i Fe(e)1758 4795 y Fd(\000)1816 4778 y Ff(~)1813 4795 y Fh(\016)r Ff(2)p Fh(m)1948 4836 y Fj(j)p Fe(T)2047 4795 y Fh(j)2083 4836 y Fl(\()p Fe(w)s Fl(\))p Fj(j)p Fe(;)515 5002 y Fl(and)j(so)h(on.)44 b(Th)m(us)1905 5255 y Fn(20)p eop %%Page: 21 21 21 20 bop 959 639 a Fj(j)p Fe(T)1058 598 y Fh(mn)1166 639 y Fl(\()p Fe(T)1275 598 y Fh(j)1311 639 y Fl(\()p Fe(w)s Fl(\)\))p Fj(j)27 b(\024)h Fe(e)1703 598 y Fd(\000)1761 581 y Ff(~)1758 598 y Fh(\016)s(mn)1901 639 y Fj(j)p Fe(T)2000 598 y Fh(j)2036 639 y Fl(\()p Fe(w)s Fl(\))p Fj(j)f(\024)h Fe(C)7 b(e)2467 598 y Fd(\000)2525 581 y Ff(~)2522 598 y Fh(\016)s Ff(\()p Fh(mn)p Ff(+)p Fh(j)t Ff(\))2807 639 y Fj(j)p Fe(w)s Fj(j)515 790 y Fl(for)32 b(all)e Fe(k)s(;)17 b(n)28 b Fj(2)g Fi(N)37 b Fj([)23 b Fl(0.)43 b Fb(\003)515 989 y Fa(Remark)23 b Fl(The)h(theorem)g(can)g (b)s(e)f(generalized)h(to)f(the)h(case)g(when)h(the)f(\014b)s(ers)515 1106 y Fe(S)581 1069 y Ff(1)662 1106 y Fl(and)43 b(the)f(base)h Fe(B)48 b Fl(are)42 b(replaced)g(b)m(y)i(some)e(manifolds)e(\(for)h (example,)515 1222 y(one)47 b(can)g(consider)g(the)g Fe(m)p Fl(-dimensional)e(tori)g Fi(T)2408 1186 y Fh(m)2525 1222 y Fl(instead)i(of)f Fe(S)3067 1186 y Ff(1)3106 1222 y Fl(\),)51 b(and)515 1338 y(demand)29 b(that)g(on)g(the)h(\014b)s(er)f Fe(w)h Fl(=)e(0)h(the)g(transformation)e(of)i(this)g(manifold)515 1454 y(is)44 b(uniquely)g(ergo)s(dic.)79 b(The)45 b("linearized")e (system)i(will)e(still)f(b)s(e)j(a)f(sk)m(ew)515 1570 y(pro)s(duct)f(of)f(this)g(uniquely)h(ergo)s(dic)f(mapping)f(of)i(the)g (manifold)d Fe(w)47 b Fl(=)e(0)515 1687 y(and)32 b(some)h(linear)e (mapping.)515 1886 y(Consider)c(no)m(w)h(a)e(more)g(general)h (situation,)f(when)i(on)f(the)h(manifold)c Fe(w)30 b Fl(=)d(0)1370 2060 y Fe(T)41 b Fl(:)28 b Fe(t)g Fj(!)f Fe(t)1748 2019 y Fd(0)1799 2060 y Fl(=)h Fe(t)22 b Fl(+)g Fe(l)j Fl(+)d Fe(G)p Fl(\()p Fe(t;)17 b(x)p Fl(\))p Fe(;)515 2234 y Fl(and)32 b(in)g(its)g(neigh)m(b)s(ourho)s(o)s(d)593 2408 y Fe(w)666 2366 y Fd(0)716 2408 y Fl(=)c(\()p Fe(A)p Fl(\()p Fe(t)p Fl(\))22 b(+)g Fe(A)1235 2423 y Ff(1)1275 2408 y Fl(\()p Fe(t;)17 b(x)p Fl(\)\))p Fe(w)25 b Fl(+)d Fe(B)1790 2423 y Ff(1)1829 2408 y Fl(\()p Fe(t;)17 b(w)s(;)g(x)p Fl(\))p Fe(;)114 b Fj(j)p Fe(B)2399 2423 y Ff(1)2438 2408 y Fl(\()p Fe(t;)17 b(w)s(;)g(x)p Fl(\))p Fj(j)27 b(\024)h Fe(c)2967 2423 y Ff(1)3006 2408 y Fj(j)p Fe(w)s Fj(j)3135 2366 y Ff(1+)p Fh(\013)3274 2408 y Fe(;)772 2581 y(t)807 2540 y Fd(0)858 2581 y Fl(=)g Fe(t)22 b Fl(+)g Fe(l)j Fl(+)d Fe(G)p Fl(\()p Fe(t;)17 b(x)p Fl(\))22 b(+)g Fe(B)1750 2596 y Ff(2)1790 2581 y Fl(\()p Fe(t;)17 b(w)s(;)g(x)p Fl(\))p Fe(;)113 b Fj(j)p Fe(B)2359 2596 y Ff(2)2398 2581 y Fl(\()p Fe(t;)17 b(w)s(;)g(x)p Fl(\))p Fj(j)27 b(\024)h Fe(c)2927 2596 y Ff(2)2967 2581 y Fj(j)p Fe(w)s Fj(j)p Fe(:)515 2732 y Fl(W)-8 b(e)38 b(supp)s(ose)g(that)f(the) h(functions)g Fe(G)p Fl(,)g Fe(A)2089 2747 y Ff(1)2166 2732 y Fl(are)g(small:)51 b Fj(j)p Fe(G)p Fj(j)35 b(\024)h Fe(\026)p Fl(,)j Fj(j)p Fe(A)p Fj(j)c(\024)h Fe(\026)515 2848 y Fl(for)c(some)g Fe(\026)c(>)f Fl(0.)515 3105 y Fa(Theorem)37 b(3.2)42 b Fk(L)-5 b(et)1466 3188 y Fc(Z)1566 3214 y Ff(1)1522 3413 y(0)1622 3323 y Fl(ln)16 b Fe(A)p Fl(\()p Fe(t)p Fl(\))p Fe(dt)28 b Fj(\024)g(\000)p Fe(\016)k(<)c Fl(0)515 3572 y Fk(for)37 b(any)h Fe(u)32 b Fj(2)i Fe(D)1135 3535 y Fh(m)1239 3572 y Fk(Then)j(for)g(any)1848 3545 y Fl(~)1844 3572 y Fe(\016)g(>)32 b Fl(0)p Fk(,)2154 3545 y Fl(~)2149 3572 y Fe(\016)37 b(<)c(\016)42 b Fk(ther)-5 b(e)37 b(exists)h(a)f(c)-5 b(onstant)515 3688 y Fe(N)38 b Fj(2)28 b Fi(N)50 b Fk(such)34 b(that)h(for)g(any)f Fe(l)c Fj(6)p Fl(=)1767 3644 y Fh(p)p 1767 3665 36 4 v 1768 3722 a(q)1812 3688 y Fk(,)35 b Fe(p;)17 b(q)31 b Fj(2)d Fi(N)9 b Fk(,)41 b Fe(q)31 b Fj(\024)d Fe(N)45 b Fk(ther)-5 b(e)35 b(ar)-5 b(e)34 b(c)-5 b(onstants)515 3818 y Fe(!)47 b(>)d Fl(0)p Fk(,)54 b Fl(~)-57 b Fe(\026)44 b(>)g Fl(0)f Fk(such)h(that)g(for)g(any)g Fe(w)s Fk(,)h Fj(j)p Fe(w)s Fj(j)e(\024)h Fe(!)t Fk(,)i(for)d(any)h Fe(\026)g Fj(\024)52 b Fl(~)-57 b Fe(\026)44 b Fk(the)515 3934 y(estimate)1527 4051 y Fj(j)p Fe(T)1626 4009 y Fh(n)1672 4051 y Fl(\()p Fe(w)s Fl(\))p Fj(j)27 b(\024)h Fe(C)7 b(e)2103 4009 y Fd(\000)2161 3992 y Ff(~)2158 4009 y Fh(\016)s(n)2239 4051 y Fj(j)p Fe(w)s Fj(j)515 4201 y Fk(holds)34 b(for)g(some)h(c)-5 b(onstant)34 b Fe(C)42 b Fk(for)35 b(any)f Fe(n)28 b Fj(2)g Fi(N)9 b Fk(.)515 4458 y Fl(Both)31 b(theorems)g(claim)e(that)i(the)h(in)m(v)-5 b(arian)m(t)30 b(manifold)e Fj(f)p Fl(\()p Fe(t;)17 b(w)s(;)g(x)p Fl(\))27 b(:)h Fe(w)i Fl(=)d(0)p Fj(g)515 4575 y Fl(is)e(attracting,)i (and)f(that)g(under)h(certain)f(conditions)f(the)i(tra)5 b(jectories)26 b(tend)515 4691 y(to)37 b(it)f(exp)s(onen)m(tially)h (fast.)58 b(Theorem)76 b(3.2)37 b(sho)m(ws)i(that)e(this)g(situation)f (is)515 4807 y(structurally)c(stable)g(\(in)g(some)g(appropriate)g(top) s(ology\).)515 5006 y(The)h(pro)s(of)f(of)g(Theorem)h(3.2)f(rep)s(eats) h(the)g(pro)s(of)f(of)g(Theorem)h(3.1.)1905 5255 y Fn(21)p eop %%Page: 22 22 22 21 bop 515 523 a Fm(4)134 b(The)56 b(relativistic)i(billiards)f(in)f (non-gra)l(vitational)716 672 y(force)46 b(\014elds)515 970 y Fl(W)-8 b(e)24 b(consider)h(\014rst)f(the)g(case,)j(when)e(the)g (particle)e(of)g(mass)h(1)g(mo)m(v)m(es)h(inside)e(a)515 1087 y(parallelepip)s(ed)e(\005)28 b Fj(2)g Fi(R)1393 1050 y Ff(3)1438 1087 y Fl(.)41 b(W)-8 b(e)24 b(assume)h(that)e(the)i (particle)d(re\015ects)k(from)d(the)515 1203 y(upp)s(er)f(and)g(lo)m(w) m(er)g(b)s(oundaries)g(of)g(\005)g(according)f(to)h(the)g(generalized)g (billiard)515 1319 y(la)m(w,)k(while)e(the)h(re\015ections)h(from)e (the)h(other)g(sides)h(are)f(the)g(classical)f(elastic)515 1435 y(ones.)57 b(Let)38 b Fe(v)i Fl(b)s(e)d(the)h(particle)e(v)m(elo)s (cit)m(y)-8 b(.)56 b(W)-8 b(e)38 b(denote)f(b)m(y)h Fe(v)2798 1450 y Fh(\027)2879 1435 y Fl(the)f(v)m(elo)s(cit)m(y)515 1552 y(comp)s(onen)m(t)h(orthogonal)e(to)i(the)g(upp)s(er)h(and)f(lo)m (w)m(er)g(b)s(oundaries,)i(and)e(b)m(y)515 1668 y Fe(v)562 1683 y Fh(\034)640 1668 y Fl(its)d(tangen)m(tial)f(comp)s(onen)m(t.)51 b(Note)36 b(that)f(while)f Fe(v)2541 1683 y Fh(\027)2620 1668 y Fl(is)h(a)g(n)m(um)m(b)s(er)g(\(one-)515 1784 y(dimensional\),)30 b Fe(v)1164 1799 y Fh(\034)1240 1784 y Fl(is)i(a)g(t)m(w)m(o-dimensional)e(v)m(ector.)515 1983 y(W)-8 b(e)37 b(supp)s(ose)h(that)f(in)f(the)i(in)m(terior)d(of)i (\005)g(the)g(particle)f(mo)m(v)m(es)i(under)g(the)515 2099 y(in\015uence)k(of)f(some)g(non-gra)m(vitational)d(force)j (\014eld,)i(giv)m(en)f(b)m(y)g(a)f(smo)s(oth)515 2216 y(b)s(ounded)e(v)m(ector)g(function)f Fe(F)14 b Fl(\()p Fe(x;)33 b Fl(_)-43 b Fe(x;)17 b(t)p Fl(\).)60 b(W)-8 b(e)39 b(assume)g(for)f(simplicit)m(y)d(that)515 2332 y(the)25 b(tangen)m(tial)f(comp)s(onen)m(t)h Fe(F)1680 2347 y Fh(\034)1748 2332 y Fl(of)g(the)h(force)f(v)m(ector)h(admits)e (the)i(follo)m(wing)515 2448 y(estimate:)1607 2564 y Fj(k)p Fe(F)1720 2579 y Fh(\034)1763 2564 y Fj(k)h Fl(=)h Fe(O)s Fl(\()p Fj(k)p Fe(v)2157 2579 y Fh(\034)2199 2564 y Fj(k)p Fl(\))515 2730 y(for)k(small)e(v)-5 b(alues)33 b(of)f Fj(k)p Fe(v)1418 2745 y Fh(\034)1461 2730 y Fj(k)p Fl(.)515 2930 y Fa(Examples)25 b Fl(1.)41 b(The)26 b(force)g(\014eld)g Fe(F)39 b Fl(is)25 b(orthogonal)f(to)i(the)g(upp)s(er)g(and)g(lo)m(w)m (er)515 3046 y(b)s(oundaries)32 b(of)g(the)h(parallelepip)s(ed)d(\005.) 515 3162 y(2.)76 b(The)44 b(particle)e(mo)m(v)m(es)i(under)h(the)f (action)e(of)h(the)h(magnetic)e(\014eld)h Fe(H)8 b Fl(,)515 3278 y(whic)m(h)37 b(is)f(directed)h(orthogonal)d(to)j(the)g(upp)s(er)g (and)f(lo)m(w)m(er)h(b)s(oundaries)f(of)515 3395 y(\005.)43 b(In)33 b(this)f(case)1202 3594 y Fe(F)42 b Fl(=)27 b Fe(e)p Fl(\()p Fe(v)f Fj(\002)d Fe(H)8 b Fl(\))p Fe(;)114 b Fj(k)p Fe(F)14 b Fj(k)27 b Fl(=)g Fe(e)p Fj(k)p Fe(v)2383 3609 y Fh(\034)2426 3594 y Fj(kk)p Fe(H)8 b Fj(k)p Fe(;)515 3793 y(e)32 b Fl(is)h(the)g(c)m(harge)g(of)f(the)h(particle.)515 3992 y(Consider)24 b(the)h(mapping)1484 3967 y(\026)1465 3992 y Fe(T)14 b Fl(,)26 b(whic)m(h)e(consists)h(of)f(t)m(w)m(o)g (successfull)h(re\015ections)515 4109 y(of)32 b(the)h(particle)e(from)g (the)i(upp)s(er)h(and)e(lo)m(w)m(er)h(b)s(oundaries)f(of)g(\005:)1507 4283 y(\026)1488 4308 y Fe(T)42 b Fl(:)27 b(\()p Fe(v)t(;)17 b(t;)g(x)p Fl(\))28 b Fj(!)f Fl(\()t(\026)-53 b Fe(v)t(;)2235 4289 y Fl(\026)2234 4308 y Fe(t;)22 b Fl(\026)-54 b Fe(x)p Fl(\))515 4507 y(The)33 b(v)m(elo)s(cit)m(y)g Fe(v)1123 4522 y Fh(\027)1194 4507 y Fe(<)27 b Fl(0)33 b(as)f(it)g(hits)g(the)h (lo)m(w)m(er)g(w)m(all,)e(th)m(us)54 b(\026)-69 b Fe(v)2701 4522 y Fh(\027)2772 4507 y Fe(<)27 b Fl(0.)515 4706 y(Let)45 b Fe(V)759 4721 y Ff(1)843 4706 y Fl(b)s(e)h(the)f(v)m(elo)s(cit)m(y)g (function)g(of)g(the)g(lo)m(w)m(er)g(w)m(all,)j(and)d Fe(V)3014 4721 y Ff(2)3098 4706 y Fl(b)s(e)h(the)515 4823 y(v)m(elo)s(cit)m(y)31 b(function)h(of)f(the)h(upp)s(er)h(w)m (all,)d(and)i(let)g Fe(l)i Fl(the)e(high)m(t)f(of)g(the)i(paral-)515 4939 y(lelepip)s(ed)e(\005.)1905 5255 y Fn(22)p eop %%Page: 23 23 23 22 bop 515 523 a Fa(Theorem)37 b(4.1)42 b Fk(Supp)-5 b(ose)34 b(that)972 644 y Fc(Z)1071 670 y Ff(1)1027 870 y(0)1128 780 y Fl(ln)1235 712 y(\(1)22 b Fj(\000)h Fe(V)1501 727 y Ff(1)1540 712 y Fl(\()p Fe(t)p Fl(\)\)\(1)f(+)g Fe(V)1953 727 y Ff(2)1992 712 y Fl(\()p Fe(t)h Fl(+)f Fe(l)r Fl(\)\))p 1235 757 1058 4 v 1235 848 a(\(1)g(+)g Fe(V)1499 863 y Ff(1)1539 848 y Fl(\()p Fe(t)p Fl(\)\)\(1)f Fj(\000)i Fe(V)1953 863 y Ff(2)1992 848 y Fl(\()p Fe(t)g Fl(+)f Fe(l)r Fl(\)\))2303 780 y Fe(dt)27 b Fl(=)h Fj(\000)p Fe(\016)t(=)p Fl(2)g Fe(<)f Fl(0)515 1050 y Fk(Then)35 b(for)h(any)1120 1024 y Fl(~)1115 1050 y Fe(\016)e(>)d Fl(0)p Fk(,)1418 1024 y Fl(~)1414 1050 y Fe(\016)j(<)c(\016)40 b Fk(ther)-5 b(e)36 b(exists)g(a)g(c)-5 b(onstant)36 b Fe(N)41 b Fj(2)30 b Fi(N)52 b Fk(such)36 b(that)515 1167 y(for)g(any)h Fe(l)d Fj(6)p Fl(=)1041 1123 y Fh(p)p 1041 1144 36 4 v 1042 1201 a(q)1086 1167 y Fk(,)k Fe(p;)17 b(q)35 b Fj(2)c Fi(N)9 b Fk(,)43 b Fe(q)36 b Fj(\024)31 b Fe(N)48 b Fk(ther)-5 b(e)37 b(is)f(a)h(c)-5 b(onstant)36 b Fe(!)f(>)c Fl(0)37 b Fk(such)g(that)515 1301 y(for)d(any)h Fe(v)t Fk(,)g Fl(1)22 b Fj(\000)g(j)p Fe(v)1218 1316 y Fh(\027)1261 1301 y Fj(j)1289 1265 y Ff(2)1356 1301 y Fj(\024)28 b Fe(!)t Fk(,)34 b(the)h(estimate)1259 1514 y Fl(1)22 b Fj(\000)g(k)p Fe(T)1550 1473 y Fh(n)1597 1514 y Fl(\()p Fe(v)t Fl(\))p Fj(k)1774 1473 y Ff(2)1840 1514 y Fj(\024)29 b Fe(C)7 b(e)2068 1473 y Fd(\000)2126 1456 y Ff(~)2123 1473 y Fh(\016)r(n)2203 1514 y Fl(\(1)22 b Fj(\000)h(j)p Fe(v)2487 1529 y Fh(\027)2530 1514 y Fj(j)2558 1473 y Ff(2)2597 1514 y Fl(\))515 1707 y Fk(holds)34 b(for)g(some)h(c)-5 b(onstant)34 b Fe(C)42 b Fk(for)35 b(any)f Fe(n)28 b Fj(2)g Fi(N)9 b Fk(.)515 1983 y Fa(Remark)28 b Fl(An)h(example)g(of)f(functions)h(satisfying)f(this)h(in)m(tegral)e (inequalit)m(y)515 2100 y(is)32 b(giv)m(en)h(in)64 b([8],)32 b(Chapter)i(1,)e(Sec.3.)515 2299 y Fa(Pro)s(of)p Fl(.)41 b(Denote)26 b Fe(w)1253 2314 y Fh(\027)1324 2299 y Fl(=)h(1)9 b Fj(\000)g Fe(v)1622 2263 y Ff(2)1618 2324 y Fh(\027)1663 2299 y Fl(.)41 b(Then)27 b Fe(w)2049 2314 y Fh(\027)2120 2299 y Fl(=)h Fe(w)12 b Fl(+)d Fj(k)p Fe(v)2488 2314 y Fh(\034)2531 2299 y Fj(k)2581 2263 y Ff(2)2620 2299 y Fl(,)27 b(where,)i(as)e(b)s(efore,)515 2415 y Fe(w)j Fl(=)d(1)22 b Fj(\000)h(k)p Fe(v)t Fj(k)1040 2379 y Ff(2)1079 2415 y Fl(.)43 b(Di\013eren)m(tiating)30 b(with)i(resp)s(ect)i(to)e (time)g Fe(t)g Fl(w)m(e)i(get)824 2608 y(_)-58 b Fe(w)863 2623 y Fh(\027)934 2608 y Fl(=)58 b(_)-58 b Fe(w)25 b Fl(+)d(2\()p Fe(v)1364 2623 y Fh(\034)1407 2608 y Fe(;)31 b Fl(_)-41 b Fe(v)1498 2623 y Fh(\034)1541 2608 y Fl(\))27 b(=)h Fe(O)s Fl(\()p Fe(w)1899 2567 y Ff(3)p Fh(=)p Ff(2)2007 2608 y Fl(\))23 b(+)f Fe(O)s Fl(\()p Fj(k)p Fe(v)2379 2623 y Fh(\034)2421 2608 y Fj(k)2471 2567 y Ff(2)2510 2532 y Fj(p)p 2593 2532 73 4 v 76 x Fe(w)s Fl(\))27 b(=)h Fe(O)s Fl(\()p Fe(w)3024 2567 y Ff(2)3021 2633 y Fh(\027)3063 2608 y Fl(\))515 2801 y(as)33 b Fj(k)p Fe(F)748 2816 y Fh(\034)791 2801 y Fj(k)27 b Fl(=)h Fe(O)s Fl(\()p Fj(k)p Fe(v)1185 2816 y Fh(\034)1227 2801 y Fj(k)p Fl(\),)k(and)h (\(15\),)f(\(16\).)515 3001 y(Consider)e(the)h(motion)d(of)i(the)h (particle)e(in)g Fi(R)2215 2964 y Ff(3)2291 3001 y Fl(under)i(the)g (in\015uence)g(of)f(the)515 3117 y(force)i(\014eld)h Fe(F)14 b Fl(.)43 b(On)33 b(an)m(y)g(time)e(in)m(terv)-5 b(al)31 b([)p Fe(t)2092 3132 y Ff(0)2132 3117 y Fe(;)17 b(t)2211 3132 y Ff(0)2273 3117 y Fl(+)22 b(2)p Fe(l)r Fl(])1314 3310 y Fe(w)1384 3325 y Fh(\027)1427 3310 y Fl(\()p Fe(t)p Fl(\))28 b(=)f Fe(w)1739 3325 y Fh(\027)1782 3310 y Fl(\()p Fe(t)1855 3325 y Ff(0)1895 3310 y Fl(\))22 b(+)g Fe(O)s Fl(\()p Fe(w)2239 3325 y Fh(\027)2281 3310 y Fl(\()p Fe(t)2354 3325 y Ff(0)2394 3310 y Fl(\))2432 3269 y Ff(3)p Fh(=)p Ff(2)2542 3310 y Fl(\))515 3586 y(As)31 b Fe(v)704 3601 y Fh(\027)747 3586 y Fl(\()p Fe(t)p Fl(\))d(=)f(1)18 b Fj(\000)1161 3547 y Ff(1)p 1161 3563 36 4 v 1161 3620 a(2)1206 3586 y Fe(w)1276 3601 y Fh(\027)1319 3586 y Fl(\()p Fe(t)p Fl(\))g(+)g Fe(O)s Fl(\()p Fe(w)1728 3601 y Fh(\027)1770 3586 y Fl(\()p Fe(t)p Fl(\))1881 3550 y Ff(2)1920 3586 y Fl(\),)31 b(the)g(particle)e (v)m(elo)s(cit)m(y)h(on)g(the)h(time)515 3702 y(in)m(terv)-5 b(al)31 b Fe(t)d Fj(2)g Fl([)p Fe(t)1086 3717 y Ff(0)1126 3702 y Fe(;)17 b(t)1205 3717 y Ff(0)1266 3702 y Fl(+)22 b(2)p Fe(l)r Fl(])33 b(is)f(estimated)g(as)1206 3944 y Fe(v)1253 3959 y Fh(\027)1296 3944 y Fl(\()p Fe(t)p Fl(\))c(=)f(1)22 b Fj(\000)1719 3876 y Fl(1)p 1719 3921 49 4 v 1719 4012 a(2)1778 3944 y Fe(w)1848 3959 y Fh(\027)1891 3944 y Fl(\()p Fe(t)1964 3959 y Ff(0)2003 3944 y Fl(\))g(+)g Fe(O)s Fl(\()p Fe(w)2347 3959 y Fh(\027)2389 3944 y Fl(\()p Fe(t)2462 3959 y Ff(0)2502 3944 y Fl(\))2540 3903 y Ff(3)p Fh(=)p Ff(2)2650 3944 y Fl(\))515 4259 y(Th)m(us)32 b(the)f(normal)e (comp)s(onen)m(t)i(of)f(the)h(particle)e(v)m(elo)s(cit)m(y)i(1)18 b Fj(\000)h Fe(v)2950 4223 y Ff(2)2946 4284 y Fh(\027)3020 4259 y Fl(do)s(es)31 b(not)515 4375 y(c)m(hange)36 b(in)e(the)h(linear) f(appro)m(ximation)e(when)k(the)g(particle)d(mo)m(v)m(es)j(in)f(the)515 4492 y(in)m(terior)c(of)h(\005.)44 b(The)33 b(mapping)1734 4466 y(\026)1715 4492 y Fe(T)46 b Fl(sends)35 b(the)e(p)s(oin)m(t)e Fe(w)2572 4507 y Fh(\027)2615 4492 y Fe(;)17 b(t)33 b Fl(to)991 4748 y(\026)-69 b Fe(w)1041 4763 y Fh(\027)1111 4748 y Fl(=)1215 4637 y Fc(\020)1284 4681 y Fl(1)22 b Fj(\000)h Fe(V)1512 4696 y Ff(1)1551 4681 y Fl(\()p Fe(t)p Fl(\))p 1284 4725 378 4 v 1285 4816 a(1)f(+)g Fe(V)1511 4831 y Ff(1)1550 4816 y Fl(\()p Fe(t)p Fl(\))1672 4637 y Fc(\021)1732 4660 y Ff(2)1771 4637 y Fc(\020)1841 4681 y Fl(1)g(+)g Fe(V)2067 4696 y Ff(2)2107 4681 y Fl(\()2146 4662 y(~)2145 4681 y Fe(t)p Fl(\))p 1841 4725 V 1841 4821 a(1)f Fj(\000)i Fe(V)2068 4836 y Ff(2)2107 4821 y Fl(\()2146 4803 y(~)2145 4821 y Fe(t)q Fl(\))2228 4637 y Fc(\021)2288 4660 y Ff(2)2327 4748 y Fe(w)2397 4763 y Fh(\027)2462 4748 y Fl(+)f Fe(O)s Fl(\()p Fe(w)2749 4707 y Ff(3)p Fh(=)p Ff(2)2746 4773 y Fh(\027)2858 4748 y Fl(\))p Fe(;)1503 4988 y Fl(\026)1501 5006 y Fe(t)28 b Fl(=)g Fe(t)22 b Fl(+)g Fe(l)r Fl(\(1)g(+)g Fe(O)s Fl(\()p Fe(w)2247 5021 y Fh(\027)2289 5006 y Fl(\)\))p Fe(;)1905 5255 y Fn(23)p eop %%Page: 24 24 24 23 bop 515 523 a Fl(see)33 b(\(8\).)515 722 y(The)g(pro)s(of)f(is)g (completed)g(b)m(y)i(applying)d(Theorem)i(3.1.)43 b Fb(\003)515 922 y Fa(Remark.)e Fl(The)28 b(teorem)f(is)g(also)g(true)h(for)f(the)g (force)h(\014eld)f Fe(F)41 b Fl(suc)m(h)29 b(that)f(the)515 1038 y(tangen)m(tial)g(comp)s(onen)m(t)i Fj(k)p Fe(F)1579 1053 y Fh(\034)1622 1038 y Fj(k)d(\024)h Fe(\026)i Fl(for)f(some)g (su\016cien)m(tly)i(small)c(constan)m(t)515 1154 y Fe(\026)p Fl(,)32 b(cf)h(Theorem)g(3.2,)f(ho)m(w)m(ev)m(er)j(the)e(pro)s(of)f(is) g(more)f(tec)m(hnical.)515 1353 y(W)-8 b(e)31 b(consider)h(no)m(w)g (the)f(general)g(case,)h(when)g(the)g(particle)e(mo)m(v)m(es)i(inside)e (a)515 1469 y(compact)d(domain)e(\005)j(of)e(an)i(arbitrary)e(form.)40 b(The)29 b(action)d(of)h(its)g(b)s(oundary)515 1586 y(\000)37 b(on)f(the)i(particles)e(is)g(giv)m(en)h(b)m(y)h(some)f(function)f Fe(f)11 b Fl(\()p Fe(\015)5 b(;)17 b(t)p Fl(\))37 b(\(see)h(the)f(In)m (tro-)515 1702 y(duction\).)50 b(As)36 b(b)s(efore,)g(the)g(normal)d (to)h(the)i(b)s(oundary)f(is)g(directed)h(inside)515 1818 y(the)h(v)m(essel)i(\005.)58 b(Supp)s(ose)38 b(that)f(in)g(the)g (in)m(terior)f(of)h(\005)g(the)h(particle)e(mo)m(v)m(es)515 1934 y(under)d(the)g(in\015uence)h(of)e(some)g(force)h(\014eld)f Fe(F)14 b Fl(.)515 2134 y(W)-8 b(e)33 b(assume)g(that)f(for)g(all)f(p)s (oin)m(ts)h Fe(\015)h Fj(2)28 b Fl(\000)k(and)h(for)f(all)e Fe(t)e Fj(2)g Fi(R)2777 2098 y Ff(1)2855 2134 y Fl(the)33 b(v)m(elo)s(cit)m(y)1680 2314 y Fe(@)5 b(f)p 1680 2358 116 4 v 1692 2449 a(@)g(t)1806 2381 y Fl(\()p Fe(\015)g(;)17 b(t)p Fl(\))27 b Fe(>)h Fl(0)p Fe(;)515 2615 y Fl(\(cf)65 b([7]\),)32 b(whic)m(h)h(means)g(that)f(the)h(gas)g(is)f(heating.)515 2898 y Fa(Theorem)37 b(4.2)42 b Fk(Ther)-5 b(e)40 b(is)h(a)f(c)-5 b(onstant)41 b Fe(u)e(>)f Fl(0)p Fk(,)43 b Fe(u)38 b(<)h Fl(1)p Fk(,)j(such)f(that)g(if)g(the)515 3014 y(p)-5 b(article)38 b(velo)-5 b(city)38 b Fj(k)p Fe(v)t Fj(k)c(\025)h Fe(u)p Fk(,)k(then)f(ther)-5 b(e)39 b(exist)f(c)-5 b(onstants)38 b Fe(\016)g(>)d Fl(0)p Fk(,)k Fe(C)i(>)34 b Fl(0)p Fk(,)515 3130 y(such)g(that)1525 3246 y Fl(1)21 b Fj(\000)i(k)p Fe(v)t Fl(\()p Fe(t)p Fl(\))p Fj(k)1957 3205 y Ff(2)2024 3246 y Fj(\024)28 b Fe(C)7 b(e)2251 3205 y Fd(\000)p Fh(\016)r(t)515 3529 y Fl(Both)27 b(Theorems)h(4.1)f(and)g(4.2)g(claim) d(that)j(under)h(the)g(corresp)s(onding)f(con-)515 3645 y(ditions)32 b(the)j(in)m(v)-5 b(arian)m(t)32 b(manifold)f Fj(M)f Fl(=)f Fj(f)p Fl(\()p Fe(x;)17 b(v)t Fl(\))30 b(:)g Fj(k)p Fe(v)t Fj(k)f Fl(=)h(1)p Fj(g)j Fl(is)g(attracting)515 3761 y(already)k(in)g(the)i(linear)d(appro)m(ximation.)57 b(This)38 b(means)g(that)g(the)g(particle)515 3877 y(energy)33 b(gro)m(ws)h(exp)s(onen)m(tially)d(fast.)515 4077 y Fa(Pro)s(of.)53 b Fl(Let)37 b Fe(L)f Fl(b)s(e)g(the)h("diameter")d(of)i(\005,)h(i.e.)53 b Fe(L)34 b Fl(=)f Fe(max)2797 4092 y Fh(\015)2833 4101 y Fg(1)2869 4092 y Fh(;\015)2925 4101 y Fg(2)2964 4077 y Fl(\()p Fe(\032)p Fl(\()p Fe(\015)3141 4092 y Ff(1)3180 4077 y Fe(;)17 b(\015)3275 4092 y Ff(2)3314 4077 y Fl(\),)515 4193 y Fe(\032)p Fl(\()p Fe(x;)g(y)t Fl(\))43 b(is)h(the)g(distance)h (b)s(et)m(w)m(een)h(the)e(t)m(w)m(o)h(p)s(oin)m(ts)f Fe(x)g Fl(and)h Fe(y)t Fl(.)77 b(As)44 b(\005)g(is)515 4309 y(compact,)32 b Fe(L)c(<)g Fj(1)p Fl(.)43 b(W)-8 b(e)33 b(will)d(need)k(the)f(follo)m(wing)515 4591 y Fa(Lemma)k(4.1)42 b Fk(Ther)-5 b(e)38 b(is)h(a)f(c)-5 b(onstant)45 b Fl(~)-55 b Fe(u)35 b(>)g Fl(0)p Fk(,)46 b Fl(~)-55 b Fe(u)35 b(<)g Fl(1)p Fk(,)k(s.t.)58 b(if)38 b(the)h(p)-5 b(article)515 4707 y(velo)g(city)34 b Fj(k)p Fe(v)t Fj(k)27 b(\025)34 b Fl(~)-55 b Fe(u)p Fk(,)34 b(then)g(for)g(some)g(c)-5 b(onstant)2275 4689 y Fl(^)2274 4707 y Fe(t)28 b(>)f Fl(0)34 b Fk(the)h(time)f(interval)f Fl(\001)p Fe(t)515 4824 y Fk(b)-5 b(etwe)g(en)33 b(any)g(two)g(suc)-5 b(c)g(essive)33 b(c)-5 b(ol)5 b(lisions)32 b(of)h(the)h(p)-5 b(article)33 b(with)g(the)h(b)-5 b(ound-)515 4940 y(ary)35 b Fl(\000)g Fk(is)f Fl(\001)p Fe(t)28 b Fj(\024)1139 4922 y Fl(^)1138 4940 y Fe(t)p Fk(.)1905 5255 y Fn(24)p eop %%Page: 25 25 25 24 bop 515 523 a Fa(Pro)s(of)35 b(of)h(Lemma)f(4.1.)44 b Fl(The)32 b(motion)d(of)h(the)i(particle)d(under)j(the)g(in\015u-)515 639 y(ence)38 b(of)e(the)i(external)f(force)g(\014eld)g(is)f(describ)s (ed)i(b)m(y)g(equations)f(\(16\).)56 b(Let)515 756 y(the)27 b(particle)e(v)m(elo)s(cit)m(y)i Fj(k)p Fe(v)t Fj(k)f Fl(b)s(e)h(close)g(to)f(1:)41 b(1)10 b Fj(\000)g(k)p Fe(v)t Fj(k)2437 719 y Ff(2)2504 756 y Fl(=)28 b Fe(w)i(<<)d Fl(1.)42 b(Consider)515 872 y(the)32 b(time)e(in)m(terv)-5 b(al)30 b([)p Fe(t)1316 887 y Ff(0)1356 872 y Fe(;)17 b(t)1435 887 y Ff(0)1494 872 y Fl(+)j(2)p Fe(L)p Fl(].)43 b(If)31 b Fe(w)s Fl(\()p Fe(t)2044 887 y Ff(0)2083 872 y Fl(\))h(is)f(small)e(enough,)j(then)g(on)f(this)515 988 y(time)g(in)m(terv)-5 b(al)1158 1174 y Fe(w)s Fl(\()p Fe(t)p Fl(\))27 b Fj(\024)i Fe(w)s Fl(\()p Fe(t)1621 1189 y Ff(0)1660 1174 y Fl(\))22 b(+)g Fe(O)s Fl(\()p Fe(w)s Fl(\()p Fe(t)2080 1189 y Ff(0)2118 1174 y Fl(\))2156 1133 y Ff(3)p Fh(=)p Ff(2)2266 1174 y Fl(\))27 b Fj(\024)i Fl(2)p Fe(w)s Fl(\()p Fe(t)2632 1189 y Ff(0)2670 1174 y Fl(\))p Fe(:)515 1360 y Fl(The)k(particle)f(v)m(elo)s(cit)m(y)g Fe(v)k Fl(can)d(b)s(e)g(estimated)f(as)1043 1547 y Fe(v)t Fl(\()p Fe(t)p Fl(\))c(=)f Fe(v)t Fl(\()p Fe(t)1460 1562 y Ff(0)1500 1547 y Fl(\))22 b(+)g Fe(O)s Fl(\()p Fe(w)s Fl(\()p Fe(t)p Fl(\)\))k(=)i Fe(v)t Fl(\()p Fe(t)2250 1562 y Ff(0)2289 1547 y Fl(\))22 b(+)g Fe(O)s Fl(\()p Fe(w)s Fl(\()p Fe(t)2709 1562 y Ff(0)2747 1547 y Fl(\)\))p Fe(:)515 1733 y Fl(Th)m(us)50 b(if)d(the)h(particle)f(v)m(elo)s(cit)m (y)h Fe(v)k Fl(is)c(close)g(to)g(1,)k(then)d(the)f(tra)5 b(jectory)515 1849 y(is)47 b(close)h(to)g(the)g(straigh)m(t)f(line.)88 b(More)48 b(precisely)-8 b(,)52 b(there)d(is)e(a)h(constan)m(t)515 1965 y Fe(u)571 1929 y Fd(0)629 1965 y Fe(>)35 b Fl(0)p Fe(;)17 b(u)889 1929 y Fd(0)947 1965 y Fe(<)36 b Fl(1)h(suc)m(h)h(that) f(if)f Fj(k)p Fe(v)t Fl(\()p Fe(t)1853 1980 y Ff(0)1892 1965 y Fl(\))p Fj(k)g Fe(>)f(u)2183 1929 y Fd(0)2243 1965 y Fl(then)j Fe(\032)p Fl(\()p Fe(x)p Fl(\()p Fe(t)2686 1980 y Ff(0)2726 1965 y Fl(\))p Fe(;)17 b(x)p Fl(\()p Fe(t)p Fl(\)\))36 b Fe(>)f(L)i Fl(for)515 2081 y(some)32 b Fe(t)c Fj(2)g Fl([)p Fe(t)978 2096 y Ff(0)1018 2081 y Fe(;)17 b(t)1097 2096 y Ff(0)1158 2081 y Fl(+)22 b(2)p Fe(L)p Fl(],)33 b Fe(x)p Fl(\()p Fe(t)p Fl(\))g(is)f(the)h(particle)f (co)s(ordinate.)515 2281 y(No)m(w)h(tak)m(e)39 b(~)-55 b Fe(u)27 b Fl(=)h Fe(max)p Fl(\()p Fe(w)s(;)17 b(u)1537 2244 y Fd(0)1560 2281 y Fl(\).)43 b Fb(\003)515 2480 y Fl(In)33 b(the)h(linear)e(appro)m(ximation)f(on)j(a)f(time)f(in)m (terv)-5 b(al)32 b([)p Fe(t;)17 b(t)23 b Fl(+)2788 2462 y(^)2787 2480 y Fe(t)p Fl(])33 b(the)h(v)m(elo)s(cit)m(y)515 2596 y Fe(w)55 b Fl(=)d(1)32 b Fj(\000)g(k)p Fe(v)t Fj(k)1109 2560 y Ff(2)1195 2596 y Fl(is)47 b(constan)m(t.)88 b(Consider)48 b(a)f(mapping)e Fe(A)53 b Fl(:)g(\()p Fe(t;)17 b(\015)5 b(;)17 b(v)t Fl(\))51 b Fj(!)515 2712 y Fl(\()p Fe(t)588 2676 y Fd(0)611 2712 y Fe(;)17 b(\015)711 2676 y Fd(0)734 2712 y Fe(;)g(v)t Fl(\),)35 b(whic)m(h)h(consists)g(of)e(one)i (re\015ection)f(from)f(the)h(b)s(oundary)h(and)f(a)515 2829 y(free)e(motion)d(till)g(the)j(next)h(collision)29 b(with)j(the)h(b)s(oundary)-8 b(.)515 3028 y(As)35 b(for)f(all)f(p)s (oin)m(ts)i Fe(\015)h Fj(2)c Fl(\000)j(and)g(for)f(all)f Fe(t)e Fj(2)h Fi(R)2238 2992 y Ff(1)2318 3028 y Fl(the)k(v)m(elo)s(cit) m(y)2861 2984 y Fh(@)t(f)p 2861 3005 83 4 v 2869 3062 a(@)t(t)2953 3028 y Fl(\()p Fe(\015)5 b(;)17 b(t)p Fl(\))32 b Fe(>)f Fl(0,)515 3160 y(there)36 b(exists)g(a)f(constan)m(t)h Fe(V)54 b(>)32 b Fl(0)j(suc)m(h)i(that)2269 3117 y Fh(@)t(f)p 2269 3138 V 2277 3195 a(@)t(t)2361 3160 y Fl(\()p Fe(\015)5 b(;)17 b(t)p Fl(\))33 b Fj(\025)f Fe(V)22 b Fl(.)52 b(Th)m(us)37 b(at)e(the)515 3277 y(re\015ection)d(from)g(the)h(b)s(oundary)g(the)g (particle)e(v)m(elo)s(cit)m(y)h(c)m(hanges)i(as)1484 3511 y(\026)-69 b Fe(w)30 b Fl(=)e Fe(W)14 b Fl(\()1822 3443 y Fe(@)5 b(f)p 1822 3488 116 4 v 1834 3579 a(@)g(t)1947 3511 y Fl(\()p Fe(\015)g(;)17 b(t)p Fl(\))p Fe(;)g(v)2249 3526 y Fh(\027)2292 3511 y Fl(\))p Fe(w)s(;)515 3789 y Fl(where)34 b Fe(W)42 b Fj(\024)1041 3762 y Fl(~)1037 3789 y Fe(\016)32 b Fl(=)1216 3678 y Fc(\020)1286 3749 y Ff(1)p Fd(\000)p Fh(V)1433 3726 y Fg(2)p 1286 3766 182 4 v 1286 3823 a Ff(1+)p Fh(V)1433 3804 y Fg(2)1477 3678 y Fc(\021)1537 3700 y Ff(2)1604 3789 y Fe(<)d Fl(1,)j(see\(12\).) 46 b(The)33 b(mapping)f Fe(A)h Fl(can)g(no)m(w)h(b)s(e)515 3928 y(written)e(as)1410 4044 y Fe(w)1483 4003 y Fd(0)1533 4044 y Fl(=)c Fe(W)14 b Fl(\()p Fe(w)s(;)j(t;)g(\015)5 b Fl(\))p Fe(w)23 b Fl(+)f Fe(o)p Fl(\()p Fe(w)s Fl(\))p Fe(:)515 4202 y Fl(There)30 b(is)f(a)h(constan)m(t)g Fe(w)1427 4217 y Ff(0)1494 4202 y Fe(>)d Fl(0,)j(suc)m(h)h(that)f(if)e Fe(w)i(<)e(w)2489 4217 y Ff(0)2528 4202 y Fl(,)i(then)g(the)g Fe(n)3027 4166 y Fh(th)3127 4202 y Fl(p)s(o)m(w)m(er)515 4318 y(of)i(the)h(mapping)e Fe(A)i Fl(can)f(b)s(e)h(estimated)f(as)1600 4505 y Fe(A)1673 4463 y Fh(n)1720 4505 y Fl(\()p Fe(w)s Fl(\))27 b Fj(\024)h Fe(C)2071 4520 y Fh(w)2132 4478 y Fl(~)2127 4505 y Fe(\016)2174 4463 y Fh(n)2221 4505 y Fe(w)515 4691 y Fl(for)k(some)g(constan)m(t)i Fe(C)1372 4706 y Fh(w)1428 4691 y Fl(.)515 4890 y(T)-8 b(o)39 b(complete)f(the)h (pro)s(of)f(of)h(the)g(theorem)g(tak)m(e)h Fe(u)e Fl(=)g Fe(max)p Fl(\()6 b(~)-55 b Fe(u)q(;)2952 4815 y Fj(p)p 3034 4815 280 4 v 3034 4890 a Fl(1)22 b Fj(\000)h Fe(w)3275 4905 y Ff(0)3314 4890 y Fl(\).)515 5006 y Fb(\003)1905 5255 y Fn(25)p eop %%Page: 26 26 26 25 bop 515 523 a Fm(5)134 b(The)52 b(generalized)i(billiards)f(in)f (the)h(gra)l(vitational)716 672 y(\014eld)515 970 y Fl(W)-8 b(e)40 b(consider)h(the)g(particle)e(free)h(motion)e(in)i(the)h (4-dimensional)c("space-)515 1087 y(time")c(with)h(some)g(metric)f Fe(g)1610 1102 y Fh(ij)1671 1087 y Fe(;)17 b(i;)g(j)36 b Fl(=)31 b(0)p Fe(;)17 b(:)g(:)g(:)f(;)h Fl(3)34 b(of)g(signature)g (\(+)p Fe(;)17 b Fj(\000)p Fe(;)g Fj(\000)p Fe(;)g Fj(\000)p Fl(\))515 1203 y(in)24 b(ev)m(ery)i(p)s(oin)m(t.)40 b(W)-8 b(e)26 b(supp)s(ose)g(that)e(the)h(particle)f(mo)m(v)m(es)i(inside)e(a) g(b)s(ounded)515 1319 y(domain)j(\005)h(in)g(the)h("space")g(v)-5 b(ariables.)41 b(The)29 b(action)f(of)g(its)g(b)s(oundary)h(\000)f(on) 515 1435 y(the)h(particle)f(is)g(giv)m(en)h(b)m(y)h(some)f(v)m (ector-function)g Fe(V)22 b Fl(\()p Fe(\015)5 b(;)17 b(t)p Fl(\))28 b(\(see)i(Section)f(2\),)515 1552 y(whic)m(h)35 b(is)f(directed)g(along)f(the)i(normal)d(to)i(the)h(b)s(oundary)g(\000) f(at)g(a)g(p)s(oin)m(t)g Fe(\015)515 1668 y Fl(\(as)28 b(ab)s(o)m(v)m(e,)i Fe(t)e Fl(=)f Fe(x)1189 1632 y Ff(0)1257 1668 y Fl(is)h(the)g("w)m(orld)g(time"\).)40 b(The)29 b(normal)d(to)i(the)h(b)s(oundary)515 1784 y(is)c(directed)i(to)e(the)i (in)m(terior)d(of)i(the)g(v)m(essel)i(\005.)41 b(W)-8 b(e)26 b(supp)s(ose)i(that)d Fe(V)d Fl(\()p Fe(\015)5 b(;)17 b(t)p Fl(\))28 b Fj(\025)515 1900 y Fe(V)572 1915 y Ff(0)639 1900 y Fl(=)f Fe(const)h(>)g Fl(0)k(is)g(also)g(directed)h (to)f(the)h(in)m(terior)e(of)h(\005.)515 2099 y(Let)c(the)g(metric)e(b) s(e)i(suc)m(h)h(that)f(for)f(some)h Fe(\017)g(>)f Fl(0)h(ev)m(ery)h (geo)s(desic,)g(suc)m(h)g(that)515 2216 y(1)22 b Fj(\000)h Fe(v)737 2180 y Ff(2)805 2216 y Fj(\024)29 b Fe(\017)p Fl(,)34 b(\(the)f(three-dimensional)e(v)m(elo)s(cit)m(y)i Fe(v)k Fl(is)32 b(de\014ned)j(b)m(y)f(\(17\),)f(see)515 2332 y(Section)g(2\),)g(whic)m(h)g(starts)h(inside)e(the)i(domain)d (\005,)i(crosses)i(the)f(b)s(oundary)515 2448 y(\000,)43 b(and)f(there)g(is)f(a)g(constan)m(t)1703 2430 y(^)1702 2448 y Fe(t)i(>)f Fl(0)f(suc)m(h)i(that)e Fj(j)p Fe(t)2500 2463 y Ff(0)2567 2448 y Fj(\000)29 b Fe(t)2708 2463 y Fh(c)2743 2448 y Fj(j)42 b(\024)2935 2430 y Fl(^)2933 2448 y Fe(t)q Fl(.)69 b(Here)42 b Fe(t)3339 2463 y Ff(0)515 2564 y Fl(is)36 b(the)i(initial)33 b(momen)m(t)j(of)h(time,)g(and)g Fe(t)2057 2579 y Fh(c)2129 2564 y Fl(is)g(the)g(momen)m(t)g(of)f(time)g (of)h(the)515 2681 y(collision.)515 2880 y(In)g(case)h(of)f(non-gra)m (vitational)c(forces)38 b(this)f(condition)f(is)g(alw)m(a)m(ys)i (ful\014lled)515 2996 y(\(see)c(Lemma)f(4.1\),)g(but)h(it)e(is)i(not)f (true)h(for)f(an)h(arbitrary)e(metric,)h(i.e.,)g(for)515 3112 y(an)f(arbitrary)g(gra)m(vitational)d(\014eld.)515 3380 y Fa(Theorem)37 b(5.1)42 b Fk(L)-5 b(et)31 b(the)f(p)-5 b(article)31 b(velo)-5 b(city)30 b(at)h Fe(t)d Fl(=)f(0)k Fk(satisfy)g(the)f(fol)5 b(lowing)515 3497 y(ine)-5 b(quality:)41 b Fl(1)6 b Fj(\000)g Fe(v)t Fl(\(0\))1306 3461 y Ff(2)1373 3497 y Fj(\024)28 b Fe(\017)p Fk(.)43 b(Then)26 b(ther)-5 b(e)28 b(ar)-5 b(e)27 b(the)h(c)-5 b(onstants)27 b Fe(\016)32 b(>)27 b Fl(0)p Fk(,)i Fe(C)35 b(>)27 b Fl(0)p Fk(,)515 3613 y(such)34 b(that)i(for)e(al)5 b(l)35 b Fe(t)28 b(>)f Fl(0)1561 3798 y(1)22 b Fj(\000)g Fe(v)t Fl(\()p Fe(t)p Fl(\))1893 3757 y Ff(2)1960 3798 y Fj(\024)28 b Fe(C)7 b(e)2187 3757 y Fd(\000)p Fh(\016)r(t)2306 3798 y Fe(:)515 4066 y Fa(Pro)s(of.)53 b Fl(The)37 b(pro)s(of)f(is)f(similar)e(to)71 b([7].)54 b(The)37 b(particle)e(energy)i Fj(E)3029 4081 y Ff(0)3104 4066 y Fl(\(18\))e(is)515 4182 y(constan)m(t)24 b(when)h(the)f(particle)e(mo)m(v)m(es)j(in)e(the)h(in)m(terior)e(of)h (the)h(domain)e(\005.)40 b(As)515 4299 y(the)e(particle)e(collides)g (with)i(the)g(b)s(oundary)-8 b(,)39 b(the)f(energy)h(transformation)515 4415 y(can)33 b(b)s(e)f(estimated)g(using)h(relation)d(\(2\):)1621 4663 y Fj(E)1683 4622 y Fd(0)1674 4688 y Ff(0)1740 4663 y Fj(\025)1856 4596 y Fl(1)22 b(+)g Fe(V)2104 4560 y Ff(2)p 1855 4640 289 4 v 1855 4732 a Fl(1)g Fj(\000)h Fe(V)2105 4703 y Ff(2)2154 4663 y Fj(E)2207 4678 y Ff(0)2246 4663 y Fe(;)515 4890 y Fj(E)577 4854 y Fd(0)568 4915 y Ff(0)644 4890 y Fl(b)s(eing)36 b(the)i(particle)e(energy)i(after)f (the)g(collision:)50 b(indeed,)38 b(the)g(energy)515 5006 y(is)d(indep)s(enden)m(t)i(of)f(the)g(co)s(ordinate)f (transformations,)g(th)m(us)i(relation)d(\(2\))1905 5255 y Fn(26)p eop %%Page: 27 27 27 26 bop 515 523 a Fl(is)30 b(also)g(v)-5 b(alid)30 b(for)g(curvilinear)g(co)s(ordinates)g(\(cf)h(Section)g(2\).)43 b(As)32 b(the)f(gra)m(vi-)515 639 y(taional)26 b(\014eld)i(is)g (constan)m(t)h(and)g(\005)g(is)f(compact,)h Fe(g)2369 654 y Ff(00)2472 639 y Fl(is)f(b)s(ounded.)42 b(Th)m(us)31 b(the)515 756 y(estimate)h(of)g(the)i(theorem)e(follo)m(ws,)g(as)h(the) h(time)d(in)m(terv)-5 b(al)32 b(b)s(et)m(w)m(een)j(ev)m(ery)515 872 y(t)m(w)m(o)e(successfull)g(collisions)d(is)i(b)s(ounded.)45 b Fb(\003)515 1247 y Fm(6)134 b(An)58 b(accelerating)j(mo)t(del)e(in)g (the)g(constan)l(t)h(force)716 1397 y(\014eld)515 1695 y Fl(W)-8 b(e)37 b(consider)h(a)f(generalization)d(of)j(the)h (accelerating)e(mo)s(del)f(in)i(the)g(con-)515 1811 y(stan)m(t)29 b(force)g(\014eld:)42 b(a)28 b(particle)g(of)g(mass)h(1)g(falls)e(v)m (ertically)g(on)i(an)g(in\014nitely-)515 1927 y(hea)m(vy)41 b(horison)m(tal)e(w)m(all.)64 b(W)-8 b(e)41 b(supp)s(ose)g(that)f(the)g (relativistic)e(factor)i(ap-)515 2044 y(p)s(ears)31 b(only)g(at)f(the)i (impact)d(with)i(the)g(w)m(all;)f(the)i(w)m(all)d(itself)h(is)g (motionless,)515 2160 y(but)35 b(it)e(acts)j(on)e(the)i(particle)d(b)m (y)j(the)f(generalized)f(billiard)d(la)m(w,)k(giv)m(en)g(b)m(y)515 2276 y(p)s(erio)s(dic)k(function)i Fe(f)11 b Fl(\()p Fe(t)p Fl(\))40 b(with)h(p)s(erio)s(d)f(1.)69 b(Ab)s(o)m(v)m(e)42 b(the)f(w)m(all)f(the)h(particle)515 2392 y(mo)m(v)m(es)35 b(with)e(the)h(constan)m(t)h(acceleration)e Fe(g)g(>)d Fl(0,)k(directed)g(orthogonal)e(to)515 2508 y(the)h(w)m(all)e(\(as)i (in)e(the)i(classical)f(case)h(of)f(the)h(free)g(fall\):)47 b(\177)-55 b Fe(x)28 b Fl(=)g Fj(\000)p Fe(g)t Fl(.)515 2708 y(Let)k(the)h(w)m(all)f(p)s(osition)f(b)s(e)h(giv)m(en)h(b)m(y)g (the)g(equation)g Fe(x)28 b Fl(=)f(0.)515 2957 y Fa(Lemma)37 b(6.1)42 b Fk(The)47 b(manifold)f Fj(M)1845 2921 y Fd(0)1919 2957 y Fl(=)51 b Fj(f)p Fe(x;)17 b(v)55 b Fl(:)c Fj(k)p Fe(v)t Fj(k)f Fl(=)h(1)d Fk(at)g Fe(x)j Fl(=)g(0)p Fj(g)c Fk(is)515 3073 y(invariant.)515 3322 y Fa(Pro)s(of.)62 b Fl(Let)39 b(the)h(particle)d(lea)m(v)m(e)j(the)f(w)m(all)f(with)g (some)h(v)m(elo)s(cit)m(y)g Fj(j)p Fe(v)t Fj(j)p Fl(.)61 b(As)515 3438 y(the)28 b(w)m(all)f(is)g(motionless,)h(the)g(particle)f (hits)g(the)i(w)m(all)e(again)f(with)i(the)g(same)515 3555 y(v)m(elo)s(cit)m(y)42 b(\(as)g(the)g(system)h(is)f (one-dimensional)d(and)j(the)h(energy)g(is)e(con-)515 3671 y(serv)m(ed\).)k(Let)33 b Fj(j)p Fe(v)t Fj(j)26 b Fl(=)i(1.)43 b(Then)34 b(after)e(the)h(impact)e(the)h(v)m(elo)s(cit)m (y)h(still)d(equals)515 3787 y(1.)43 b(Th)m(us)34 b Fj(M)1001 3751 y Fd(0)1057 3787 y Fl(is)e(in)m(v)-5 b(arian)m(t.)42 b Fb(\003)515 3986 y Fl(If)34 b Fj(j)p Fe(v)t Fj(j)c Fl(=)h(1,)k(then)g(the)g(time)e(b)s(et)m(w)m(een)j(t)m(w)m(o)f (successiv)m(e)j(collisions)31 b(with)k(the)515 4102 y(w)m(all)c(equals)i(2)p Fe(=g)t Fl(.)515 4302 y(W)-8 b(e)34 b(in)m(tro)s(duce)g(the)g(mapping)e Fe(T)43 b Fl(:)30 b(\()p Fe(v)t(;)17 b(t)p Fl(\))29 b Fj(!)h Fl(\()p Fe(v)2301 4266 y Fd(0)2324 4302 y Fe(;)17 b(t)2403 4266 y Fd(0)2426 4302 y Fl(\))34 b(in)f(the)h(follo)m(wing)d(w)m(a)m(y)-8 b(.)515 4418 y(Supp)s(ose)30 b(that)e(at)h(the)g(time)f(momen)m(t)g Fe(t)h Fl(the)g(particle)f(lea)m(v)m(es)h(the)h(w)m(all)d(with)515 4534 y(the)47 b(v)m(elo)s(cit)m(y)f Fe(v)56 b(>)c Fl(0.)85 b(Then)48 b Fe(t)1767 4498 y Fd(0)1838 4534 y Fl(is)e(the)h(momen)m(t)f (of)g(time)g(of)g(the)h(next)515 4650 y(collision)26 b(and)j Fe(v)1123 4614 y Fd(0)1174 4650 y Fe(>)e Fl(0)i(is)f(the)i(v)m (elo)s(cit)m(y)f(that)g(the)g(particle)f(equires)i(after)f(the)515 4767 y(next)k(collision)d(with)i(the)h(w)m(all:)962 4971 y Fe(t)997 4930 y Fd(0)1048 4971 y Fl(=)27 b Fe(t)c Fl(+)f(2)p Fe(v)t(=g)102 b Fl(\(mo)s(d)32 b(1\))p Fe(;)114 b(v)2142 4930 y Fd(0)2193 4971 y Fl(=)2306 4904 y Fe(v)26 b Fj(\000)d Fl(2)p Fe(V)43 b Fl(+)22 b Fe(V)2805 4868 y Ff(2)2844 4904 y Fe(v)p 2306 4948 589 4 v 2307 5039 a Fl(1)g Fj(\000)h Fl(2)p Fe(V)e(v)26 b Fl(+)c Fe(V)2855 5011 y Ff(2)2905 4971 y Fe(;)1905 5255 y Fn(27)p eop %%Page: 28 28 28 27 bop 515 530 a Fl(where)34 b Fe(V)49 b Fl(=)1016 486 y Fh(d)-12 b(f)p 1016 507 66 4 v 1018 564 a(dt)1092 530 y Fe(t)1127 494 y Fd(0)1151 530 y Fl(.)515 729 y(As)37 b(b)s(efore,)i(w)m(e)f(denote)g(b)m(y)g Fe(T)1667 693 y Fh(n)1714 729 y Fl(\()p Fe(v)t Fl(\),)g Fe(T)1977 693 y Fh(n)2023 729 y Fl(\()p Fe(t)p Fl(\))f(the)h Fe(v)t Fj(\000)f Fl(and)h Fe(t)p Fj(\000)p Fl(co)s(ordinate)e(of)515 845 y(the)d Fe(n)741 809 y Fh(th)844 845 y Fl(p)s(o)m(w)m(er)g(of)f (the)h(mapping)e Fe(T)42 b Fl(:)28 b(\()p Fe(v)t(;)17 b(t)p Fl(\))27 b Fj(!)g Fl(\()p Fe(v)2411 809 y Fd(0)2434 845 y Fe(;)17 b(t)2513 809 y Fd(0)2536 845 y Fl(\).)515 1127 y Fa(Theorem)37 b(6.1)42 b Fk(Supp)-5 b(ose)29 b(that)h Fl(2)p Fe(=g)j Fk(is)d(a)f(r)-5 b(ational)30 b(numb)-5 b(er)29 b(and)h(that)g(ther)-5 b(e)515 1243 y(is)37 b(a)h(moment)f Fe(\034)11 b Fk(,)39 b(such)e(that)1647 1199 y Fh(d)-12 b(f)p 1647 1220 V 1649 1278 a(dt)1723 1243 y Fl(\()p Fe(\034)11 b Fl(\))33 b Fe(>)g Fl(0)p Fk(.)53 b(Then)37 b(on)h(the)f(set)h(of)g(the)f(initial)515 1359 y(c)-5 b(onditions)24 b Fl(\()p Fj(j)p Fe(v)t Fj(j)i(\024)j Fl(1)p Fe(;)17 b(t)99 b Fl(\(mo)s(d)32 b(1\)\))25 b Fk(ther)-5 b(e)25 b(is)g(a)g(subset)g(of)g(p)-5 b(ositive)25 b(L)-5 b(eb)g(esgue)515 1476 y(me)g(asur)g(e,)34 b(such)h(that)g Fj(j)p Fe(T)1442 1440 y Fh(n)1488 1476 y Fl(\()p Fe(v)t Fl(\))p Fj(j)27 b(!)h Fl(1)515 1757 y Fa(Pro)s(of.)66 b Fl(W)-8 b(e)40 b(assume)h(for)f(simplicit)m(y)d(that)j(2)p Fe(=g)j Fl(=)e Fe(n)p Fl(,)h Fe(n)f Fj(2)g Fi(N)9 b Fl(.)72 b(W)-8 b(e)40 b(\014rst)515 1873 y(pro)m(v)m(e)24 b(the)e(follo)m(wing) e(lemma.)38 b(Consider)23 b(a)f(smo)s(oth)g(mapping)f Fe(A)28 b Fl(:)g(\()p Fe(x;)17 b(y)t Fl(\))26 b Fj(!)515 1990 y Fl(\()p Fe(x)608 1954 y Fd(0)631 1990 y Fe(;)17 b(y)727 1954 y Fd(0)750 1990 y Fl(\))36 b(of)h(the)g(plane)f Fi(R)1442 1954 y Ff(2)1525 1990 y Fl(to)g(itself,)h(suc)m(h)h(that)f (all)e(the)i(p)s(oin)m(ts)f Fe(y)i Fl(=)d(0)i(are)515 2106 y(the)c(stationary)f(p)s(oin)m(ts)g(for)g(this)g(mapping:)1051 2304 y Fe(x)1106 2263 y Fd(0)1157 2304 y Fl(=)27 b Fe(x)c Fl(+)f Fe(B)1510 2319 y Ff(1)1550 2304 y Fl(\()p Fe(x;)17 b(y)t Fl(\))p Fe(;)113 b(y)1969 2263 y Fd(0)2019 2304 y Fl(=)28 b Fe(a)p Fl(\()p Fe(x)p Fl(\))p Fe(y)d Fl(+)d Fe(B)2550 2319 y Ff(2)2590 2304 y Fl(\()p Fe(x;)17 b(y)t Fl(\))p Fe(;)515 2503 y Fl(where)34 b(the)f(functions)f Fj(j)p Fe(B)1487 2518 y Ff(1)1526 2503 y Fj(j)c(\024)g Fe(C)1757 2518 y Ff(1)1796 2503 y Fj(j)p Fe(y)t Fj(j)p Fl(,)j Fj(j)p Fe(B)2064 2518 y Ff(2)2103 2503 y Fj(j)d(\024)g Fe(C)2334 2518 y Ff(2)2373 2503 y Fj(j)p Fe(y)t Fj(j)2481 2467 y Ff(2)2552 2503 y Fl(as)k Fj(j)p Fe(x)p Fj(j)p Fe(;)17 b Fj(j)p Fe(y)t Fj(j)26 b Fe(<)i Fl(1.)515 2784 y Fa(Lemma)37 b(6.2)42 b Fk(L)-5 b(et)37 b(on)g(the)g(interval)f Fe(x)c Fj(2)g Fl(\()p Fe(x)2210 2799 y Ff(1)2250 2784 y Fe(;)17 b(x)2349 2799 y Ff(2)2389 2784 y Fl(\))31 b Fj(2)h Fl([)p Fj(\000)p Fl(1)p Fe(;)17 b Fl(1])37 b Fk(the)g(function) 515 2901 y Fj(j)p Fe(a)p Fl(\()p Fe(x)p Fl(\))p Fj(j)27 b(\024)h Fe(\033)k(<)c Fl(1)p Fk(.)42 b(Then)27 b(the)h(e)-5 b(quilibria)27 b Fl(\()p Fe(x;)17 b Fl(0\))p Fk(,)28 b Fe(x)h Fj(2)f Fl(\()p Fe(x)2567 2916 y Ff(1)2606 2901 y Fe(;)17 b(x)2705 2916 y Ff(2)2745 2901 y Fl(\))28 b Fk(ar)-5 b(e)27 b(Lyapunov)515 3017 y(stable)34 b(and)h(asymptotic)-5 b(al)5 b(ly)34 b(stable)h(with)f(r)-5 b(esp)g(e)g(ct)35 b(to)g Fe(y)t Fk(.)515 3298 y Fa(Pro)s(of)g(of)g(Lemma)g(6.2.)43 b Fl(Let)30 b(0)e Fj(2)g Fl(\()p Fe(x)1995 3313 y Ff(1)2035 3298 y Fe(;)17 b(x)2134 3313 y Ff(2)2173 3298 y Fl(\).)43 b(W)-8 b(e)31 b(sho)m(w)h(that)e(the)h(equilib-)515 3415 y(rium)e(\(0,0\))h(is)g(Ly)m(apuno)m(v)i(stable.)42 b(Consider)31 b(a)f Fe(\016)t Fl(-neigh)m(b)s(ourho)s(o)s(d)f(of)h(zero,)515 3531 y(whic)m(h)h(lies)g(inside)f(the)i(in)m(terv)-5 b(al)30 b(\()p Fe(x)1846 3546 y Ff(1)1885 3531 y Fe(;)17 b(x)1984 3546 y Ff(2)2024 3531 y Fl(\).)43 b(Let)31 b Fe(x)d Fj(2)g Fl(\()p Fj(\000)p Fe(\016)n(;)17 b(\016)t Fl(\).)44 b(Consider)32 b(the)515 3657 y(iterations)f(of)i(the)g (mapping)f Fe(A)p Fl(.)45 b(F)-8 b(or)32 b(an)m(y)39 b(~)-54 b Fe(\033)32 b(>)d(\033)t Fl(,)38 b(~)-54 b Fe(\033)32 b(<)d Fl(1)j(there)i(is)f(a)3155 3630 y(~)3150 3657 y Fe(\016)g(>)28 b Fl(0)515 3773 y(suc)m(h)34 b(that)e(when)i Fj(j)p Fe(y)t Fj(j)26 b Fe(<)1443 3747 y Fl(~)1439 3773 y Fe(\016)36 b Fl(and)d Fe(x)28 b Fj(2)g Fl(\()p Fe(x)1978 3788 y Ff(1)2018 3773 y Fe(;)17 b(x)2117 3788 y Ff(2)2156 3773 y Fl(\),)33 b Fj(j)p Fe(y)2334 3737 y Fd(0)2356 3773 y Fj(j)27 b Fe(<)33 b Fl(~)-54 b Fe(\033)t Fj(j)p Fe(y)t Fj(j)p Fl(.)42 b(Let)33 b Fj(j)p Fe(y)t Fj(j)26 b Fe(<)3168 3747 y Fl(~)3163 3773 y Fe(\016)t Fl(.)515 3972 y(Supp)s(ose)47 b(that)e(at)h Fe(n)k Fj(\024)h Fe(N)56 b Fl(the)46 b Fe(n)1877 3936 y Fh(th)1993 3972 y Fl(p)s(o)m(w)m(er)h (of)f(the)g(mapping)e Fe(A)3084 3936 y Fh(n)3131 3972 y Fl(\()p Fe(x)p Fl(\))51 b Fj(2)515 4088 y Fl(\()p Fe(x)608 4103 y Ff(1)648 4088 y Fe(;)17 b(x)747 4103 y Ff(2)786 4088 y Fl(\),)51 b(and,)g(consequen)m(tly)-8 b(,)52 b Fj(j)p Fe(A)1852 4052 y Fh(n)1899 4088 y Fl(\()p Fe(y)t Fl(\))p Fj(j)f Fe(<)58 b Fl(~)-55 b Fe(\033)2293 4052 y Fh(n)2340 4088 y Fj(j)p Fe(y)t Fj(j)p Fl(,)50 b(and)d(as)g Fe(n)53 b Fl(=)f Fe(N)42 b Fl(+)32 b(1)515 4205 y Fe(A)588 4168 y Fh(N)7 b Ff(+1)745 4205 y Fl(\()p Fe(x)p Fl(\))40 b Fe(=)-61 b Fj(2)28 b Fl(\()p Fe(x)1091 4220 y Ff(1)1131 4205 y Fe(;)17 b(x)1230 4220 y Ff(2)1270 4205 y Fl(\).)43 b(Then)1003 4403 y Fj(j)p Fe(A)1104 4362 y Fh(N)7 b Ff(+1)1261 4403 y Fl(\()p Fe(x)p Fl(\))p Fj(j)27 b(\024)i(j)p Fe(x)p Fj(j)22 b Fl(+)g Fe(C)1854 4418 y Ff(1)1893 4403 y Fj(j)p Fe(y)t Fj(j)e Fl(+)i Fe(:)17 b(:)g(:)22 b Fl(+)g Fe(C)2424 4418 y Ff(1)2463 4403 y Fj(j)p Fe(A)2564 4362 y Fh(N)2631 4403 y Fl(\()p Fe(y)t Fl(\))p Fj(j)27 b(\024)964 4650 y(\024)h Fe(\016)e Fl(+)c Fe(C)1306 4665 y Ff(1)1345 4650 y Fj(j)p Fe(y)t Fj(j)p Fl(\(1)f(+)27 b(~)-54 b Fe(\033)26 b Fl(+)c Fe(:)17 b(:)g(:)k Fl(+)27 b(~)-54 b Fe(\033)2131 4609 y Fh(N)2199 4650 y Fl(\))27 b Fj(\024)h Fe(\016)e Fl(+)c Fe(C)2606 4665 y Ff(1)2650 4624 y Fl(~)2646 4650 y Fe(\016)2787 4583 y Fl(1)p 2703 4627 218 4 v 2703 4732 a(1)g Fj(\000)2878 4705 y Fl(~)2873 4732 y Fe(\016)515 4890 y Fl(This)30 b(estimate)g(do)s(es)h(not)f(dep)s(end)i(on)f Fe(N)10 b Fl(.)43 b(If)30 b(the)h(constan)m(ts)h Fe(\016)n(;)2923 4864 y Fl(~)2918 4890 y Fe(\016)j Fl(are)30 b(small)515 5006 y(enough,)37 b(then)f Fe(A)1180 4970 y Fh(n)1227 5006 y Fl(\()p Fe(x)p Fl(\))e Fj(2)g Fl(\()p Fe(x)1585 5021 y Ff(1)1624 5006 y Fe(;)17 b(x)1723 5021 y Ff(2)1763 5006 y Fl(\))36 b(for)f(all)f Fe(n)f Fj(2)h Fi(N)9 b Fl(,)42 b(if)35 b Fe(x)e Fj(2)h Fl(\()p Fj(\000)p Fe(\016)n(;)17 b(\016)t Fl(\).)54 b(But)36 b(in)1905 5255 y Fn(28)p eop %%Page: 29 29 29 28 bop 515 523 a Fl(this)35 b(case)h Fj(j)p Fe(A)1018 487 y Fh(n)1065 523 y Fl(\()p Fe(y)t Fl(\))p Fj(j)e Fl(deca)m(ys)j(exp) s(onen)m(tially)d(fast:)49 b Fj(j)p Fe(A)2503 487 y Fh(n)2550 523 y Fl(\()p Fe(y)t Fl(\))p Fj(j)31 b(\024)38 b Fl(~)-54 b Fe(\033)2906 487 y Fh(n)2953 523 y Fj(j)p Fe(y)t Fj(j)33 b Fl(\(whic)m(h)515 658 y(means)28 b(asymptotic)g(stabilit)m(y\),)g (and)h Fj(j)p Fe(A)2036 621 y Fh(n)2083 658 y Fl(\()p Fe(x)p Fl(\))p Fj(j)e(\024)h Fe(\016)19 b Fl(+)14 b Fe(C)2596 673 y Ff(1)2693 601 y(~)2690 618 y Fh(\016)p 2645 635 124 4 v 2645 701 a Ff(1)p Fd(\000)2738 683 y Ff(~)2735 701 y Fh(\016)2779 658 y Fl(,)29 b(whic)m(h)g(pro)m(v)m(es)515 785 y(Ly)m(apuno)m(v)34 b(stabilit)m(y)-8 b(.)42 b Fb(\003)515 984 y Fl(Consider)23 b(no)m(w)g(the)h("ligh)m(t")c(tra)5 b(jectory)24 b(with)e(the)h(initial)c(conditions)j Fe(v)32 b Fl(=)27 b(1,)515 1101 y Fe(t)39 b Fl(=)g Fe(\034)11 b Fl(.)65 b(This)39 b(tra)5 b(jectory)40 b(is)f(a)g Fk(stationary)i(p) -5 b(oint)39 b Fl(of)g(the)h(mapping)e Fe(T)14 b Fl(,)41 b(as)515 1217 y(w)m(e)35 b(assumed,)g(that)f(2)p Fe(=g)f Fl(=)d Fe(n)p Fl(,)35 b(moreo)m(v)m(er,)g(all)d(the)i(p)s(oin)m(ts)g (\()p Fe(v)g Fl(=)c Fj(\000)p Fl(1)p Fe(;)17 b(t)p Fl(\))34 b(are)515 1333 y(stationary)-8 b(.)515 1532 y(There)33 b(is)e(an)h(in)m(terv)-5 b(al)30 b(\()p Fe(\034)i Fj(\000)21 b Fe(\033)n(;)c(\034)33 b Fl(+)20 b Fe(\033)t Fl(\))32 b(suc)m(h)h(that)f Fe(d)-16 b(f)11 b(=dt)p Fl(\()p Fe(t)p Fl(\))27 b Fe(>)g Fl(0)32 b(for)f(all)f Fe(t)e Fj(2)515 1649 y Fl(\()p Fe(\034)c Fj(\000)13 b Fe(\033)n(;)k(\034)25 b Fl(+)13 b Fe(\033)t Fl(\).)41 b(But)29 b(this)e(means)h(that)g(the)h (mapping)d Fe(T)42 b Fl(is)27 b(asymptotically)515 1765 y(stable)e(with)f(resp)s(ect)j(to)d Fe(v)29 b Fl(in)24 b(the)i(linear)d(appro)m(ximation)g(for)i(an)m(y)g(\014xed)h Fe(t)g Fl(in)515 1881 y(this)32 b(in)m(terv)-5 b(al.)42 b(Th)m(us)34 b(the)e(conditions)g(of)f(Lemma)g(1)h(are)h(ful\014lled,)d (and)j(the)515 1997 y("ligh)m(t")25 b(tra)5 b(jectories)26 b Fe(v)32 b Fl(=)27 b(1,)h Fe(t)g Fj(2)g Fl(\()p Fe(\034)22 b Fj(\000)11 b Fe(\033)n(;)17 b(\034)23 b Fl(+)11 b Fe(\033)t Fl(\))26 b(are)h Fk(asymptotic)-5 b(al)5 b(ly)26 b Fl(stable)515 2113 y(in)46 b Fe(v)t Fl(.)87 b(Ob)m(viously)47 b(the)g("basin)g(of)f (attraction")g(has)h(p)s(ositiv)m(e)g(Leb)s(esgue)515 2230 y(measure)33 b(on)f(\()p Fj(j)p Fe(v)t Fj(j)27 b(\024)h Fl(1)p Fe(;)114 b(t)100 b Fl(\(mo)s(d)32 b(1\)\).)43 b Fb(\003)515 2429 y Fa(Remark.)d Fl(Under)25 b(the)f(conditions)f(of)h (Theorem)g(6.1)f(the)i(in)m(v)-5 b(arian)m(t)22 b(set)j Fj(M)3356 2393 y Fd(0)515 2545 y Fl(ma)m(y)34 b(not)h(b)s(e)g (attracting!)50 b(Indeed,)37 b(if)d(there)h(is)g(suc)m(h)h(momen)m(t)e (of)h(time)e Fe(\034)3312 2560 y Ff(1)3352 2545 y Fl(,)515 2661 y(that)736 2618 y Fh(d)-12 b(f)p 736 2638 66 4 v 738 2696 a(dt)812 2661 y Fl(\()p Fe(\034)892 2676 y Ff(1)931 2661 y Fl(\))28 b Fe(<)f Fl(0,)33 b(then)g(the)f(equilibria)e Fe(v)h Fl(=)d(1)p Fe(;)17 b(t)27 b Fl(=)h Fe(t)2505 2676 y Ff(1)2572 2661 y Fj(2)g Fl(\()p Fe(\034)2746 2676 y Ff(1)2808 2661 y Fj(\000)22 b Fe(\033)2962 2676 y Ff(1)3002 2661 y Fe(;)17 b(\034)3088 2676 y Ff(1)3149 2661 y Fl(+)k Fe(\033)3301 2676 y Ff(1)3341 2661 y Fl(\))515 2778 y(of)32 b(the)i(mapping)d Fe(T)47 b Fl(are)33 b Fk(unstable)g Fl(\(the)g(constan)m(t)h Fe(\033)2495 2793 y Ff(1)2568 2778 y Fl(is)f(c)m(hosen)i(suc)m(h)f(that)525 2850 y Fh(d)-12 b(f)p 525 2871 V 527 2928 a(dt)600 2894 y Fl(\()p Fe(t)p Fl(\))28 b Fe(<)g Fl(0)k(for)g Fe(t)c Fj(2)g Fl(\()p Fe(\034)1310 2909 y Ff(1)1372 2894 y Fj(\000)23 b Fe(\033)1527 2909 y Ff(1)1566 2894 y Fe(;)17 b(\034)1652 2909 y Ff(1)1714 2894 y Fl(+)22 b Fe(\033)1867 2909 y Ff(1)1907 2894 y Fl(\)\).)515 3176 y Fa(Theorem)37 b(6.2)42 b Fk(L)-5 b(et)43 b Fl(2)p Fe(=g)i Fk(satisfy)e(the)g(fol)5 b(lowing)41 b(c)-5 b(ondition:)60 b Fl(2)p Fe(=g)45 b Fj(6)p Fl(=)d Fe(p=q)515 3292 y Fk(for)34 b(any)h(inte)-5 b(gers)35 b Fe(p;)17 b(q)31 b(>)c Fl(0)p Fk(,)35 b(such)g(that)g Fe(q)c Fj(\024)d Fe(N)10 b Fk(.)46 b(Supp)-5 b(ose)34 b(that)1409 3420 y Fc(Z)1509 3446 y Ff(1)1464 3645 y(0)1565 3555 y Fl(ln)1672 3488 y(1)22 b Fj(\000)h Fe(d)-16 b(f)11 b(=dt)p Fl(\()p Fe(t)p Fl(\))p 1672 3532 510 4 v 1673 3624 a(1)22 b(+)g Fe(d)-16 b(f)11 b(=dt)p Fl(\()p Fe(t)p Fl(\))2192 3555 y Fe(dt)27 b(<)h Fl(0)p Fe(:)515 3809 y Fk(Then,)34 b(if)g Fe(N)46 b Fk(is)34 b(su\016ciently)h(lar)-5 b(ge,)34 b(the)h(invariant)g(set)515 3925 y Fj(M)635 3889 y Fd(0)685 3925 y Fl(=)28 b Fj(f)p Fl(\()p Fe(v)t(;)17 b(t)99 b Fl(\(mo)s(d)32 b(1\)\))27 b(:)h Fe(v)k Fl(=)27 b(1)p Fj(g)35 b Fk(is)f(attr)-5 b(acting.)515 4207 y Fa(Pro)s(of.)43 b Fl(Let)33 b Fe(w)d Fl(=)d(1)22 b Fj(\000)h Fe(v)1461 4171 y Ff(2)1500 4207 y Fl(.)44 b(The)33 b(transformation)e Fe(T)46 b Fl(tak)m(es)34 b(\()p Fe(w)s(;)17 b(t)p Fl(\))31 b(to)665 4459 y Fe(w)738 4418 y Fd(0)789 4459 y Fl(=)892 4349 y Fc(\020)962 4392 y Fl(1)21 b Fj(\000)i Fe(V)p 962 4436 250 4 v 963 4527 a Fl(1)e(+)h Fe(V)1221 4349 y Fc(\021)1280 4371 y Ff(2)1320 4459 y Fe(w)i Fl(+)e Fe(O)s Fl(\()p Fe(w)1701 4418 y Ff(2)1739 4459 y Fl(\))p Fe(;)114 b(t)1953 4418 y Fd(0)2005 4459 y Fl(=)27 b Fe(t)22 b Fl(+)g(2)p Fe(=g)j Fl(+)d Fe(O)s Fl(\()p Fe(w)s Fl(\))99 b(\(mo)s(d)32 b(1\))p Fe(;)515 4716 y Fl(where)i Fe(V)49 b Fl(=)1016 4673 y Fh(d)-12 b(f)p 1016 4693 66 4 v 1018 4751 a(dt)1092 4716 y Fl(\()p Fe(t)1165 4680 y Fd(0)1188 4716 y Fl(\).)515 4916 y(No)m(w)33 b(Theorem)g(6.2)f(follo)m(ws)f(from) h(Theorem)g(3.1.)44 b Fb(\003)1905 5255 y Fn(29)p eop %%Page: 30 30 30 29 bop 515 523 a Fm(References)564 764 y Fl([1])41 b(G.)32 b(Birkho\013.)g Fk(Dynamic)-5 b(al)34 b(Systems)p Fl(.)e(AMS,)h(New)h(Y)-8 b(ork,)33 b(1927.)564 932 y([2])41 b(L.A.Bunimo)m(vic)m(h.)30 b(Billiards)d(that)i(are)h(close)g(to)g (scattering)g(billiards.)708 1048 y(\(Russian\))i Fk(Mat.)k(Sb.)e (\(N.S.\))e Fa(94)h Fl(\(136\))f(\(1974\),)f(49{73.)564 1217 y([3])41 b(V.V.Kozlo)m(v,)47 b(D.V.T)-8 b(reshc)m(h)m(\177)-46 b(ev.)47 b Fk(Bil)5 b(liar)-5 b(ds.)44 b(A)h(genetic)g(intr)-5 b(o)g(duction)708 1333 y(to)31 b(the)g(dynamics)f(of)g(systems)h(with)f (imp)-5 b(acts.)27 b Fl(T)-8 b(ranslations)28 b(of)f(Math-)708 1449 y(ematical)59 b(Monographs,)70 b(89.)61 b(American)g(Mathematical) f(So)s(ciet)m(y)-8 b(,)708 1565 y(Pro)m(vidence,)34 b(RI,)f(1991.)e (viii+171)f(pp.)564 1734 y([4])41 b(T.Kr)s(\177)-51 b(uger,)30 b(L.D.Pust)m(yl'nik)m(o)m(v)i(and)e(S.E.T)-8 b(roub)s(etzk)m(o)m(y)g(.) 33 b(Acceleration)708 1850 y(of)f(b)s(ouncing)g(balls)f(in)h(external)h (\014elds.)g Fk(Nonline)-5 b(arity)32 b Fa(8)h Fl(\(1995\),)708 1966 y(397{410.)564 2134 y([5])41 b(L.D.Landau,)27 b(E.M.Lifshitz.)f Fk(The)j(Classic)-5 b(al)27 b(the)-5 b(ory)29 b(of)g(Fields.)c Fl(P)m(erg-)708 2251 y(amon)32 b(press,)i(Oxford,)e(1962.)564 2419 y([6])41 b(A.P)m(oincar)m(\023)-46 b(e.)27 b(R)m(\023)-46 b(e\015exions)28 b(sur)f(la)f(th)m(\023)-46 b(eorie)27 b(cin)m(\023)-46 b(etique)27 b(des)g(gaz.)g Fk(J.)j(Phys.)708 2535 y(The)-5 b(or)g(et.)34 b(et)h(Appl.)e Fl(\(4\))f Fa(5)g Fl(\(1906\),)g(349-403)564 2704 y([7])41 b(L.D.Pust)m(yl'nik)m (o)m(v.)c(The)f(la)m(w)g(of)f(en)m(trop)m(y)i(increase)f(and)g (generalized)708 2820 y(billiards.)30 b(Russian)i(Math.)h(Surv)m(eys)i (54)d(\(1999\),)g(no.)g(3,)h(650{651.)564 2988 y([8])41 b(L.D.Pust)m(yl'nik)m(o)m(v.)31 b(P)m(oincar)m(\023)-46 b(e)30 b(mo)s(dels,)f(rigorous)g(justi\014cation)f(of)i(the)708 3104 y(second)35 b(la)m(w)e(of)g(thermo)s(dynamics)f(from)g(mec)m (hanics,)i(and)g(the)g(F)-8 b(ermi)708 3221 y(acceleration)30 b(mec)m(hanism.)g Fk(R)n(ussian)i(Math.)i(Surveys)d Fa(50)g Fl(\(1995\),)f(no.)708 3337 y(1,)i(145{189.)564 3505 y([9])41 b(L.D.Pust)m(yl'nik)m(o)m(v.)62 b(Stable)e(and)h(oscillating)d (motions)h(in)h(nonau-)708 3621 y(tonomous)f(dynamical)f(systems.)j(I)s (I.)f(\(Russian\))f Fk(T)-7 b(rudy)60 b(Moskov.)708 3737 y(Mat.)37 b(Ob)-5 b(\024)-45 b(s)n(\024)e(c.)34 b Fa(34)h Fl(\(1977\),)f(3{103.)g(English)g(transl.)g(in)h(T)-8 b(rans.Mosco)m(w)708 3854 y(Math.So)s(c.1978,no)32 b(2.)515 4022 y([10])41 b(L.D.Pust)m(yl'nik)m(o)m(v.)47 b(A)f(new)h(mec)m (hanism)f(for)f(particle)g(acceleration)708 4138 y(and)36 b(a)f(relativistic)e(analogue)i(of)g(the)h(F)-8 b(ermi-Ulam)31 b(mo)s(del.)k Fk(The)-5 b(or)g(et.)708 4254 y(and)34 b(Math.)h(Phys.)e Fa(77)f Fl(\(1988\),)g(no.)h(1,)f(1110{1115)515 4423 y([11])41 b(Y)-8 b(a.G.Sinai.)25 b(Dynamical)f(systems)29 b(with)d(elastic)g(re\015ections.)i(Ergo)s(dic)708 4539 y(prop)s(erties)56 b(of)f(disp)s(ersing)g(billiards.)e Fk(R)n(uss.)j(Math.)h(Surv.)e Fa(25)p Fl(,)62 b(2,)708 4655 y(\(1970\),)32 b(137{189.)515 4824 y([12])41 b(Y)-8 b(a.G.Sinai)32 b(\(ed.\).)j Fk(Dynamic)-5 b(al)34 b(Systems)i(2.)e Fl(Springer-V)-8 b(erlag,)33 b(1989,)708 4940 y(281)f(pp.)1905 5255 y Fn(30)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0206280733202--