99-81 Ulrich Mutze

Predicting Classical Motion Directly from the Action Principle (305K, Postscript) Mar 19, 99

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Abstract. Time-discrete non-relativistic classical dynamics for $d$ degrees of freedom is here formulated in a new way: Discrete approximations to the action (i.e. the time integral of the Lagrangian) determine the new state from a given one, and no differential equations or discretized differential equations are being used. The basis for this dynamical law is a formulation of initial conditions by position and $d+1$ velocities. The mean value of these velocities corresponds to the ordinary initial velocity. The dynamical law works as follows: for an initial configuration we consider $d+1$ rectilinear paths corresponding to the $d+1$ velocities of the initial condition. After a time step $\tau$ all paths change their velocity in such a manner, that they re-unite after a second time step $\tau$. The \emph{principle of stationary action} is then applied to these paths to determine the position of the merging point. The $d+1$ new velocities are taken as the velocities associated with the sub-paths of the second time step. The method is as straightforward and as careless about achieving numerical accuracy as the Euler discretization of differential equations. It turns out, however, that it is closely related---both in explicit representation and in performance---to the more powerful modern \emph{symplectic integration methods} for Hamiltonian systems.

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