 9981 Ulrich Mutze
 Predicting Classical Motion Directly from the Action Principle
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Mar 19, 99

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Abstract. Timediscrete nonrelativistic 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
reunite 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 subpaths 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 relatedboth in
explicit representation and in performanceto the more
powerful modern \emph{symplectic integration methods} for
Hamiltonian systems.
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