 99271 Ulrich Mutze
 Predicting Classical Motion Directly from the Action Principle II
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Jul 16, 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 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 related both in
explicit representation and in performance  to the more powerful
symplectic integration methods for Hamiltonian systems. If the spread
of the velocities is adjusted suitably, the method also accounts for
the lowest order quantum correction to the motion of the center of a
wave packet.
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