A GEOMETRIC DISCRETE MODEL FOR THE SECOND ORDER SYSTEMS

摘要

lfM is a manifold, then M!2=M×M is a model for TM, the tangent space of At (or the space of velocities) and M~3=M×M×M is a model for T~2M, the second order tangent space of M, or the second osculator space (the space of accelerations). A second order discrete Lagrangian is a Junction L: M~3→IR. The discrete Euler-Lagrange equation of L can be given by a variational principle and the solutions can be put in correspondence with generalized trajectories in T~*M. The Legendre map ofL can be also considered. One study some properties of right and left hyperregular discrete Lagrangians. Some examples of second order discrete Lagrangian systems are considered. Some new facts, that are not encountered in the first order case, are discussed.