A Note on Symplectic Algorithms

1. It is well known that the symplectic algorithms [1][2] for the finite dimensional Hamiltonian systems are very powerful and successful in numerical calculations in comparison with other various non-symplectic computational schemes since the symplectic schemes preserve the symplectic structure in certain sense. On the other hand, the Lagrangian formalism is quie useful for the Hanmiltonian systems. since the both are important at least in the equal fooding. Therefore, it should be not useless to establish the symplectic algorithms in the Lagrangian formalism. As a matter of fact, the Lagrangian formalism is more or less earier to generalized to the infinite dimensional systems such as classical field theory. In this note we present the symplectic geometry and symplectic algorithm in the Lagrangian formalism in addition to the Hamiltonian one for the finite dimensional Hamiltonian systems with the help of the Euler-Lagrange (EL) cohomological concepts introduced very recently by the authors in [5]. In the course of numerical calculation, the ”time” t ∈ R is always discretized, usually with equal spacing h = ∆t: