Abstract
In this paper we apply some higher order symplectic numerical methods to analyze the dynamics of 3-site Toda lattices (reduced to relative coordinates). We present benchmark numerical simulations that has been generated from the HOMsPY (Higher Order Methods in Python) library. These results provide detailed information of the underlying Hamiltonian system. These numerical simulations reinforce the claim that the symplectic numerical methods are highly accurate qualitatively and quantitatively when applied not only to Hamiltonian of the Toda lattices, but also to other physical models. Excepting exactly integrable models, these symplectic numerical schemes are superior, efficient, energy preserving and suitable for a long time integrations, unlike standard non-symplectic numerical methods which lacks preservation of energy (and other constants of motion, when such exist).
Highlights
Hamiltonian equations of motion belong to a class of ordinary differential equations (ODEs) which in general are difficult or mostly impossible to solve analytically
The autonomous Hamiltonian equations of motion constitute a system of first order ordinary differential equations, q_a 1⁄4 @H ; @pa p_a 1⁄4 À
Since the symplectic solvers have been widely accepted to be superior than the conventional numerical methods for solving the Hamiltonian systems, Mushtaq et al [5] constructed a well behaved class of higher order symplectic integrators schemes based on the extensions of the Stormer-Verlet scheme for Hamiltonians like Eq (1)
Summary
Hamiltonian equations of motion belong to a class of ordinary differential equations (ODEs) which in general are difficult or mostly impossible to solve analytically. The standard non-symplectic numerical integrators, that have been used to solve general initial value problems numerically, do not preserve this qualitative behaviour, or the constants of motion for the system. Since the symplectic solvers have been widely accepted to be superior than the conventional numerical methods for solving the Hamiltonian systems, Mushtaq et al [5] constructed a well behaved class of higher order symplectic integrators schemes based on the extensions of the Stormer-Verlet scheme for Hamiltonians like Eq (1) The structure of the rest of this paper is as follows: In Section 2 we review the Toda lattice models These are integrable, nonlinear systems that have a number of extra constants of motions beyond standard ones like energy and momentum.
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