Jolt factorization of pendulum map

Abstract

Abstract. In this paper, we apply the symplectic integration method using jolt factorization described in an earlier paper to the symplectic map describing the nonlinear pendulum Hamiltonian. We compare results obtained with this method with those obtained using nonsymplectic methods and demonstrate that our results are much better.

Solvable Map Method for Integrating Nonlinear Hamiltonian Systems

“…This however violates the symplectic condition. Though this method is justifiable in short term tracking, it does not work well in long term tracking as the non-symplecticity can lead to spurious damping or even chaotic behaviour which is not present in the original system [10]. Therefore, we refactorize M in terms of simpler symplectic maps that can be evaluated both exactly and quickly.…”

“…Several symplectic integration methods have been discussed in literature [1,2,3,4,5,6,7,8,9,10,11]. Methods using Lie algebraic perturbation theory which give maps where the final values of variables are explicit functions of initial ones offer several advantages.…”

Polynomial Map Factorization of Symplectic Maps

“…1 Several symplectic integration algorithms have been proposed in the literature. [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] Some of these directly use the Hamiltonian whereas others use the symplectic map 22,23 representing the nonlinear Hamiltonian system. For complicated systems like the Large Hadron Collider which has thousands of elements, using individual Hamiltonians for each element can drastically slow down the integration process.…”