Simpson’s Variational Integrator for Systems with Quadratic Lagrangians

Juan Antonio Rojas-Quintero (1, 2) , François Dubois (3, 4) , José Guadalupe Cabrera-Díaz (1)

1 TecNM - Tecnológico Nacional de México
2 LS2N - Laboratoire des Sciences du Numérique de Nantes
3 LMO - Laboratoire de Mathématiques d'Orsay
4 LMSSC - Laboratoire de Mécanique des Structures et des Systèmes Couplés

This contribution proposes a variational symplectic integrator aimed at linear systems issued from the least action principle. An internal quadratic finite-element interpolation of the state is performed at each time step. Then, the action is approximated by Simpson’s quadrature formula. The implemented scheme is implicit, symplectic, and conditionally stable. It is applied to the time integration of systems with quadratic Lagrangians. The example of the linearized double pendulum is treated. Our method is compared with Newmark’s variational integrator. The exact solution of the linearized double pendulum example is used for benchmarking. Simulation results illustrate the precision and convergence of the proposed integrator.

Mots clés: Ordinary Differential Equations; Oscillator; Numerical analysis; Symplectic scheme

Domaines: Mathématiques [math]; Informatique [cs]

Juan Antonio Rojas-Quintero, François Dubois, José Guadalupe Cabrera-Díaz. Simpson’s Variational Integrator for Systems with Quadratic Lagrangians. Axioms, 2024, 13 (4), pp.255. ⟨10.3390/axioms13040255⟩. ⟨hal-04548211⟩