Discrete Lagrangian mechanics for nonseparable nonsmooth systems

David Pekarek*, Todd D. Murphey

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


We consider event-driven schemes for the simulation of nonseparable mechanical systems subject to holonomic unilateral constraints. Systems are modeled in discrete time using variational integrator (VI) theory, by which equations of motion follow from discrete variational principles. For smooth dynamics, VIs are known to exactly conserve a discrete symplectic form and a modified Hamiltonian function. The latter of these conservation laws can play a pivotal role in stabilizing the energy behavior of collision simulations. Previous efforts to leverage modified Hamiltonian conservation have been limited to integrators using the Störmer-Verlet method on separable, nonsmooth Hamiltonian mechanical systems. We generalize the existing approach to the family of all VIs applied to nonseparable, potentially nonconservative Lagrangian mechanical systems. We examine the properties of the resulting integrators relative to other structured collision simulation methods in terms of conserved quantities, trajectory errors as a function of initial condition, and required computation time. Interestingly, we find that the modified collision Verlet algorithm (MCVA) using the Störmer-Verlet integrator defined as a composition method leads to the best accuracy. Although relative to this method, the VI-based generalized MCVA method offers computational savings when collisions are particularly sparse.

Original languageEnglish (US)
Pages (from-to)440-463
Number of pages24
JournalInternational Journal for Numerical Methods in Engineering
Issue number6
StatePublished - Feb 10 2016


  • Backwards error analysis
  • Double pendulum
  • Geometric integration
  • Hamiltonian systems
  • Impact dynamics
  • Lagrangian mechanics
  • Störmer-Verlet

ASJC Scopus subject areas

  • Numerical Analysis
  • General Engineering
  • Applied Mathematics


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