Mechanical integrators derived from a discrete variational principle.(English)Zbl 0963.70507
Summary: Many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion. In this paper, we present a procedure to construct time-stepping algorithms that approximate the flow of continuous ODEs for mechanical systems by discretizing Hamilton’s principle rather than the equations of motion. The discrete equations share similarities to the continuous equations by preserving invariants, including the symplectic form and the momentum map. We first present a formulation of discrete mechanics along with a discrete variational principle. We then show that the resulting equations of motion preserve the symplectic form, and that this formulation of mechanics leads to conservation laws from a discrete version of Noether’s theorem. We then use the discrete mechanics formulation to develop a procedure for constructing symplectic-momentum mechanical integrators for Lagrangian systems with holonomic constraints. We apply the construction procedure to the rigid body and the double spherical pendulum to demonstrate numerical properties of the integrators.
MSC:
| 70-08 | Computational methods for problems pertaining to mechanics of particles and systems |
| 70H25 | Hamilton’s principle |
| 65L99 | Numerical methods for ordinary differential equations |
| 70E15 | Free motion of a rigid body |
| 70F20 | Holonomic systems related to the dynamics of a system of particles |
| 70H03 | Lagrange’s equations |
Keywords:
rigid body integration;discretized Hamilton’s principle;time-stepping algorithms;discrete version of Noether’s theorem;symplectic-momentum mechanical integrators;Lagrangian systems with holonomic constraints;rigid body;double spherical pendulumSoftware:
MathematicaCite
References:
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