A Brief Introduction to Variational Integrators

Abstract

In this chapter, a brief introduction to the formulation of variational methods for finite-dimensional Lagrangian systems is presented. To this end, the first two sections focus on describing the Lagrangian and Hamiltonian points of view of mechanics for systems evolving on manifolds. Special attention is paid to the construction of the Lagrangian function and to the role of Hamilton's variational principle in the deduction of the balance equations. The relation between the symmetries of the Lagrangian function and the existence of invariants of the dynamics along with the symplectic nature of the flow are also addressed. In the third section, the discussion turns towards the formulation of a time-discrete analogue of the theory. The cornerstone of such a construction is given by a discrete analogue of Hamilton's variational principle which provides a systematic procedure to construct discrete approximations to the exact trajectory of a mechanical system on both the configuration space and the phase space. The approximation properties and the geometric characteristics of the resulting discrete trajectories are explained. Finally, we apply the variational methodology to construct symplectic and momentum-conserving time integrators for two problems of practical interest in engineering and science.

The GGL Variational Principle for Constrained Mechanical Systems

“…Fig. 16 in Lew & Mata [10]), H(t) oscillates around its true value and the energy error remains stable. Furthermore, as the gyroscopic top is subject to external forces acting in the e e e 3 -direction, the symmetry of the system reduces to a conservation of the angular momentum about the e e e 3 -axis such that L 3 = constant.…”

A framework for data‐driven structural analysis in general elasticity based on nonlinear optimization: The dynamic case

“…At this point, there is no necessity to specify n e since it depends on the structural model considered, which for now remains unspecified. The cost function (1) has to be minimized under the following constraints: (i) the compatibility equation that enforces the equivalence between strain variables and displacement‐based strains at time instant , in which is the vector of generalized coordinates and Q stands for the configuration manifold; (ii) the discrete balance equation that establishes the dynamic equilibrium, for instance, we chose an approximation inspired by a family of variational integrators 13‐20 that renders the dynamic equilibrium at time instant t i as …”

“…in which q i+ 1 2 ∈ Q ⊂ R m+n is the vector of generalized coordinates and Q stands for the configuration manifold; (ii) the discrete balance equation that establishes the dynamic equilibrium, for instance, we chose an approximation inspired by a family of variational integrators [13][14][15][16][17][18][19][20] that renders the dynamic equilibrium at time instant t i as…”

Simulating squirmers with volumetric solvers

“…Because (t) belongs to the special rotation group the manifold of possible configurations of the squirmer is Q = ℝ d × (d) [5,43,62] and the body's Eulerian velocity B is given by where c (t) =̇c(t) is the translational velocity and (t) is the pseudovector of angular velocities in the spatial frame. It relates to (t) and ̇( t) by where the isomorphism sk [⋅] between vectors and skewsymmetric matrices has been introduced.…”