Total Variation and Variational Symplectic-Energy- Momentum integrators

In 1980’s, Lee proposed an energy-preserving discrete mechanics with variable time steps by taking (discrete) time as dynamical variable [2, 3, 4]. On the other hand, motivated by the symplectic property of Lagrangian mechanics, a version of discrete Lagrangian mechanics has been devoloped and variational integrators that preserve discrete symplectic two form have been obtained [11, 12, 15, 16, 17]. However, variational integrators obtained in this way have fixed time steps and consequently they are not energypreserving in general. Obviously, the energy-preserving discrete mechanics and variational integrator are more preferable since solutions of the Euler-Lagrange equations of conservative continuous systems are not only symplectic but also energy-preserving. In order to attain this goal, some discrete mechanics with discrete energy conservation and symplectic variational integrators are needed to study. Recently, Kane, Marsden and Ortiz have employed appropriate time steps to conserve a defined discrete energy and developed what they called symplectic-energy-momentum preserving variational integrators in [10]. Although their approach is more or less related to Lee’s discrete mechanics, but the discrete energy preserving condition is not derived by the variational principle. The purpose of this letter is to generalize or improve these approaches as well as to explore the relation among discrete total variation, Lee’s discrete mechanics and Kane-Marsden-Ortiz’s integrators. We will present a discrete total variation calculus with variable time steps and a discrete mechanics that is discretely symplectic, energy preserving and has the correct continuous limit. In fact, this discrete variation calculus and mechanics are a generalization of Lee’s discrete mechanics in symplectic-preserving sense and can directly derive the variational symplectic-energy-momentum integrators by Kane, Marsden and Ortiz. This letter is organized as follows. In the next section, we remind total variation calculus for continuous mechanics. In section 3, we present a discrete total variation calculus with variable time steps and a discrete mechanics, derive Kane-Marsden-Ortiz’s integrators and explore the relation among our approach, Lee’s discrete mechanics and Kane-Marsden-Ortiz’s approach. We finish with some conclusions and remarks in Section 4. Before ending this section, we recall very briefly the ordinary variational principle in Lagrangian mechanics for later use. Suppose Q denotes the extended configuration space with coordinates (t, q) and Q the first prolongation of Q with coordinates (t, q, q̇) [13]. Here t denotes time and q, i = 1, 2, · · · , n denote the position. Consider a Lagrangian L : Q → R. The corresponding action functional is