2014/10/21 by Leonardo Colombo, Colombo, Leonardo, Sebastián J. Ferraro +3
Mathematics · #49J15 (Secondary) #70G45 (Primary) #70Hxx #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.1410.5766
openalex publication_date 2014/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Numerical methods that preserve geometric invariants of the system, such as\nenergy, momentum or the symplectic form, are called geometric integrators. In\nthis paper we present a method to construct symplectic-momentum integrators for\nhigher-order Lagrangian systems. Given a regular higher-order Lagrangian\nL colon T(k)Q\→\ℝ with k\≥ 1, the resulting discrete\nequations define a generally implicit numerical integrator algorithm on\nT(k-1)Q\× T(k-1)Q that approximates the flow of the higher-order\nEuler--Lagrange equations for L. The algorithm equations are called\nhigher-order discrete Euler--Lagrange equations and constitute a variational\nintegrator for higher-order mechanical systems. The general idea for those\nvariational integrators is to directly discretize Hamilton's principle rather\nthan the equations of motion in a way that preserves the invariants of the\noriginal system, notably the symplectic form and, via a discrete version of\nNoether's theorem, the momentum map.\n We construct an exact discrete Lagrangian Lde using the locally unique\nsolution of the higher-order Euler--Lagrange equations for L with boundary\nconditions. By taking the discrete Lagrangian as an approximation of Lde,\nwe obtain variational integrators for higher-order mechanical systems. We apply\nour techniques to optimal control problems since, given a cost function, the\noptimal control problem is understood as a second-order variational problem.\n