Error analysis of variational integrators of unconstrained Lagrangian systems

Patrick, George W.; Cuell, Charles

doi:10.1007/s00211-009-0245-3

Cited by 42 publications

(93 citation statements)

References 14 publications

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“…Variational error analysis. Now we rewrite the result of Patrick [25] and Marsden and West [21] for the particular case of a Lagrangian L d : T Q × T Q → R. Following [21,26], we have the next result about the order of our variational integrator. Note that given a discrete Lagrangian L d : T Q × T Q → R its order can be calculated by expanding the expressions for L d (q(0),q(0), q(h),q(h), h) in a Taylor series in h and comparing this to the same expansions for the exact Lagrangian.…”

Section: Relationship Between Discrete and Continuous Variational Sysmentioning

confidence: 99%

Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal Control

2016

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Numerical methods that preserve geometric invariants of the system, such as energy, momentum or the symplectic form, are called geometric integrators. In this paper we present a method to construct symplectic-momentum integrators for higher-order Lagrangian systems. Given a regular higher-order Lagrangian L : T (k) Q → R with k ≥ 1, the resulting discrete equations define a generally implicit numerical integrator algorithm on T (k−1) Q × T (k−1) Q that approximates the flow of the higher-order Euler-Lagrange equations for L. The algorithm equations are called higher-order discrete Euler-Lagrange equations and constitute a variational integrator for higher-order mechanical systems. The general idea for those variational integrators is to directly discretize Hamilton's principle rather than the equations of motion in a way that preserves the invariants of the original system, notably the symplectic form and, via a discrete version of Noether's theorem, the momentum map.We construct an exact discrete Lagrangian L e d using the locally unique solution of the higher-order Euler-Lagrange equations for L with boundary conditions. By taking the discrete Lagrangian as an approximation of L e d , we obtain variational integrators for higher-order mechanical systems. We apply our techniques to optimal control problems since, given a cost function, the optimal control problem is understood as a second-order variational problem. Contents 24 References 24Mathematics Subject Classification (2010): 70G45, 70Hxx, 49J15.

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Section: Relationship Between Discrete and Continuous Variational Sysmentioning

confidence: 99%

Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal Control

2016

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“…Since we start from a variational SOdE, we have a corresponding discrete Lagrangian L h d : Q × Q −→ R. Our aim is to find, using methods of backward error analysis, a continuous Lagrangian function such that the corresponding exact discrete Lagrangian is close to L h d up to any chosen order of accuracy. This implies that the flow of the corresponding continuous SODE is also close to the original discrete system up to the same order of accuracy, see [30].…”

Section: 2mentioning

confidence: 80%

The inverse problem of the calculus of variations for discrete systems

Barbero-Lin͂án¹,

Puiggalí²,

Ferraro³

et al. 2018

J. Phys. A: Math. Theor.

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We develop a geometric version of the inverse problem of the calculus of variations for discrete mechanics and constrained discrete mechanics. The geometric approach consists of using suitable Lagrangian and isotropic submanifolds. We also provide a transition between the discrete and the continuous problems and propose variationality as an interesting geometric property to take into account in the design and computer simulation of numerical integrators.

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“…An analysis of the stability properties of asynchronous VI's can be found in (Fong et al 2008). See (Focardi and Maria-Mariano 2008) for a converge analysis of asynchronous VI's in linear elastodynamics and (Patrick and Cuell 2009) for a complete error analysis.…”

Section: Variational Integrationmentioning

confidence: 98%

A Brief Introduction to Variational Integrators

Lew

2016

Structure-Preserving Integrators in Nonlinear Structural Dynamics and Flexible Multibody Dynamics

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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.

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Error analysis of variational integrators of unconstrained Lagrangian systems

Cited by 42 publications

References 14 publications

Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal Control

Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal Control

The inverse problem of the calculus of variations for discrete systems

A Brief Introduction to Variational Integrators

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