Discrete Hamilton-Pontryagin mechanics and generating functions on Lie groupoids

Stern, Ari

doi:10.4310/jsg.2010.v8.n2.a5

Cited by 14 publications

(14 citation statements)

References 19 publications

(20 reference statements)

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“…In particular, the constrained variational formulation of continuous Lie-Poisson reduction [6] appears to be related to the Hamilton-Pontryagin variational principle [27]. It would be interesting to develop discrete Lie-Poisson reduction [18] from the Hamiltonian perspective, in the context of the discrete Hamilton-Pontryagin principle [16,25]. • Extensions to Multisymplectic Hamiltonian PDEs.…”

Section: Discussionmentioning

confidence: 99%

Discrete Hamiltonian variational integrators

Leok

Zhang²

2011

IMA Journal of Numerical Analysis

113

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We derive a variational characterization of the exact discrete Hamiltonian, which is a Type II generating function for the exact flow of a Hamiltonian system, by considering a Legendre transformation of Jacobi's solution of the Hamilton-Jacobi equation. This provides an exact correspondence between continuous and discrete Hamiltonian mechanics, which arise from the continuous-and discrete-time Hamilton's variational principle on phase space, respectively. The variational characterization of the exact discrete Hamiltonian naturally leads to a class of generalized Galerkin Hamiltonian variational integrators that includes the symplectic partitioned Runge-Kutta methods. This extends the framework of variational integrators to Hamiltonian systems with degenerate Hamiltonians, for which the standard theory of Lagrangian variational integrators cannot be applied. We also characterize the group invariance properties of discrete Hamiltonians that lead to a discrete Noether's theorem.

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Section: Discussionmentioning

confidence: 99%

Discrete Hamiltonian variational integrators

Leok

Zhang²

2011

IMA Journal of Numerical Analysis

113

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“…The variational principle for (14), in which the configuration variables, velocities and momenta are varied independently while the reconstruction equations are treated as constraints, is a particular example of the Hamiltonian-Pontryagin principle (see [16]). A version of the Hamilton-Pontryagin principle specific to Lie groups can be found in [2]; see also [1].…”

Section: The Deformation Energymentioning

confidence: 99%

“…which can be viewed as the discrete counterpart of (14). Here, g k ∈ SE(2), ξ k ∈ se(2) and µ k ∈ se(2) * are independent variables.…”

Section: The Deformation Energymentioning

confidence: 99%

Relative geodesics in the special Euclidean group

Holm

Noakes²,

Vankerschaver

2013

Proc. R. Soc. A.

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We introduce a measurement (the discrepancy) of the minimum energy needed to transform from a standard parametrized planar curve c 0 to an observed curve c 1 . To this end, we say that a curve of transformations in the special Euclidean group SE (2) is admissible if it maps the source curve to the target curve under the point-wise action of SE (2) on the plane. After endowing the group SE(2) with a leftinvariant metric, we define a relative geodesic in SE(2) to be a critical point of the energy functional associated to the metric, over all admissible curves. The discrepancy is then defined as the value of the energy of the minimizing relative geodesic. In the first part of the paper, we derive a scalar ODE which is a necessary condition for a curve in SE(2) to be a relative geodesic, and we discuss some of the properties of the discrepancy. In the second part of the paper, we consider discrete curves, and by means of a variational principle, we derive a system of discrete equations for the relative geodesics. We finish with several examples.

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“…Let L : G → R be a discrete Lagrangian on a Lie groupoid G ⇒ M . Then, S L = dL(G) ⊆ T * G is clearly a Lagrangian submanifold of the symplectic manifold T * G and S L is also an implicit difference equation on the cotangent groupoid T * G ⇒ A * G. If we apply the forward integrability algorithm to S L , then there is an interesting relation with the discrete Euler-Lagrange equations for L (see [18,25] and the Appendix A for more details on discrete Lagrangian Mechanics on Lie groupoids), which we describe in this section. Proposition 4.1.…”

Section: Discrete Dynamics In Implicit Formmentioning

confidence: 99%

Discrete dynamics in implicit form

Ponte¹,

Marrero²,

Diego³

et al. 2013

Discrete &Amp; Continuous Dynamical Systems - A

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A notion of implicit difference equation on a Lie groupoid is introduced and an algorithm for extracting the integrable part (backward or/and forward) is formulated. As an application, we prove that discrete Lagrangian dynamics on a Lie groupoid G may be described in terms of Lagrangian implicit difference equations of the corresponding cotangent groupoid T * G. Other situations include finite difference methods for time-dependent linear differentialalgebraic equations and discrete nonholonomic Lagrangian systems, as particular examples. Dedicated to Ernesto Lacomba on the occasion of his 65th birthday

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Discrete Hamilton-Pontryagin mechanics and generating functions on Lie groupoids

Cited by 14 publications

References 19 publications

Discrete Hamiltonian variational integrators

Discrete Hamiltonian variational integrators

Relative geodesics in the special Euclidean group

Discrete dynamics in implicit form

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