Geometric Hamilton–Jacobi theory for systems with external forces

León, Manuel de; Valcázar, Manuel Lainz; Asier, López-Gordón,

doi:10.1063/5.0073214

Cited by 5 publications

(4 citation statements)

References 58 publications

(54 reference statements)

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“…This is the discrete analogue of the first statement from our reduction lemma [6, lemma 15] (see also references [7,21]). The first statement was previously found by Marsden and West [23, theorem 3.2.1].…”

Section: If This Equation Holds F D Is ξ-Invariant If and Only Ifmentioning

confidence: 63%

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Discrete Hamilton–Jacobi theory for systems with external forces

León

Valcázar

Asier

2022

J. Phys. A: Math. Theor.

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This paper is devoted to discrete mechanical systems subject to external forces. We introduce a discrete version of systems with Rayleigh-type forces, obtain the equations of motion and characterize the equivalence for these systems. Additionally, we obtain a Noether’s theorem and other theorem characterizing the Lie subalgebra of symmetries of a forced discrete Lagrangian system. Moreover, we develop a Hamilton-Jacobi theory for forced discrete Hamiltonian systems. These results are useful for the construction of so-called variational integrators, which, as we illustrate with some examples, are remarkably superior to the usual numerical integrators such as the Runge-Kutta method.

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Section: If This Equation Holds F D Is ξ-Invariant If and Only Ifmentioning

confidence: 63%

“…Remark 1. It is worth noting that we have changed the sign criteria for discrete forces with respect to our previous papers [6,7,21], in order to be consistent with Lew, Marsden, Ortiz and West's criteria [20,23].…”

Section: Lagrangian Systems With External Forcesmentioning

confidence: 99%

Discrete Hamilton–Jacobi theory for systems with external forces

León

Valcázar

Asier

2022

J. Phys. A: Math. Theor.

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“…There is a nice way to employ external forces to Hamiltonian dynamics. For the dynamical systems admitting some external forces, geometric HJ theory is studied in [5,50,99].…”

Section: An Incomplete Literature On Geometric Hamilton-jacobi Theorymentioning

confidence: 99%

“…on the cotangent bundle, called Darboux coordinates, on cotangent bundle T * Q such that the canonical one form turns out to be Θ Q = p i dq i , whereas the canonical symplectic two-form in (50) becomes…”

Section: Space Of Hamiltonian Vector Fields (Revisited)mentioning

confidence: 99%

Reviewing the geometric Hamilton–Jacobi theory concerning Jacobi and Leibniz identities

Esen¹,

León²,

Valcázar³

et al. 2022

J. Phys. A: Math. Theor.

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In this survey, we review the classical Hamilton–Jacobi theory from a geometric point of view in different geometric backgrounds. We propose a Hamilton-Jacobi equation for different geometric structures attending to one particular characterization: whether they fulfill the Jacobi and Leibniz identities simultaneously, or if at least they satisfy one of them. In this regard, we review the case of time-dependent ($t$-dependent in the sequel) and dissipative physical systems as systems that fulfill the Jacobi identity but not the Leibnitz identity. Furthermore, we review the contact-evolution Hamilton-Jacobi theory as a split off the regular contact geometry, and that actually satisfies the Leibniz rule instead of Jacobi. Furthermore, we include a novel result, which is the Hamilton-Jacobi equation for conformal Hamiltonian vector fields as a generalization of the well-known Hamilton-Jacobi on a symplectic manifold, that is retrieved in the case of a zero conformal factor. The interest of a geometric Hamilton–Jacobi equation is the primordial observation that if a Hamiltonian vector field $X_H$ can be projected into a configuration manifold by means of a $1$-form $dW$, then the integral curves of the projected vector field $X_{H}^{dW}$ can be transformed into integral curves of $X_H$ provided that $W$ is a solution of the Hamilton-Jacobi equation. Geometrically, the solution of the Hamilton-Jacobi equation plays the role of a Lagrangian submanifold of a certain bundle. Exploiting these features in different geometric scenarios we propose a geometric theory for multiple physical systems depending on the fundamental identities that their dynamic satisfies. Different examples are pictured to reflect the results provided, being all of them new, except for one that is reassessment of a previously considered example.

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