Systems of Hamilton-Jacobi equations

Cambronero, Julio; Álvarez, Javier Pérez

doi:10.1080/14029251.2019.1640473

Cited by 3 publications

(3 citation statements)

References 9 publications

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“…On the basis of the canonicalization, some important methods to solve motion differential equations (such as the Hamilton-Jacobi method [25][26][27][28][29][30][31][32] and the symplectic algorithm [33] ) have been proposed.…”

Section: Introductionmentioning

confidence: 99%

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Quasi-canonicalization for linear homogeneous nonholonomic systems*

Wang

Jinchao

et al. 2020

Chinese Phys. B

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For conservative linear homogeneous nonholonomic systems, there exists a cotangent bundle with the symplectic structure dπμ ∧ dξμ , in which the motion equations of the system can be written into the form of the canonical equations by the set of quasi-coordinates πμ and quasi-momenta ξμ . The key to construct this cotangent bundle is to define a set of suitable quasi-coordinates πμ by a first-order linear mapping, so that the reduced configuration space of the system is a Riemann space with no torsion. The Hamilton–Jacobi method for linear homogeneous nonholonomic systems is studied as an application of the quasi-canonicalization. The Hamilton–Jacobi method can be applied not only to Chaplygin nonholonomic systems, but also to non-Chaplygin nonholonomic systems. Two examples are given to illustrate the effectiveness of the quasi-canonicalization and the Hamilton–Jacobi method.

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Section: Introductionmentioning

confidence: 99%

“…From the geometric point of view, the canonicalization shows that a holonomic system has a natural symplectic structure. [27][28][29][30][31][32][33][34] If the symplectic form on the cotangent bundle of a holonomic system is recorded as Ω = dq µ ∧ dp µ , the Hamilton's equations (2) can be written as i X Ω = dH.…”

Section: Introductionmentioning

confidence: 99%

Quasi-canonicalization for linear homogeneous nonholonomic systems*

Wang

Jinchao

et al. 2020

Chinese Phys. B

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“…A new solution paradigm is proposed for the finite-horizon optimal control problem originally proposed in [23,24]. More specifically, sufficient conditions of optimality are rigorously established that require the solution of a system of Hamilton-Jacobi PDEs [26]. Similar conditions are proved for the extension of the same problem to an infinite horizon.…”

Section: Introductionmentioning

confidence: 99%

Optimal Propagating Fronts Using Hamilton-Jacobi Equations

et al. 2019

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The optimal handling of level sets associated to the solution of Hamilton-Jacobi equations such as the normal flow equation is investigated. The goal is to find the normal velocity minimizing a suitable cost functional that accounts for a desired behavior of level sets over time. Sufficient conditions of optimality are derived that require the solution of a system of nonlinear Hamilton-Jacobi equations. Since finding analytic solutions is difficult in general, the use of numerical methods to obtain approximate solutions is addressed by dealing with some case studies in two and three dimensions.

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The Lagrange–Charpit Theory of the Hamilton–Jacobi Problem

Álvarez

2021

Mediterr. J. Math.

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The Lagrange–Charpit theory is a geometric method of determining a complete integral by means of a constant of the motion of a vector field defined on a phase space associated to a nonlinear PDE of first order. In this article, we establish this theory on the symplectic structure of the cotangent bundle $$T^{*}Q$$ T ∗ Q of the configuration manifold Q. In particular, we use it to calculate explicitly isotropic submanifolds associated with a Hamilton–Jacobi equation.

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Systems of Hamilton-Jacobi equations

Cited by 3 publications

References 9 publications

Quasi-canonicalization for linear homogeneous nonholonomic systems*

Quasi-canonicalization for linear homogeneous nonholonomic systems*

Optimal Propagating Fronts Using Hamilton-Jacobi Equations

The Lagrange–Charpit Theory of the Hamilton–Jacobi Problem

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