A geometric Hamilton–Jacobi theory on a Nambu–Jacobi manifold

León, Manuel de; Sardón, Cristina

doi:10.1142/s0219887819400073

Cited by 5 publications

(2 citation statements)

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“…Takhtajan identity in the case of Nambu mechanics) then, one arrives at Nambu-Jacobi manifolds. The geometric HJ for these systems can be found in [65].…”

Section: An Incomplete Literature On Geometric Hamilton-jacobi Theorymentioning

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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“…Takhtajan identity in the case of Nambu mechanics) then, one arrives at Nambu-Jacobi manifolds. The geometric HJ for these systems can be found in [65].…”

Section: An Incomplete Literature On Geometric Hamilton-jacobi Theorymentioning

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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“…An unifying Hamilton-Jacobi theory for almost-Poisson manifolds was developed in reference [47]. The Hamilton-Jacobi theory has also been generalized to Hamiltonian systems with non-canonical symplectic structures [54], non-Hamiltonian systems [57] locally conformally symplectic manifols [23], Nambu-Poisson and Nambu-Jacobi manifolds [38,39], Lie algebroids [36] and implicit differential systems [22]. The applications of Hamilton-Jacobi theory include the relation between classical and quantum mechanics [6,11,52], information geometry [14,15], control theory [58] and the study of phase transitions [34].…”

Section: Introductionmentioning

confidence: 99%

Geometric Hamilton-Jacobi theory for systems with external forces

de León,

Lainz,

López-Gordón

2021

Preprint

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In this paper, we develop a Hamilton-Jacobi theory for forced Hamiltonian and Lagrangian systems. We study the complete solutions, particularize for Rayleigh systems and present some examples. Additionally, we present a method for the reduction and reconstruction of the Hamilton-Jacobi problem for forced Hamiltonian systems with symmetry. Furthermore, we consider the reduction of the Hamilton-Jacobi problem for a Čaplygin system to the Hamilton-Jacobi problem for a forced Lagrangian system.

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