Jacobi group symmetry of Hamilton's mechanics
type: Item
instance of: scholarly article
title: Jacobi group symmetry of Hamilton's mechanics
author: Stephen G. Low
author: Rutwig Campoamor Stursberg
published in: Journal of Geometry and Physics
publication date: 2024-09-02T00:00:00Z
review text: This paper is devoted to Jacobi group symmetry of Hamilton's mechanics. The formulation of Hamilton's equations is on a \(2n\) dimensional symplectic manifold denoted \((\mathbb{P}^{\circ}, \omega^{\circ})\), where \(\omega^{\circ}\) is a closed, nondegenerate 2-form. This is the phase space with local momentum-position coordinates \(y\in\mathbb{R}^{2n}\), and \(q , p \in \mathbb{R}^{n}\). It is known that the symplectic 2-form locally can be expressed, \(\omega^{\circ}=\delta_{i,j} dp^i\wedge dq^j\) (\(i,j=1,\ldots ,n\)). The Hamiltonian vector field \(X_{H^{\circ}}\) is defined by \(i_{X_{H^{\circ}}} \omega^{\circ}=dH^{\circ}\), \(H^{\circ}:\mathbb{P}^{\circ}\to \mathbb{R}:y\rightarrowtail H^{\circ}(y)\), where \(H^{\circ}\) is the Hamiltonian that is not time-dependent and the map \(i_{X}\) maps a vector field \(X\) to a 1-form \(\theta \), \(X\rightarrowtail \theta =i_X\omega^{\circ}\). This defines an isomorphism between tangent and cotangent spaces. The triple \((\mathbb{P}^{\circ}, \omega^{\circ},X_{H^{\circ}})\) defines a Hamiltonian system for which the flows \(\phi^{\circ}\) of \(X_H\) satisfy Hamilton's equations: \N\[\N\phi^{\circ}:\mathbb{R}\otimes \mathbb{P}^{\circ}\to \mathbb{P}^{\circ}:(t,y) \rightarrowtail \phi^{\circ} (t,y), \ \frac{d\phi^{\circ}(t,y)}{dt}= X_{H^{\circ}}(\phi^{\circ}(t,y)).\N\]\NIn the present paper the authors describe an alternative formulation of Hamilton's systems on the extended phase space \((\mathbb{P},\omega )\). The new formulation begins with the invariance of Newtonian time rather than the Hamiltonian. This is expressed as the geometric structure of a degenerate orthogonal metric \(\lambda \) and results in the expression of Hamilton's equations in terms of a Jacobi geometry, \((\mathbb{P},\omega ,\lambda )\), based on the Jacobi group. The new idea requires to define a symplectomorphisms \(\varrho :\mathbb{P}\to \mathbb{P}\) that leave invariant the 2-form \(\varrho^{\ast}(\omega )=\omega \). Then their Jacobian matrice…
reviewed by: Dimitar A. Kolev
Mathematics Subject Classification ID: 37J11
Mathematics Subject Classification ID: 37J06
Mathematics Subject Classification ID: 37J39
Mathematics Subject Classification ID: 53D05
Mathematics Subject Classification ID: 53D22
zbMATH DE Number: 7904720
zbMATH Keywords: Jacobi group
zbMATH Keywords: Hamilton equations
zbMATH Keywords: symplectic geometry
zbMATH Keywords: noninertial symmetry
zbMATH Keywords: Weyl-Heisenberg group
zbMATH Keywords: Jacobi morphisms
MaRDI profile type: Publication
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DOI: 10.1016/J.GEOMPHYS.2024.105249
description: scientific article; zbMATH DE number 7904720