Jacobi group symmetry of Hamilton's mechanics (Q6596119)

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 matrices take values in the symplectic group, that is, \(\biggl[\frac{\partial\varrho (z)} {\partial z}\biggr] \in \mathrm{Sp}(2n+2)\), \(z=(q,p,\varepsilon ,t)\), \(z\in \mathbb{R}^{2n+2}\), \(q,p\in\mathbb{R}^n\), \(\varepsilon ,t\in\mathbb{R}\). The extended Hamiltonian function is defined by \(K:\mathbb{P}\to \mathbb{R}:z \rightarrowtail K(z)\). Thus the triple \((\mathbb{P},\omega ,X_K)\) defines a Hamiltonian system for which the flows \(\varphi \) of \(X_K\) satisfy Hamilton's equations. Note that here the considered systems are nonrelativistic so that the time \(t\) is invariant Newton time. Then the Hamilton function has the form \(K(q,p,\varepsilon ,t)=H(q,p,t) -\varepsilon \). The time component of the flow \(d\varphi_t/d\tau =1\) implies that \(t = \varphi_t(\tau ,z) =\tau \), where the integration constant is set equal to zero. The energy component \(\varepsilon = \varphi_{\varepsilon}(t,z)=H(q,p,t)\), and the constraint \(K(z)=0\). The alternative formulation requires that the above stated symplectomorphisms also leave a degenerate orthogonal line element \(\lambda = dt^2\) invariant, that is, \(\varrho^{\ast} (\lambda ) = \lambda \). This line element defines so called `Newtonian time'. It is shown that its invariance restricts the Jacobian matrix to be an element of a subgroup of \(\mathrm{Sp}(2n+2)\) that is the Jacobi group \(\mathcal{J}a(n) \equiv \mathcal{H}\mathrm{Sp}(2n)\). The Jacobi group \(\mathcal{H}\mathrm{Sp}(2n)\) is the semidirect product of a symplectic group \(\mathrm{Sp}(2n)\) and a Weyl-Heisenberg group \(\mathcal{H}(n)\) that is the normal subgroup. The Weyl-Heisenberg group has parameters of velocity \(v\in \mathbb{R}^n\), force, \(f\in \mathbb{R}^n\) and power \(r\in\mathbb{R}\). The following equalities hold for these diffeomorphisms (Jacobimorphisms) \(\varrho\): \(\varrho^{\ast}(\omega ) = \omega \) and \(\varrho^{\ast}(\lambda ) = \lambda \). The semidirect product structure of the Jacobi group means that the diffeomorphism can always be decomposed into two diffeomorphisms \(\sigma \) and \(\rho \) so that \(\varrho =\rho\circ\sigma \), \(\tilde{z} = \sigma (z)\). Here the authors show that the Jacobimorphisms \(\rho \) locally satisfy Hamilton's equations. The main result is that the diffeomorphisms \(\varrho ::\mathbb{P}\to \mathbb{P}\), that leave \(\omega \) and \(\lambda \) invariant, are Jacobimorphisms as their Jacobian matrix takes values in the Jacobi group. The semidirect product structure enables the Jacobimorphisms to be represented as \(\varrho =\rho\circ\sigma \); \(\sigma \) are the usual canonical transformations on position momentum phase space \((\mathbb{P}^{\circ}, \omega^{\circ})\) and the Jacobian matrix expression for \(\rho \) taking values in the Weyl-Heisenberg group, locally correspond to Hamilton's equations. The maximal symmetry group of Hamilton's equations is the group \(\mathcal{H}\mathrm{Sp}(2n)\times \mathbb{Z}_2\), where \(\mathcal{H}\mathrm{Sp}(2n)\) is the Jacobi group and \(\mathbb{Z}_2\) is the discrete time reversal symmetry.