Canonical transformations and Hamiltonian evolutionary systems (Q2865444)

The author considers the Lagrangian field theory for some physical fields. It is known that in many cases there exists Poisson brackets defined on the space of local functionals of a fiber bundle. Three different cases of interest are investigated. These cases depend on the dimensions of the vector bundles characterizing the theories and the corresponding Hamiltonian differential operators. It is known that the evolutionary systems of Hamiltonian type have the form \(\partial u/\partial t = {\mathcal{D}}\delta\mathcal{H}\), where \(\mathcal{H}\) is a local functional called the Hamiltonian, \(\mathcal{D}\) is a Hamiltonian differential operator, \(\delta \) is the variational derivative. The solutions of the Hamiltonian system are sections of the fiber bundle. Recall that the variational derivative of a local functional \({\mathcal{P}}=\int_{M}Pv\) is given by \({\mathbf{E}}_a=(-D)_I\partial P/\partial u_I^a\), where \(D_i=\displaystyle\frac{\partial }{\partial x^i}+u_{iJ}^a\displaystyle\frac{\partial } {\partial u_{J}^a}\), \(I=\{i_1,i_2,\ldots , i_k\}\), \(k>0\), \(D_I=(D_{i_1}\circ\cdots\circ D_{i_k})\), and \(P\) is some density, that is, a local function representing \(\mathcal{P}\), and \(v=fdx^1\wedge\cdots \wedge dx^n\) is a volume element on the base manifold \(M\). An interesting result for the first case where \(\dim M=1\), \(\dim E=2\), \({\mathcal{D}}=\omega D_x\) with \(\omega\in \mathrm{Loc}_E\) -- the algebra of local functions on \(J^{\infty }E\) -- is that the induced transformation \(\Psi \) on functionals is canonical, i.e., \(\{\Psi ({\mathcal{P}}), \Psi ({\mathcal{Q}})\}=\Psi (\{{\mathcal{P}},{\mathcal{Q}}\})\) for \({\mathcal{P}},{\mathcal{Q}}\in {\mathcal{F}}\) if and only if \(D_x(\text{det}\psi_M\partial \psi_E/\partial u)=0\), and \(\omega\circ j\psi = \omega (\text{det}\psi_M)^2(\partial \psi_E/\partial u)^2\). Here, \({\mathcal{F}}\) is the space of functionals, and \(j\psi :J^{\infty }E\to J^{\infty }E\) is the jet prolongation of the automorphism \(\psi :E\to E\). Furthermore, the author studies other two cases with different dimensions: dim\(M=n\), dim\(E=m+n\), \({\mathcal{D}}=\omega^{abi}D_i\), and the case dim\(M=1\), dim\(E=2\), \({\mathcal{D}}=\omega^{i}D_i\). The necessary and sufficient conditions for a transformation on the space of local functionals to be canonical are established. The conservation laws are also discussed. Some examples illustrating the theory are given as well.