On the geometrical theory of higher-order Hamilton spaces (Q2733943)

The higher order Hamilton space is defined by Legendre transformations of \(k\)-th order Lagrange space: Leg: \(L^{(k)n}\to H^{(k)n}\), where NEWLINE\[NEWLINEL^{(k)n}= \bigl(M,L(x,y^{(1)}, \dots,y^{(k-1)}, y^{(k)})\bigr),\;H^{(k)n}= \bigl(M,H(x,y^{(1)}, \dots,y^{(k-1)},p) \bigr),NEWLINE\]NEWLINE Leg: \((x,y^{(1)}, \dots, y^{(k-1)}, y^{(k)})\in T^kM \mapsto(x,y^{(1)}, \dots,y^{(k-1)}, p)\in T^{* k}M\) and \(p_i= {1\over 2}{\partial L\over \partial y^{(k)i}}.\)NEWLINENEWLINENEWLINEThe mapping Leg is a local diffeomorphism between \(T^kM\) and \(T^{*k}M\). The fundamental metric tensor \(g^{ij}(x,y^{(1)},\dots,y^{(k-1)},p)\) of \(H^{(k)n}\) and the invariant vector field \(z^{(k)i}\) are determined. Furthermore, the Poisson brackets \(\{f,g\}\), \(\{f,g\}_\alpha\) and the natural presymplectic structures are defined.NEWLINENEWLINEFor the entire collection see [Zbl 0966.00031].