The tangent groups of a Lie group and gauge invariance in Lagrangian dynamical systems (Q2711772)

The author explains the genesis of Dirac systems with first-class constraints with the help of tangent Lie groups. First, he considers some properties of tangent bundle \(TG\) and second tangent bundle \(T^2G\) of a Lie group \(G\). The Tulczyjew differential operator is used to construct the first- and the second-order prolongation of multiplication law, as well as bases of left-invariant 1-forms on \(TG\) and \(T^2G\). By means of these forms and Maurer-Cartan equations, the author expresses the structural constants of \(TG\) and \(T^2G\). Next, assuming that a smooth manifold \(Q\) admits \(G\)-action, he constructs \(TG\)-action on \(Q\) and \( T^2G\)-action on \(TQ\). Lastly, the technique developed is applied to coordinate results of \textit{D. M. Gitman} and \textit{I. V. Tyutin} [Canonical quantization of fields with constraints, Nauka: Moscow (1986)] (who derive Hamiltonian system with first-class constraints from degenerate Lagrangian one-invariant with respect to certain gauge transformation) and to geometrical approach of \textit{G. Mendella, G. Marmo} and \textit{W. M. Tulchyjew } [J. Phys. A, Math. Gen. 28, 149--163 (1995; Zbl 0703.70026)] which uses the notion of characteristic foliation.NEWLINENEWLINEFor the entire collection see [Zbl 0937.00046].