Tangent bundle geometry for Lagrangian dynamics.

*(English)*Zbl 0536.58004The main purpose of this paper is to explain the geometrical background of recent results concerning the inverse problem of Lagrangian mechanics. The author starts with an extensive survey of various intrinsic constructions on the tangent bundle of a manifold. Most importantly, he discusses an intrinsically defined type \((1,1)\) tensor field \(S\) on \(TM\), with vanishing Nijenhuis tensor and with the property that \((\mathcal L_\Gamma S)^ 2=\text{identity}\), for every second-order differential equation field \(\Gamma\). The latter property enables him to define two projection operators \(P\) and \(Q\) which give rise to vertical and horizontal distributions of vector fields on \(TM\). The tensor field \(S\) is shown to play a key role in the formulation of Lagrangian mechanics and a concise geometrical version is described of the so-called Helmholtz conditions for the existence of a Lagrangian for given second order equations. In the latter context, necessary algebraic conditions, which were derived by M. Henneaux [Ann. Phys. 140, 45–64 (1982; Zbl 0501.70020)] and by the reviewer [J. Phys. A 15, 1503–1517 (1982; Zbl 0537.70018)] are here reproduced in an intrinsic way through a recursive process involving \(P\) and \(Q\). The author further gives new insight into the possible existence of alternative Lagrangians and into the classification of symmetries with related first integrals. The final section of the paper lists coordinate expressions of all previously explained intrinsic results.

Reviewer: W. Sarlet