Symmetries and constants of the motion in the Lagrangian formalism on \$TQ\$: beyond point transformations

Marmo, G and Mukunda, N (1986) Symmetries and constants of the motion in the Lagrangian formalism on \$TQ\$: beyond point transformations. In: Il Nuovo Cimento - B, 92 (1). pp. 1-12.

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Abstract

Most treatments of symmetries in Lagrangian mechanics are confined to the class of point transformations on a tangent bundle. The purpose of the present paper is to give a more complete account of the theory by allowing the symmetry transformations to be velocity dependent. Within this broader framework, the similarity between the Lagrangian and the Hamiltonian approach to symmetries becomes more apparent. First it is observed that with an arbitrary vector field \$X\$ on a tangent bundle \$TQ\$ and any second-order vector field \$D\$ on \$TQ\$, one can associate a particular vector field \$X(D)=X+v[D, X]\$, where \$v\$ denotes the vertical endomorphism. This yields a natural extension of the complete lift construction for vector fields defined on \$Q\$, to which it actually reduces in case \$X\$ is projectable. Given a Lagrangian \$\scr L\$ on \$TQ\$, a vector field \$X\$ is called an infinitesimal symmetry of \$\scr L\$ if there exists a function \$F\$ such that for each second-order vector field \$D\$ one has \$L_{X(D)}\scr L=L_DF\$ (where \$L\$ denotes Lie derivation). It is then shown that each infinitesimal symmetry of \$\scr L\$ determines a constant of the motion, even if \$\scr L\$ is degenerate. This is a generalized version of Noether's theorem which, provided \$\scr L\$ is regular, also admits a converse, i.e., to each first integral of the system one can associate an infinitesimal symmetry of \$\scr L\$ in the above sense.

Item Type: Journal Article Copyright of this article belongs to Italian Physical Society. Division of Physical & Mathematical Sciences > Centre for Theoretical Studies (Ceased to exist at the end of 2003) jobish pachat 16 Aug 2004 10 Jan 2012 07:02 http://eprints.iisc.ac.in/id/eprint/977