Abstract
Tulczyjew's triples are constructed for the Schmidt-Legendre transformations of both second and third-order Lagrangians. Symplectic diffeomorphisms relating the Ostrogradsky-Legendre and the Schmidt-Legendre transformations are derived. Several examples are presented.
Highlights
A geometrization of Schmidt-Legendre transformation of the higher order Lagrangians is proposed by building a proper Tulczyjew’s triplet
The dynamics of a system can either be formulated by a Lagrangian function on the tangent bundle of a configuration space or by a Hamiltonian function on the cotangent bundle [1, 4]
At the late 70’s, Tulczyjew showed that the dynamics can be represented as a Lagrangian submanifold of certain symplectic manifold on higher order bundles [7, 56, 57, 61, 62]
Summary
The dynamics of a system can either be formulated by a Lagrangian function on the tangent bundle of a configuration space or by a Hamiltonian function on the cotangent bundle [1, 4]. In classical mechanics a Lagrangian density is a function of positions and velocities, it is possible to find theories involving Lagrangian densities depending on the higher order derivatives as well. In such cases, to pass the Hamiltonian picture, it is a tradition to employ the Ostrogradsky-Legendre transformation [44]. The last section, will be reserved for several examples including Pais-Uhlenberg, Sarıoglu-Tekin and Clement Lagrangians
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