On the geometry of the Schmidt-Legendre transformation

Abstract

A geometrization of Schmidt-Legendre transformation of the higher order Lagrangians is proposed by building a proper Tulczyjew's triplet. The symplectic relation between Ostrogradsky-Legendre and Schmidt-Legendre transformations is obtained. Several examples are presented.

A Hamilton–Jacobi formalism for higher order implicit Lagrangians

“…where the triangle is the special symplectic structure presented as the right wing of the Tulczyjew triple (22). Here, D is the image of S under the musical mapping…”

“…Then, we define the adapted symplectic diffeomorphism Ξ T k−1 Q and Ω ♭ T k−1 Q from the symplectic diffeomorphism Ξ Q and Ω ♭ Q in the first order Tulczyjew triple (22). Accordingly, they are computed as…”

“…Nonetheless, starting from an implicit differential equation on T T * T k−1 Q, by the projection T π T k−1 Q , we arrive at a submanifold in T T k−1 Q. Hence, we need to make use of Tulcyjew's triple (22) to pass from the Lagrangian to Hamiltonian pictures T T k−1 Q and T * T k−1 Q through some morphisms. In this section we develop a geometric Hamilton-Jacobi theory for higher order implicit differential equations using two different approaches.…”

“…Ostrogradsky approach is based on the idea that consecutive time derivatives of initial coordinates form new coordinates. In Schmidt's approach, the acceleration is defined as a new coordinate instead of the velocity [3,22,57] and Lagrange multipliers do not make an appearance, that is the advantage of the method. Instead, the Lagrangian function is modified by adding a gauge term such that the associated energy function contains additional terms, but no Lagrange multipliers.…”

Reductions of topologically massive gravity I: Hamiltonian analysis of second order degenerate Lagrangians

“…We call AQ as acceleration bundle [29]. Comparing the definitions of T 2 Q and AQ given in (9) and (11), respectively, one immediately observes that AQ is also a submanifold of T 2 Q.…”