Regularity properties of fiber derivatives associated with higher-order mechanical systems

Abstract

The aim of this work is to study fiber derivatives associated to Lagrangian and Hamiltonian functions describing the dynamics of a higher-order autonomous dynamical system. More precisely, given a function in T * T (k−1) Q, we find necessary and sufficient conditions for such a function to describe the dynamics of a kth-order autonomous dynamical system, thus being a kth-order Hamiltonian function. Then, we give a suitable definition of (hyper)regularity for these higher-order Hamiltonian functions in terms of their fiber derivative. In addition, we also study an alternative characterization of the dynamics in Lagrangian submanifolds in terms of the solutions of the higher-order Euler-Lagrange equations.

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

“…Here, F is an arbitrary function depending on (q (1) , q (3) ,q (1) , r). A direct calculation proves that the first order Euler-Lagrange equations generated by L 3 on T (AQ × M ) is equivalent to the second order Euler-Lagrange equations (19) only if the matrix [∂ 2 F/∂q (1) ∂r] is non-degenerate. In order to satisfy this condition, one may simply choose the auxiliary function as F =q (1) · r. Let us proceed with this choice.…”

The variational discretization of the constrained higher-order Lagrange-Poincaré equations

“…Next, we consider the HO case (28). According to (27) and (30), we can find local coordinates [q 0 , q 1 , ..., q k ] G in U (k+1) × G k given by…”

“…Variational integrators for constrained HO Lagrange-Poincaré equations: Next, we consider the HO case (28). According to ( 27) and ( 30), we can find local coordinates [q 0 , q 1 , ..., q k ] G in U (k+1) × G k given by (p 0 , ..., p k , g 0 , g 1 , ...., g k−1 ) ,…”

On the geometry of the Schmidt-Legendre transformation

“…Define two bundle structures by considering the base manifold as AQ, namely the trivial bundle AQ × M and T 2 Q. Here, the manifold M is the one presented in the subsection (4.5) whereas the bundle structure of T 2 Q is the one described in (15). Accordingly, we take the Whitney product of these two bundle and arrive T 2 Q × AQ (AQ × M) with local coordinates (Q, A,Q, r).…”

“…For the system whose configuration is a Lie group, we additionally refer [14,24,25], and for field theories, see [63]. We cite a recent study on the geometry of higher order theories in terms of Tulczyjew's triplet [15].…”