Higher order mechanics on graded bundles

TL;DR

This paper develops a geometric framework for higher order mechanics on graded bundles using weighted algebroids, deriving phase equations and Euler-Lagrange equations in both Lagrangian and Hamiltonian formalisms.

Contribution

It introduces a novel geometric approach to higher order mechanics on graded bundles with weighted algebroids, including the Tulczyjew triple and equations for singular Lagrangians.

Findings

Derived phase equations from arbitrary Lagrangians and Hamiltonians.

Established geometric derivation of higher order Euler-Lagrange equations.

Applied framework to invariant Lagrangians on Lie groupoids.

Abstract

In this paper we develop a geometric approach to higher order mechanics on graded bundles in both, the Lagrangian and Hamiltonian formalism, via the recently discovered weighted algebroids. We present the corresponding Tulczyjew triple for this higher order situation and derive in this framework the phase equations from an arbitrary (also singular) Lagrangian or Hamiltonian, as well as the Euler-Lagrange equations. As important examples, we geometrically derive the classical higher order Euler-Lagrange equations and analogous reduced equations for invariant higher order Lagrangians on Lie groupoids.