Higher order mechanics on graded bundles

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.

28 pages, comments welcomed. Minor typos and editing artefacts corrected. No major revision

References in corpus (5)

• Avoiding Dark Energy with 1/R Modifications of Gravity
• Tangent Lifts of Poisson and Related Structures
• $Q$-manifolds and Higher Analogs of Lie Algebroids
• Tulczyjew Triples in Higher Derivative Field Theory
• Higher order Lagrange-Poincaré and Hamilton-Poincaré reductions

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• A novel approach to contact Hamiltonians and contact Hamilton-Jacobi theory
• Remarks on Contact and Jacobi Geometry
• Foundations of ghost stability
• Queer Poisson brackets
• Graded Bundles in the Category of Lie Groupoids
• Polarisation of Graded Bundles
• VB-structures and generalizations
• Introduction to graded bundles
• Duality for graded manifolds
• Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms
• Connections Adapted to Non-Negatively Graded Structures
• On the concept of a filtered bundle