AV-differential geometry: Euler-Lagrange equations

Abstract

A general, consistent and complete framework for geometrical formulation of mechanical systems is proposed, based on certain structures on affine bundles (affgebroids) that generalize Lie algebras and Lie algebroids. This scheme covers and unifies various geometrical approaches to mechanics in the Lagrangian and Hamiltonian pictures, including time-dependent lagrangians and hamiltonians. In our approach, lagrangians and hamiltonians are, in general, sections of certain $\mathbb{R}$-principal bundles, and the solutions of analogs of Euler-Lagrange equations are curves in certain affine bundles. The correct geometrical and frame-independent description of Newtonian Mechanics is of this type.

20 pages

References in corpus (4)

• Lagrangian submanifolds and dynamics on Lie algebroids
• Lagrangian submanifolds and dynamics on Lie affgebroids
• AV-differential geometry and Newtonian mechanics
• An affine framework for analytical mechanics

Cited by in corpus (12)

• Dirac Algebroids in Lagrangian and Hamiltonian Mechanics
• Variational calculus with constraints on general algebroids
• Higher order mechanics on graded bundles
• The Tulczyjew triple for classical fields
• Classical field theories of first order and lagrangian submanifolds of premultisymplectic manifolds
• Tulczyjew Triples in Higher Derivative Field Theory
• Time-dependent Mechanics and Lagrangian submanifolds of Dirac manifolds
• Partial Differential Hamiltonian Systems
• Reduction of symplectic principal $\mathbb{R}$-bundles
• Wigner's Theorem and geometry of extreme positive maps
• The Schroedinger operator as a generalized Laplacian
• Affine bundles are affine spaces over modules