Variational calculus with constraints on general algebroids

Abstract

Variational calculus on a vector bundle E equipped with a structure of a general algebroid is developed, together with the corresponding analogs of Euler-Lagrange equations. Constrained systems are introduced in the variational and in the geometrical setting. The constrained Euler-Lagrange equations are derived for analogs of holonomic, vakonomic and nonholonomic constraints. This general model covers majority of first-order Lagrangian systems which are present in the literature and reduces to the standard variational calculus and the Euler-Lagrange equations in Classical Mechanics for E=TM.

23 pages, a few references added, the version to appear in J. Phys. A: Math. Theor

References in corpus (4)

• Lagrangian submanifolds and dynamics on Lie algebroids
• Classical field theory on Lie algebroids: Variational aspects
• AV-differential geometry: Euler-Lagrange equations
• Lagrangian submanifolds and dynamics on Lie affgebroids

Cited by in corpus (25)

• Dirac Algebroids in Lagrangian and Hamiltonian Mechanics
• Nonholonomic Constraints: a New Viewpoint
• A novel approach to contact Hamiltonians and contact Hamilton-Jacobi theory
• Linear duals of graded bundles and higher analogues of (Lie) algebroids
• Dirac--Lie systems and Schwarzian equations
• Hamilton-Jacobi Theory for Degenerate Lagrangian Systems with Holonomic and Nonholonomic Constraints
• Quasi-Lie Brackets and the Breaking of Time-Translation Symmetry for Quantum Systems Embedded in Classical Baths
• Modular classes of skew algebroid relations
• Pre-Courant Algebroids
• Tulczyjew triples: from statics to field theory
• On Dirac structures admitting a variational approach
• Bundle-theoretic methods for higher-order variational calculus
• VB-structures and generalizations
• Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings
• Dirac Structures in Vakonomic Mechanics
• Nonassociative analogs of Lie groupoids
• Gauge equivalence and conserved quantities for Lagrangian systems on Lie algebroids
• Modular classes revisited
• Lifting statistical structures
• Duality for graded manifolds
• Discrete mechanics on unitary octonions
• The category of ${\mathbb{Z}}_2^n$-supermanifolds
• Remarks on generalized Lie algebroids and related concepts
• Jacobi algebroids and Jacobi sigma models
• Hamilton-Dirac systems for charged particles in gauge fields