Dirac Algebroids in Lagrangian and Hamiltonian Mechanics

Abstract

We present a unified approach to constrained implicit Lagrangian and Hamiltonian systems based on the introduced concept of Dirac algebroid. The latter is a certain almost Dirac structure associated with the Courant algebroid on the dual $E^{\ast }$ to a vector bundle $E$. If this almost Dirac structure is integrable (Dirac), we speak about a Dirac-Lie algebroid. The bundle $E$ plays the role of the bundle of kinematic configurations (quasi-velocities), while the bundle $E^{\ast }$ - the role of the phase space. This setting is totally intrinsic and does not distinguish between regular and singular Lagrangians. The constraints are part of the framework, so the general approach does not change when nonholonomic constraints are imposed, and produces the (implicit) Euler-Lagrange and Hamilton equations in an elegant geometric way. The scheme includes all important cases of Lagrangian and Hamiltonian systems, no matter if they are with or without constraints, autonomous or non-autonomous etc., as well as their reductions; in particular, constrained systems on Lie algebroids. we prove also some basic facts about the geometry of Dirac and Dirac-Lie algebroids.

30 pages

References in corpus (14)

• Higher vector bundles and multi-graded symplectic manifolds
• Lie algebroids and Poisson-Nijenhuis structures
• Lagrangian submanifolds and dynamics on Lie algebroids
• Algebroids - general differential calculi on vector bundles
• Classical field theory on Lie algebroids: Variational aspects
• Lagrangian Reduction, the Euler--Poincaré Equations, and Semidirect Products
• Variational calculus with constraints on general algebroids
• Nonholonomic Constraints: a New Viewpoint
• AV-differential geometry: Poisson and Jacobi structures
• Lie algebroid structures and Lagrangian systems on affine bundles
• AV-differential geometry: Euler-Lagrange equations
• Lagrangian submanifolds and dynamics on Lie affgebroids
• AV-differential geometry and Newtonian mechanics
• A general framework for nonholonomic mechanics: Nonholonomic Systems on Lie affgebroids

Cited by in corpus (25)

• A novel approach to contact Hamiltonians and contact Hamilton-Jacobi theory
• Dirac--Lie systems and Schwarzian equations
• Brackets
• Higher order mechanics on graded bundles
• Modular classes of skew algebroid relations
• The Tulczyjew triple for classical fields
• Reduced dynamics and Lagrangian submanifolds of symplectic manifolds
• Multiplicative Dirac structures
• Tulczyjew triples: from statics to field theory
• On Dirac structures admitting a variational approach
• Geometry of quantum dynamics in infinite dimension
• VB-structures and generalizations
• Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings
• Nonassociative analogs of Lie groupoids
• Dirac Structures in Vakonomic Mechanics
• Modular classes revisited
• Lifting statistical structures
• Discrete mechanics on unitary octonions
• The category of ${\mathbb{Z}}_2^n$-supermanifolds
• Remarks on generalized Lie algebroids and related concepts
• Dirac Reduction for Nonholonomic Mechanical Systems and Semidirect Products
• The Tulczyjew triple in mechanics on a Lie group
• Higher Lie and Leibniz algebras
• The interplay between Dirac systems, Morse families and Interconnection
• Quantum Systems as Lie Algebroids