Lagrangian submanifolds and dynamics on Lie algebroids
By Manuel de León, Juan Carlos Marrero, E. Martínez. 2004
228 citations recorded by the source corpus.
In some previous papers, a geometric description of Lagrangian mechanics on Lie algebroids has been developed. In this topical review, we give a Hamiltonian description of mechanics on Lie algebroids. In addition, we introduce the notion of a Lagrangian submanifold of a symplectic Lie algebroid and we prove that the Lagrangian (Hamiltonian) dynamics on Lie algebroids may be described in terms of Lagrangian submanifolds of symplectic Lie algebroids. The Lagrangian (Hamiltonian) formalism on Lie algebroids permits us to deal with Lagrangian (Hamiltonian) functions not defined necessarily on tangent (cotangent) bundles. Thus, we may apply our results to the projection of Lagrangian (Hamiltonian) functions which are invariant under the action of a symmetry Lie group. As a consequence, we obtain that Lagrange–Poincare (Hamilton–Poincare) equations are the Euler–Lagrange (Hamilton) equations associated with the corresponding Atiyah algebroid. Moreover, we prove that Lagrange–Poincare (Hamilton–Poincare) equations are the local equations defining certain Lagrangian submanifolds of symplectic Atiyah algebroids.
Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/55fec26d5fa565445dee03bc15919887a03006e3
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Citations in the corpus
- Calculus on Lie algebroids, Lie groupoids and Poisson manifolds (2008, background): “Lie groupoids and Lie algebroids are now an active domain of r esearch, with applications in various parts of mathematics [51, 4, 2, 29, 30, 5].”