Dirac structures and Lagrangian systems on tangent bundles (Symmetry and Singularity of Geometric Structures and Differential Equations)

Abstract

In mechanics, a Dirac structure, which is the unified notion of symplectic and Poisson structures, has been widely used to formulate mechanical systems with nonholonomic constraints, electric circuits as well as thermodynamic systems. In particular, the induced Dirac structure on the cotangent bundle from a given constraint distribution plays an essential role in the context of implicit Lagrangian and Hamiltonian systems. However, there has been almost no research on the Dirac geometry associated to the tangent bundle TQ, although it may be relevant with regular Lagrangian systems. In this paper, we introduce an induced Dirac structure on TQ, called a Lagrangian Dirac structure. For the regular case, we finally show that one can define a Lagrange-Dirac system on TQ.