Induced Symplectic Structures and Holonomic Lagrangian Mechanical Systems

Abstract

This paper presents a geometric approach to holonomic mechanical systems in the context of Lagrangian systems. First, it is shown how a standard regular Lagrangian system can be established in the context of symplectic geometry; namely, how a symplectic structure on the tangent bundle of a configuration manifold can be induced from the canonical symplectic structure on the cotangent bundle by using the Legendre transformation and also how a second-order Lagrangian vector field can be developed by an energy function associated to a given Lagrangian as well as the induced symplectic structure on the tangent bundle. Second, it is demonstrated that Lagrangian systems with holonomic constraints can be formulated in the context of the induced symplectic structure by combining constraint distributions with the second-order Lagrangian vector field. Further, it is shown how the standard Lagrangian system can be also understood in the context of an induced Dirac structure on the tangent bundle. Finally, a holonomic Lagrangian system is illustrated by a planar linkage system, together with a local expression of differential algebraic equations (DAE) of index 3.