Geometric calculus for second-order differential equations and generalizations of the inverse problem of Lagrangian mechanics

We review the main features of the geometric calculus which has been introduced over the past 15 years in the study of second-order ordinary differential equations and then explain how a recently introduced generalization of the inverse problem of Lagrangian mechanics can be very concisely dealt with by this calculus in an intrinsic way. This paper is an account of the lecture with the same title presented at the ICDVC-2010 Conference in Hangzhou, May 12–14 (2010).

► Tangent bundle geometry is the general framework for this paper.
► We review the main ingredients of the calculus along the tangent bundle projection.
► A connection associated to second-order dynamics provides important concepts and tools.
► Generalizations of the inverse problem of Lagrangian mechanics illustrate the theory.