On the differential geometry of the Euler-Lagrange equations, and the inverse problem of Lagrangian dynamics

Abstract

The conditions for a system of second-order differential equations to be derivable from a Lagrangian-the conditions of self-adjointness, in the terminology of Santilli (1978) and others-are related, in the time-independent case, to the differential geometry of the tangent bundle of configuration space. These conditions are simply expressed in terms of the horizontal distribution which is associated with any vector field representing a system of second-order differential equations. Necessary and sufficient conditions for such a vector field to be derivable from a Lagrangian may be stated as the existence of a two-form with certain properties: it is interesting that it is a deduction, not an assumption, that this two-form is closed and thus defines a symplectic structure. Some other differential geometric properties of Euler-Lagrange second-order differential equations are described.