Freedom of h(2)-variationality and metrizability of sprays

In this paper we are investigating variational homogeneous second order differential equations by considering the question of how many different variational principles exist for a given spray. We focus our attention on h(2)-variationality; that is, the regular Lagrange function is homogeneous of degree two in the directional argument. Searching for geometric objects characterizing the degree of freedom of h(2)-variationality of a spray, we show that the holonomy distribution generated by the tangent direction to the parallel translations can be used to calculate it. As a working example, the class of isotropic sprays is considered.