Room 326

Frobenius integrability and metrizability problem

Ebtsam H. Taha

HRI

The inverse problem of Lagrangian mechanics requires to decide whether or not a given system of second order ordinary differential equations (SODE or semi-spray) can be derived from a variational principle. In general, the problem is far from being solved. It has been completely solved in dimension 1 by Darboux and in dimension 2 by Douglas. In the case when the given SODE is homogeneous (i.e., a spray), the problem is known as the Finsler metrizability problem.

For a 2-dimensional non-flat spray we associate a Berwald frame and a 3-dimensional distribution that we call the Berwald distribution. The Frobenius integrability of the Berwald distribution characterises the Finsler metrizability of the given spray. In the integrable case, the sought after Finsler function is provided by a closed, homogeneous 1-form from the annihilator of the Berwald distribution. The main advantage of our approach is its constructive character providing a well- defined simple algorithm to test the existence of the required Finsler structure. We also provide some examples that reveal the effectiveness of our method.