On the geometry of higher order Lagrange spaces

A Lagrange space of order k greater than or equal to 1 is the space of accelerations of order k endowed with a regular Lagrangian. For theses spaces we discuss: certain natural geometrical structures, variational problem associated to a given regular Lagrangian and the induced semispray, nonlinear connection, metrical connections. Special attention is paid to the prolongations of the Riemannian and Finslerian structures. In the end we sketch the geometry of time-dependent Lagrangian. The geometry, which we have developed, is directed to Mechanicists and Physicists. The paper is a brief survey of our results in the higher order geometry. For details we refer to the monograph [3].