Zermello's conditions and energies of higher order in generalized Lagrange-Hamilton spaces

Irena Čomić

Many significant geometers contributed to the generalization of Riemann spaces in different directions. An almost complete list of them can be found in Miron's books. Here are mention [1], [2], [17] and [18] in which Hamilton and Finsler spaces are examined, further [3-9], where generalized Hamilton spaces are studied; [10] is most connected with the subject, and in [11-16] this problem also appears. Zermello's condition in Miron's $Osc^kM$ was examined in [6]. Here, the Zermello's conditions are given in Lagrange-Hamilton spaces, introduced in [9] and presented at the Workshop on Finsler Geometry 2009, Debrecen. It is proved that for a fundamental function for which the Zermello's conditions are satisfied all energies of higher order are equal to zero.