FINSLER GEOMETRY IN CLASSICAL MECHANICS AND IN BIANCHI COSMOLOGICAL MODELS

A gauge-invariant approach to study the dynamical behaviour of general Lagrangian systems is illustrated. The method is able to manage syste ms whose potential depends both on coordinates and velocities (possibl y, on time), using a geometrical description. The manifold in which ti le dynamical systems live is a Finslerian space in which the conformal factor is a positively homogeneous function of first degree in the ve locities (the homogeneous Lagrangian of the system). This method is a generalization of geometrodynamical approaches based on Riemannian man ifolds, since it allows the study of a wider class of dynamical system s. Moreover, it is well suited to treat conservative systems with few degrees of freedom and peculiar dynamical systems whose Lagrangian is not ''standard'', such as the one describing the so-called Mixmaster U niverse. We present the method and apply it to some cases of interest: 1) systems with N degrees of freedom described by conservative potent ials, 2) Bianchi I); Cosmological Models (Mixmaster Universe), 3) the restricted three-body problem. The second example is particularly enli ghtening as the introduction of Finsler geometry overcomes the critica l problems which cause the Jacobi (Riemannian) metric to fail.