  The local geometry of the parameter space of a quantum system is described by
the quantum metric tensor and the Berry curvature, which are two fundamental
objects that play a crucial role in understanding geometrical aspects of
condensed matter physics. We consider classical integrable systems and report a
new approach to obtain the classical analogs of the quantum metric tensor and
the Berry curvature. An advantage of this approach is that it can be applied to
a wide variety of classical systems corresponding to quantum systems with
bosonic and fermionic degrees of freedom. Our approach arises from the
semiclassical approximation of the Berry curvature and the quantum metric
tensor in the Lagrangian formalism. We also exploit this semiclassical
approximation to establish, for the first time, the relation between the
quantum metric tensor and its classical counterpart. We illustrate and validate
our approach by applying it to five systems: the generalized harmonic
oscillator, the symmetric and linearly coupled harmonic oscillators, the
singular Euclidean oscillator, and a spin-half particle in a magnetic field.
Finally, we mention some potential applications of this approach and possible
generalizations that can be of interest in the field of condensed matter
physics
