  A powerful mathematical method for the investigation of the properties of
dynamical systems is represented by the Kosambi-Cartan-Chern (KCC) theory. In
this approach the time evolution of a dynamical system is described in
geometric terms, treating the solution curves of a dynamical system by
geometrical methods inspired by the geodesics theory of Finsler spaces. In
order to geometrize the dynamical evolution one introduces a non-linear and a
Berwald type connection, respectively, and thus the properties of any dynamical
system are described in terms of five geometrical invariants, with the second
one giving the Jacobi stability of the system. The Jacobi (in)stability is a
natural generalization of the (in)stability of the geodesic flow on a
differentiable manifold endowed with a metric (Riemannian or Finslerian) to the
{\it non-metric} setting. Usually, the KCC theory is formulated by reducing the
dynamical evolution equations to a set of second order differential equations.
In this paper we present an alternative view on the KCC theory, in which the
theory is applied to a first order dynamical system. After introducing the
general framework of the KCC theory, we investigate in detail the properties of
the two dimensional autonomous dynamical systems. The relationship between the
linear stability and the Jacobi stability is also established. As a physical
application of the formalism we consider the geometrization of Hamiltonian
systems with one degree of freedom, and their stability properties.
