A geodesical approach for the harmonic oscillator

The harmonic oscillator (HO) is present in all contemporary physics, from elementary classical mechanics to quantum field theory. It is useful in general to exemplify techniques in theoretical physics. In this work, we use a method for solving classical mechanic problems by first transforming them to a free particle form and using the new canonical coordinates to reparametrize its phase space. This technique has been used to solve the one-dimensional hydrogen atom and also to solve for the motion of a particle in a dipolar potential. Using canonical transformations we convert the HO Hamiltonian to a free particle form which becomes trivial to solve. Our approach may be helpful to exemplify how canonical transformations may be used in mechanics. Besides, we expect it will help students to grasp what they mean when it is said that a problem has been transformed into another completely different one. As, for example, when the Kepler problem is transformed into free (geodesic) motion on a spherical surface.