Cosmossos said:It's the classic expression (H=p^2/2m+kx^2/2)
A canonical transformation is a mathematical transformation that preserves the form of Hamilton's equations, which describe the motion of a dynamical system. In other words, it maps one set of canonical variables (position and momentum) to another set of canonical variables, while still satisfying the Hamiltonian equations of motion.
Canonical transformations are important because they allow us to simplify the description of a dynamical system by choosing a set of canonical variables that are more convenient to work with. This can make the equations of motion easier to solve and can reveal underlying symmetries and conserved quantities in the system.
There are two types of canonical transformations: point transformations and generating function transformations. Point transformations simply change the variables without introducing any new functions, while generating function transformations involve introducing a new function that transforms the original variables into the new ones.
To determine if a transformation is canonical, we can use the Poisson bracket. If the Poisson bracket of the transformed coordinates is equal to the Poisson bracket of the original coordinates, then the transformation is canonical. In other words, if the transformed Hamiltonian equations of motion have the same form as the original equations, then the transformation is canonical.
Some examples of canonical transformations include the transformation from Cartesian coordinates to polar coordinates, the transformation from position and momentum to action-angle variables, and the transformation from the Hamiltonian of a free particle to that of a harmonic oscillator. These transformations help simplify the equations of motion and reveal new insights into the dynamics of the system.