# Canonical Transformations

## Abstract

One of the benefits of the Lagrangian approach to mechanical systems is that we can choose the generalized coordinates as we please. We have seen that once we select a set of coordinates \(q_i\) we can define the generalized momenta \(p_i\) according to Eq. ( 1.26) and form a Hamiltonian ( 1.28). We could also have chosen a different set of generalized coordinates \(Q_i=Q_i(q_k, t)\), expressed the Lagrangian as a function of \(Q_i\), used Eqs. ( 1.26) and ( 1.28), and obtained a different set of momenta \(P_i\) and a different Hamiltonian \(H'(Q_i,P_i, t)\). Although mathematically different, these two representations are physically equivalent — they describe the same dynamics of our physical system. Understanding the freedom that we have in the choice of the conjugate variables for a Hamiltonian is important: a judicious choice of the variables could allow us to simplify the description of the system dynamics.