The transition from configuration space to phase space can be achieved through a Legendre transformation, particularly when velocity is expressed in terms of momenta. A function f(q, \dot{q}) in configuration space corresponds to a function g(q, p) in phase space, defined as g(q, p) = \dot{q}(p) p - f(q, \dot{q}(p)). This transformation effectively relates the variables of the two spaces, allowing for a clear mapping of dynamics. Understanding this relationship is crucial for analyzing physical systems in both configuration and phase spaces. The discussion emphasizes the importance of the Legendre transformation in this context.
