Quantum Mechanics and Group Theory

where xi is the the Cartesian coordinate of a point particle of mass m, Fi is the external force applied to the particle, and the force is assumed to be conservative, the negative gradient of a potential energy function V (x). Newtonian mechanics for rigid bodies can be formulated in a similar way, and this approach can even be extended to relativistic point mechanics. The calculus of variations uses functionals, functions of functions, F [f ]. In a functional, the independent “variable” is a whole function; the functional F maps a given function to a number: F [f ] : f → F . Typically, the F [f ] is constructed as an integral over an integrand involving some function f . The calculus of such functionals varies the independent function f to see what happens to the functional value F . The variations are denoted by δ : f → f + δf maps to F → F + δF. In classical mechanics, the path in space xi(t) is the independent function. The key functional is the action S[x, ẋ] :