Analytical Mechanics - Hamiltonian-Jacobi Mechanics

Hamiltonian-Jacobi Mechanics

Canonical transformations

The invariance of the Hamiltonian (under addition of the partial time derivative of an arbitrary function of p, q, and t)allows the Hamiltonian in one set of coordinates q and momenta p to be transformed into a new set Q = Q(q, p, t) and P = P(q, p, t), in four possible ways:

With the restriction on P and Q such that the transformed Hamiltonian system is:

the above transformations are called canonical transformations, each function G_n is called a generating function of the "nth kind" or "type-n". The transformation of coordinates and momenta can allow simplification for solving Hamilton's equations for a given problem.

The Poisson bracket

The choice of Q and P is completely arbitrary, but not every choice leads to a canonical transformation. A simple test to check if a transformation q → Q and p → P is canonical is to calculate the Poisson bracket, defined by:

and if it's unity:

for all i = 1, 2,...N, then the transformation is canonical, else it is not.

Calculating the total derivative of an arbitrary function A = A(q, p, t) and substituting Hamilton's equations into the result leads to:

The Poission bracket has a foundation in quantum mechanics: Dirac's canonical quantization replaces the Poisson bracket with the commutator of quantum operators, denoted by hats (^):

and the previous equation in A is closely related to the equation of motion in the Heisenberg picture.

The Hamilton-Jacobi equation

By setting the canonically transformed Hamiltonian K = 0, and the type-2 generating function equal to Hamilton's principle function (also the action ) plus an arbitrary constant C:

the generalized momenta become:

and P is constant, then the Hamiltonian-Jacobi equation (HJE) can be derived from the type-2 canonical transformation:

where H is the Hamiltonian as before:

Another function named after Hamilton is Hamilton's characteristic function

used to solve the HJE by additive separation of variables for a time-independent Hamiltonian H.

The study of the solutions of the Hamilton-Jacobi equations leads naturally to the study of symplectic manifolds and symplectic topology. In this formulation, the solutions of the Hamilton–Jacobi equations are the integral curves of Hamiltonian vector fields.