Hamilton's canonical equations

From Wikiversity
Jump to navigation Jump to search


be a vector of generalized coordinates, ( might itself be a vector of positions and a vector of momenta, both of them having the same size)

be a vector differential operator,

be a Givens rotation matrix for a counterclockwise right-angle rotation. (NB: This is a square root of , where is the identity 2×2 matrix; this is a matric version of the imaginary unit.) Let H denote the Hamiltonian (function for total energy).


where overdot is Newton’s fluxional notation for time derivative.

This is a compactified form of Hamilton’s canonical equations (of motion). It could also be re-expressed (trivially) as

To see that it is the pair of canonical equations, unpack the symbolism:


An immediate consequence of the canonical equation is that


that is,


This means (geometrically) that g moves perpendicularly to H’s gradient, so as to conserve H.

a simple example[edit | edit source]


This corresponds to a simple harmonic oscillator, where m is mass and k is a spring constant. The first term is kinetic energy and the second term is potential energy (of displacement from the stable equilibrium).


Note how deriving H by q returns a multiple of q; q is an “eigen-derivator” of H, as it were. ( follows from being true in general (for conservative forces), where U denotes potential energy; is Newton’s second law, and m may be assumed to be constant)

On the other hand,

and note how deriving H by p returns a multiple of p, so p is also an “eigen-derivator” of H.