# Hamilton's canonical equations

Let

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).

Then

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:

thus

An immediate consequence of the canonical equation is that

- ,

that is,

so

```
```

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

## a simple example[edit | edit source]

Let

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).

Then

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*.