View source for Hamilton's canonical equations

Let
:<math> g := \begin{pmatrix} q \\ p \end{pmatrix} </math>

be a vector of generalized coordinates, (<math>q</math> might itself be a vector of positions and <math>p</math> a vector of momenta, both of them having the same size)

:<math> \nabla := \begin{pmatrix} {\partial \over \partial q} \\ {\partial \over \partial p} \end{pmatrix} </math>

be a vector differential operator,

:<math> i := \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} </math>

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

Then
:<math> \dot g = \begin{pmatrix} \dot q \\ \dot p \end{pmatrix} </math>

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

 <math>\nabla H = i \dot g </math>

This is a compactified form of Hamilton’s canonical equations (of motion). It could also be re-expressed (trivially) as 
:<math> \dot g = -i \nabla H </math>

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

:<math> \begin{pmatrix} {\partial H \over \partial q} \\ {\partial H \over \partial p} \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} \dot q \\ \dot p \end{pmatrix} = \begin{pmatrix} -\dot p \\ \dot q\end{pmatrix} </math>
thus
:<math> {\partial H \over \partial q} = -\dot p </math>
:<math> {\partial H \over \partial p} = \dot q </math>

An immediate consequence of the canonical equation is that
:<math> \dot g \cdot i \dot g = 0</math>,
that is,
:<math> \begin{pmatrix} \dot q \\ \dot p\end{pmatrix} \cdot \begin{pmatrix} -\dot p \\ \dot q\end{pmatrix} = -\dot q \dot p + \dot p \dot q = 0</math>
so
 <math> \dot g \cdot \nabla H = 0 </math>

This means (geometrically) that ''g'' moves perpendicularly to <i>H</i>’s gradient, so as to conserve ''H''.

== a simple example ==
Let
:<math> H = {1\over 2 m} p^2 + {k \over 2} q^2 </math>
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
:<math>{\partial H \over \partial q} = k q = -F = -\dot p = -m \ddot q </math>
Note how deriving ''H'' by ''q'' returns a multiple of ''q''; ''q'' is an “eigen-derivator” of ''H'', as it were. (<math>k q = -F</math> follows from <math>{\partial U \over \partial q} = -F</math> being true in general (for conservative forces), where ''U'' denotes potential energy; <math>F = \dot p</math> is Newton’s second law, and ''m'' may be assumed to be constant)

On the other hand,
:<math>{\partial H \over \partial p} = {p \over m} = {m \dot q \over m} = \dot q </math>
and note how deriving ''H'' by ''p'' returns a multiple of ''p'', so ''p'' is also an “eigen-derivator” of ''H''.

[[Category:Advanced Classical Mechanics]]