# Changes

General tidy up

In classical dynamics, the Hamiltonian is defined to be

<math>H(q_i, p_i) =\sum_i p_i \dot{q_i} - L </math>

where <math>q_i</math> are the generalised coordinates and <math>p_i</math> are the canonically conjugate [[momentum|momenta]] ~~conjugate to ~~for these coordinates, and <math>L</math> is the [[Lagrangian]]. ~~For many problems the Hamiltonian is the same ~~The canonically conjugate momentum can be found as ~~the energy.~~:

:<math>\dot p_i = -\frac{\partial H}{\partial q_i}</math>

===Example===

For a [[mass]] <math>m</math> attached to a [[Hooke's Law|spring]] of ~~[[~~spring constant~~]] ~~<math>k</math> extended by a distance <math>x</math>~~, ~~. Therefore the [[Lagrangian]] is <math>L=\frac{1}{2} m\dot{x}^~~2/~~2-\frac{1}{2} kx^~~2/~~2</math>~~. ~~ The canonically conjugate momentum is <math>p = \frac{\partial L}{\partial \dot {x}} = m \dot {x}</math> and so

<math>

H = p\dot{x} - L

</math>

</math>

which is the familiar expression for the energy of a simple ~~[[~~harmonic oscillator~~]]~~.

The equations of motion are:

:<math>\dot x =~~\frac{\partial H}{\partial p_i} = p/m</math>.

Inserting <math>p = m \dot x</math> this into the first equation, we get <math>m \ddot x = -kx</math>. This is ~~just ~~same as if we had used Newton's second law, <math>F = ma</math>.

== Quantum mechanics ==

The Hamiltonian for many quantum mechanical systems can be obtained by writing down a corresponding classical Hamiltonian and promoting all of the coordinates and momenta to operators. The non-relativistic Hamiltonian is <math>\hat{H} = \frac{\hat{p}^2}{2m} + V</math> with <math>\hat{p}</math> being the [[momentum (physics)|momentum]] operator, <math>m</math> the [[mass (science)|mass]] and <math>V</math> the potential. The quantum mechanical Hamiltonian is of central importance to the [[Schrodinger equation]].

[[Category:Mechanics]]

[[Category:Physics]]