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→Classical mechanics
where <math>q_i</math> are the generalised coordinates and <math>p_i</math> are the [[momentum|momenta]] conjugate to these coordinates, and <math>L</math> is the [[Lagrangian]]. For many problems the Hamiltonian is the same as the energy.
where <math>q_i</math> are the generalised coordinates and <math>p_i</math> are the [[momentum|momenta]] conjugate to these coordinates, and <math>L</math> is the [[Lagrangian]]. For many problems the Hamiltonian is the same as the energy.
The Hamilton equations are:
:<math>\dot p_i = -\frac{\partial H}{\partial q_i}</math>
:<math>\dot q_i =~~\frac{\partial H}{\partial p_i}.</math>
===Example===
===Example===
For a [[mass]] <math>m</math> attached to a [[spring]] of [[spring constant]] <math>k</math> extended by a distance <math>x</math>, <math>L=m\dot{x}^2/2-kx^2/2</math> and so
For a [[mass]] <math>m</math> attached to a [[spring]] of [[spring constant]] <math>k</math> extended by a distance <math>x</math>, <math>L=m\dot{x}^2/2-kx^2/2</math>. The momentum <math>p = \frac{\partial L}{\partial \dot x} = m \dot x</math> and so
<math>
<math>
H=m\dot{x}^2-L
H = p\dot{x} - L
</math>
</math>
<math>
<math>
H=\frac{m\dot{x}}{2}+\frac{kx^2}{2}
H = \frac{p^2}{2m}+\frac{kx^2}{2}
</math>
</math>
which is the familiar expression for the energy of a simple [[harmonic oscillator]].
which is the familiar expression for the energy of a simple [[harmonic oscillator]].
The equations of motion are:
:<math>\dot p = -\frac{\partial H}{\partial x} = -kx</math>
:<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 Newton's second law, F = ma.
== Quantum mechanics ==
== Quantum mechanics ==