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In classical dynamics, the Hamiltonian is defined to be

In classical dynamics, the Hamiltonian is defined to be

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

<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 [[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 canonically conjugate [[momentum|momenta]] for these coordinates, and <math>L</math> is the [[Lagrangian]]. The canonically conjugate momentum can be found as:

The Hamilton equations are:

 

 

Hamilton's equations are:

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

:<math>\dot p_i = -\frac{\partial H}{\partial q_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>. The momentum <math>p = \frac{\partial L}{\partial \dot x} = m \dot x</math> and so

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 - \frac{1}{2} kx^2</math>

 

The canonically conjugate momentum is

 

<math>p = \frac{\partial L}{\partial \dot{x}} = m \dot{x}</math>

 

and so

<math>

<math>

H = p\dot{x} - L

H = p \dot{x} - L

</math>

</math>

</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:

The equations of motion are:

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

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

== Quantum mechanics ==

== 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 quantum mechanical Hamiltonian is of central importance to the [[Schrodinger equation]].

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

 

 

 

The quantum mechanical Hamiltonian is of central importance to the [[Schrodinger equation]].

[[Category:Physics]]

[[Category:Physics]]