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Hamiltonian

546 bytes added, 19:59, 16 December 2016
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.:
The <math>p_i = \frac{\partial L}{\partial \dot{q_i}}</math> For many problems the Hamiltonian is the same as the total [[energy]] of the system. Hamilton 's equations are:
:<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]]
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