# Changes

## Hamiltonian

546 bytes added, 19:59, 16 December 2016
General tidy up
In classical dynamics, the Hamiltonian is defined to be
$H(q_i, p_i) =\sum_i p_i \dot{q_i} - L$
where $q_i$ are the generalised coordinates and $p_i$ are the canonically conjugate [[momentum|momenta]] conjugate to for these coordinates, and $L$ is the [[Lagrangian]]. For many problems the Hamiltonian is the same The canonically conjugate momentum can be found as the energy.:
The $p_i = \frac{\partial L}{\partial \dot{q_i}}$ For many problems the Hamiltonian is the same as the total [[energy]] of the system. Hamilton 's equations are:
:$\dot p_i = -\frac{\partial H}{\partial q_i}$
===Example===
For a [[mass]] $m$ attached to a [[Hooke's Law|spring]] of [[spring constant]] $k$ extended by a distance $x$, . Therefore the [[Lagrangian]] is $L=\frac{1}{2} m\dot{x}^2/2-\frac{1}{2} kx^2/2$.  The canonically conjugate momentum is $p = \frac{\partial L}{\partial \dot {x}} = m \dot {x}$  and so
$H = p\dot{x} - L$
[/itex]
which is the familiar expression for the energy of a simple [[harmonic oscillator]].
The equations of motion are:
:$\dot x =~~\frac{\partial H}{\partial p_i} = p/m$.
Inserting $p = m \dot x$ this into the first equation, we get $m \ddot x = -kx$. This is just same as if we had used Newton's second law, $F = ma$.
== 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  $\hat{H} = \frac{\hat{p}^2}{2m} + V$ with $\hat{p}$ being the [[momentum (physics)|momentum]] operator, $m$ the [[mass (science)|mass]] and $V$ the potential. The quantum mechanical Hamiltonian is of central importance to the [[Schrodinger equation]].
[[Category:Mechanics]]
[[Category:Physics]]
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