Editing Angular momentum (section)

=== Hamiltonian formalism ===
Equivalently, in [[Hamiltonian mechanics]] the Hamiltonian can be described as a function of the angular momentum. As before, the part of the kinetic energy related to rotation around the z-axis for the ''i''th object is:
<math display="block">\frac{1}{2} {I_z}_i { {\omega_z}_i}^2 = \frac{ { {L_z}_i}^2}{2 {I_z}_i}</math>
which is analogous to the energy dependence upon momentum along the z-axis, <math>\frac{ { {p_z}_i}^2}{ {2m}_i}</math>.

Hamilton's equations relate the angle around the z-axis to its conjugate momentum, the angular momentum around the same axis:
<math display="block">\begin{align}
  \frac{d{\theta_z}_i}{dt} &= \frac{\partial \mathcal{H} }{\partial {L_z}_i} = \frac{ {L_z}_i}{ {I_z}_i} \\
       \frac{d{L_z}_i}{dt} &= -\frac{\partial \mathcal{H} }{\partial {\theta_z}_i} = -\frac{\partial V}{\partial {\theta_z}_i}
\end{align}</math>

The first equation gives
<math display="block">{L_z}_i = {I_z}_i \cdot { {\dot{\theta}_z}_i} = {I_z}_i \cdot {\omega_z}_i</math>

And so we get the same results as in the Lagrangian formalism.

Note, that for combining all axes together, we write the kinetic energy as:
<math display="block">
  E_k = \frac{1}{2}\sum_i \frac{|\mathbf{p}_i|^2}{2m_i}
      = \sum_i \left(\frac{ {p_r}_i^2}{2m_i} + \frac{1}{2} {\mathbf{L}_i}^\textsf{T}{I_i}^{-1} \mathbf{L}_i\right)
</math>
where ''p''<sub>r</sub> is the momentum in the radial direction, and the [[moment of inertia#Inertia matrix in different reference frames|moment of inertia is a 3-dimensional matrix]]; bold letters stand for 3-dimensional vectors.

For point-like bodies we have:
<math display="block">E_k = \sum_i \left(\frac{ {p_r}_i^2}{2m_i} + \frac{|{\mathbf{L}_i}|^2}{2m_i {r_i}^2}\right)</math>

This form of the kinetic energy part of the Hamiltonian is useful in analyzing [[central potential]] problems, and is easily transformed to a [[quantum mechanical]] work frame (e.g. in the [[hydrogen atom]] problem).