In the phase space formulation, the constant of motion \(G\) given by Noether's theorem, expressed in terms of coordinates and momenta generates the infinitesimal symmetry transformation.
$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$ A canonical transformation is a change of variables \((q,p) \rightarrow (Q,P)\) in phase space such that the Hamiltonian form of equations of motion is preserved. Depending choice of independent variables we have four special cases of canonical transformations., Generating functions for the four cases are introduced and details of the four cases are discussed.
The definition finite and infinitesimal canonical transformation are given. Using the action principle we define the generator of a canonical transformation. $\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}\newcommand{\Label}[1]{\label{#1}}$
The angular momentum of a rigid body is given by where \(\mathbf I\) is moment of inertia tensor and \(\vec \omega \) is the angular velocity.\begin{eqnarray} \vec{L}&=&\int dv \rho(\vec{X})\vec{X}\times(\vec{\omega}\times\vec{X})\\ &=&\int dV \rho(\vec{X})\Big[(\vec{X}\cdot\vec{X})\vec{\omega}-(\vec{X}\cdot\vec{\omega} )\vec{X}\Big] \end{eqnarray} or \(\vec L=\mathbf I\, \vec \omega\).
||Message]
