Editing Canonical transformation (section)

==== Rotation ====
Consider an orthogonal system for an N-particle system:

<math display="block">\begin{array}{l}{{\mathbf q=\left(x_{1},y_{1},z_{1},\ldots,x_{n},y_{n},z_{n}\right),}}\\ {{\mathbf p=\left(p_{1x},p_{1y},p_{1z},\ldots,p_{n x},p_{n y},p_{n z}\right).}}\end{array}</math>

Choosing the generator to be: <math>G=L_{z}=\sum_{i=1}^{n}\left(x_{i}p_{i y}-y_{i}p_{i x}\right) </math> and the infinitesimal value of <math> \alpha = \delta \phi </math>, then the change in the coordinates is given for x by:

<math display="block">\begin{array}{c}
{\delta x_{i}=\{x_{i},G\}\delta\phi=\displaystyle\sum_{j} \{x_{i},x_{j}p_{j y}-y_{j}p_{j x}\}\delta\phi=\displaystyle\sum_{j}(\underbrace{\{x_{i},x_{j}p_{j y}\}}_{=0} -{ \{x_{i},y_{j}p_{j x}\}}})\delta\phi\\
{{=\displaystyle -\sum_{j} y_{j} \underbrace{\{x_i,p_{jx}\}}_{=\delta_{ij}}\delta\phi=- y_{i} \delta \phi}}  
\end{array}  </math>

and similarly for y:

<math display="block">\begin{array}{c}

\delta y_{i}=\{y_{i},G\}\delta\phi=\displaystyle\sum_{j}\{y_{i},x_{j}p_{j y}-y_{j}p_{j x}\}\delta\phi=\displaystyle\sum_{j}(\{y_{i},x_{j}p_{j y}\}-\underbrace{ \{y_{i},y_{j}p_{j x}\}}_{=0})\delta \phi\\ 
{=\displaystyle\sum_{j}x_{j}\underbrace{\{y_i,p_{jy}\}}_{=\delta_{ij}} \delta\phi=x_{i}\delta\phi\,,} 
\end{array} </math>

whereas the z component of all particles is unchanged: <math display="inline"> \delta z_{ i }=\left\{z_{i},G\right\}\delta\phi=\sum_{j}\left\{z_{i},x_{j}p_{j y}-y_{j}p_{j x}\right\}\delta \phi =0</math>.

These transformations correspond to rotation about the z axis by angle <math>\delta \phi </math> in its first order approximation. Hence, repeated application of the infinitesimal canonical transformation generates a rotation of system of particles about the z axis. If the Hamiltonian is invariant under rotation about the z axis, the generator, the component of angular momentum along the axis of rotation, is an invariant of motion.<ref name=":1" />