09 May 2026

Poisson Theorems and Angular Momentum

Poisson theorem, canonical invariance, cyclic coordinates, and angular momentum brackets.

msc semester-i classical-mechanics poisson-bracket poisson-theorem angular-momentum

Previous: Poisson Brackets

This note keeps only the syllabus essentials: Poisson theorems, canonical invariance, cyclic coordinates, and angular momentum brackets.

Poisson theorem

If $f$ and $g$ are constants of motion, then their Poisson bracket is also a constant of motion:

\[\boxed{ \frac{df}{dt}=0, \quad \frac{dg}{dt}=0 \quad\Rightarrow\quad \frac{d}{dt}\{f,g\}=0. }\]

For functions with no explicit time dependence, this follows from

\[\{f,H\}=0, \qquad \{g,H\}=0,\]

and the Jacobi identity.

Cyclic coordinates

If $q_k$ is absent from $H$, then

\[\frac{\partial H}{\partial q_k}=0.\]

Hamilton’s equation gives

\[\dot p_k=-\frac{\partial H}{\partial q_k}=0.\]

Equivalently,

\[\{p_k,H\}=0.\]

So $p_k$ is conserved.

Canonical transformation test

A transformation $(q,p)\to(Q,P)$ is canonical when

\[\boxed{ \{Q_i,Q_j\}=0, \qquad \{P_i,P_j\}=0, \qquad \{Q_i,P_j\}=\delta_{ij}. }\]

This is often the quickest test for canonicity.

Angular momentum brackets

For

\[\mathbf L=\mathbf r\times\mathbf p,\]

the components satisfy

\[\boxed{ \{L_x,L_y\}=L_z, \qquad \{L_y,L_z\}=L_x, \qquad \{L_z,L_x\}=L_y. }\]

In compact form,

\[\boxed{ \{L_i,L_j\}=\epsilon_{ijk}L_k. }\]

These brackets show that angular momentum components generate rotations.

Central force

For a central potential,

\[H=\frac{\mathbf p^2}{2m}+V(r),\]

rotational symmetry gives

\[\boxed{\{L_i,H\}=0.}\]

Therefore all components of angular momentum are conserved.

Main points

• Poisson theorem says constants of motion are closed under Poisson brackets.
• A cyclic coordinate gives a conserved conjugate momentum.
• Canonical transformations preserve fundamental Poisson brackets.
• Angular momentum components satisfy ${L_i,L_j}=\epsilon_{ijk}L_k$.
• Central forces conserve angular momentum.

Practice questions

1. State and prove Poisson’s theorem.
2. Show that a cyclic coordinate gives a conserved momentum.
3. Test whether $Q=p$, $P=-q$ is canonical.
4. Derive ${L_x,L_y}=L_z$.
5. Show that angular momentum is conserved for $V=V(r)$.