10 May 2026

Conservation Theorems and Infinitesimal Generators

Symmetry properties, conservation theorems, infinitesimal generators, and Hamiltonian form of Noether's idea.

msc semester-i classical-mechanics symmetry noether infinitesimal-generators

The syllabus asks for conservation theorems, symmetry properties, and infinitesimal generators. The key idea is that a continuous symmetry gives a conserved quantity. If a transformation changes the description but not the physics, there is usually a quantity that remains constant during motion.

This principle is seen first through cyclic coordinates and then more systematically through generators and Poisson brackets.

Conservation from Lagrange equations

For a Lagrangian $L(q_i,\dot q_i,t)$, the conjugate momentum is

\[p_i=\frac{\partial L}{\partial\dot q_i}.\]

If $q_i$ is cyclic, then

\[\frac{\partial L}{\partial q_i}=0,\]

and Lagrange’s equation gives

\[\dot p_i=0.\]

So $p_i$ is conserved.

Thus a missing coordinate in the Lagrangian is not a small detail. It is a signal of a symmetry and a corresponding conservation law.

Standard symmetry-conservation pairs

Symmetry	Conserved quantity
Time translation	Energy
Space translation	Linear momentum
Rotation	Angular momentum

This is the practical content of Noether’s theorem for classical mechanics.

Infinitesimal canonical transformation

A transformation close to identity may be written as

\[Q_i=q_i+\delta q_i, \qquad P_i=p_i+\delta p_i.\]

It is generated by a phase-space function $G(q,p,t)$:

\[\boxed{ \delta q_i=\epsilon\frac{\partial G}{\partial p_i}, \qquad \delta p_i=-\epsilon\frac{\partial G}{\partial q_i}. }\]

For any function $f(q,p,t)$,

\[\boxed{ \delta f=\epsilon\{f,G\}. }\]

Thus $G$ is called the infinitesimal generator.

Examples of generators

If

\[G=p_x,\]

then

\[\delta x=\epsilon, \qquad \delta p_x=0.\]

So linear momentum generates translation.

If

\[G=L_z=xp_y-yp_x,\]

then $G$ generates rotations about the $z$-axis.

If

\[G=H,\]

then the transformation is time evolution:

\[\delta q_i=\epsilon\dot q_i, \qquad \delta p_i=\epsilon\dot p_i.\]

Hamiltonian form of conservation

For a quantity $G$ with no explicit time dependence,

\[\frac{dG}{dt}=\{G,H\}.\]

If the Hamiltonian is invariant under the transformation generated by $G$, then

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

Therefore

\[\boxed{\frac{dG}{dt}=0.}\]

The generator is conserved.

Rotation generated by $L_z$

Take

\[G=L_z=xp_y-yp_x.\]

For an infinitesimal transformation,

\[\delta f=\epsilon\{f,G\}.\]

Then

\[\delta x=\epsilon\{x,L_z\} =\epsilon\frac{\partial L_z}{\partial p_x} =-\epsilon y,\]

and

\[\delta y=\epsilon\{y,L_z\} =\epsilon\frac{\partial L_z}{\partial p_y} =\epsilon x.\]

Thus $L_z$ rotates the coordinates in the $x$-$y$ plane. If the Hamiltonian is rotationally invariant, then

\[\{L_z,H\}=0,\]

so $L_z$ is conserved.

Main points

• Cyclic coordinates imply conserved momenta.
• Symmetry and conservation are paired by Noether’s idea.
• Infinitesimal canonical transformations are generated by $G$.
• The change of any phase-space function is $\delta f=\epsilon{f,G}$.
• If ${G,H}=0$, then $G$ is conserved.

Practice questions

1. State the conservation law associated with a cyclic coordinate.
2. List the conserved quantities associated with time translation, space translation, and rotation.
3. Derive $\delta f=\epsilon{f,G}$.
4. Show that $p_x$ generates translation in $x$.
5. Explain why an invariant Hamiltonian gives a conserved generator.