Susskind Classical Mechanics - Symmetries and conservation laws
A symmetry is a transformation that leaves the physics unchanged. In Susskind’s Lagrangian language, a continuous symmetry leaves the Lagrangian unchanged to first order.
Noether pattern
For an infinitesimal transformation:
q_i → q_i + ε f_i(q)if:
δL = 0then the conserved quantity is:
Q = Σ_i p_i f_i(q)Three core examples
| Symmetry | Conserved quantity | Notes |
|---|---|---|
| Space translation | Linear momentum | The laws do not care where the isolated system is placed. |
| Rotation | Angular momentum | The laws do not care how the isolated system is oriented. |
| Time translation | Energy/Hamiltonian | The laws do not care when the isolated experiment starts. |
Cyclic coordinates
If q_i does not appear in L, then shifting q_i is a symmetry. Its conjugate momentum is conserved:
∂L/∂q_i = 0 ⇒ dp_i/dt = 0This is a concrete, easy-to-spot version of the larger symmetry principle.
Generator viewpoint
In Hamiltonian mechanics, a conserved quantity also generates the corresponding symmetry through Poisson brackets:
δF = ε {F, G}If G generates a symmetry of H, then G is conserved. This is developed in Susskind Lecture 10 - Poisson Brackets and Angular Momentum.
Common pitfalls
- Thinking symmetry means a pretty shape rather than invariance of the dynamics.
- Forgetting that time-translation symmetry is different from coordinate translation symmetry.
- Assuming conservation laws survive external time-dependent forcing.
- Missing hidden symmetries because the current coordinates obscure them.