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Susskind Lecture 7 - Symmetries and Conservation Laws

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Susskind Lecture 7 - Symmetries and Conservation Laws

Lecture 7 is Susskind’s Noether bridge: conservation laws are not isolated miracles. They come from symmetries of the Lagrangian.

What counts as a symmetry

Susskind uses the active viewpoint. A transformation does not merely relabel coordinates; it moves the system to a nearby configuration.

A continuous infinitesimal transformation has the form:

q_i → q_i + ε f_i(q)

The transformation is a symmetry if the Lagrangian does not change to first order:

δL = 0

That phrase matters: the symmetry is about the Lagrangian’s value for every allowed configuration, not about one lucky trajectory.

The conserved quantity

For a symmetry q_i → q_i + ε f_i(q), the conserved quantity is:

Q = Σ_i p_i f_i(q)

where:

p_i = ∂L/∂q_dot_i

Along a path satisfying the Euler-Lagrange equations:

dQ/dt = 0

This is the practical core of Noether’s theorem in the book’s language.

Translation symmetry gives momentum

If shifting every particle by the same amount leaves L unchanged, then total momentum is conserved.

Example: if a two-coordinate system has a potential depending only on q_1 - q_2, then shifting both q_1 and q_2 together does not change the potential. The conserved quantity is p_1 + p_2.

This reframes Newton’s third-law momentum conservation as a deeper fact about space: the laws do not care where the entire isolated system is placed.

Rotation symmetry gives angular momentum

If a particle in the x-y plane has a potential depending only on distance from the origin, rotations leave the Lagrangian invariant.

For rotation about the z axis, the conserved angular momentum component is:

L_z = x p_y - y p_x

For a fully rotationally invariant system, all components of angular momentum are conserved.

Double pendulum lesson

The double pendulum illustrates why the Lagrangian method is not just elegant. It gives a mechanical recipe:

  1. Choose coordinates that specify the configuration.
  2. Compute total kinetic energy T.
  3. Compute potential energy V if needed.
  4. Form L = T - V.
  5. Apply Euler-Lagrange for each coordinate.
  6. Look for symmetries to identify conserved quantities.

This avoids explicitly solving for internal rod tensions, which is often painful in Newtonian force language.

Common pitfalls

  • Confusing active transformations with passive coordinate relabeling.
  • Checking symmetry only at one point instead of checking the Lagrangian generally.
  • Assuming every conserved quantity is obvious as ordinary momentum or energy. Some are hidden combinations.
  • Forgetting that a cyclic coordinate is a special case of a continuous symmetry.
  • Treating angular momentum conservation as automatic. It requires rotational symmetry.

See also

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