Susskind Classical Mechanics - Gauge invariance

Gauge invariance appears in Susskind’s classical mechanics course through the vector potential for magnetic fields. It is a classical preview of a principle that dominates later quantum mechanics and field theory.

Vector potential

Magnetic fields obey:

∇ · B = 0

A convenient way to guarantee this is to write:

B = ∇ × A

The auxiliary field A is the vector potential.

Gauge transformation

The same B is produced by many A fields. If s(x) is any scalar field:

A → A + ∇s

then B is unchanged because the curl of a gradient is zero.

Why A is still necessary

The magnetic Lorentz force depends on B:

F_magnetic = (e/c) v × B

But the Lagrangian uses A:

L = (1/2)m v^2 + (e/c) A · v - e Φ

A gauge transformation changes the action by an endpoint term, so the fixed-endpoint stationary-action equations do not change.

Canonical versus mechanical momentum

p_canonical = m v + (e/c) A
p_mechanical = m v

Mechanical momentum is gauge invariant and directly observable. Canonical momentum is the phase-space variable needed for Hamiltonian mechanics.

Practical lesson

Gauge-dependent variables can be indispensable in the formalism even though measured physics is gauge invariant. Do not throw A away just because it is not unique.

Common pitfalls

• Treating a gauge choice as a physical observable.
• Expecting canonical momentum to be gauge invariant.
• Forgetting that the Hamiltonian must be written in canonical variables.
• Thinking “not unique” means “not useful.”

See also

• Susskind Classical Mechanics MOC
• Susskind Lecture 11 - Electric and Magnetic Forces
• Susskind Classical Mechanics - Hamiltonian and Hamilton equations
• Electromagnetism MOC