Susskind The Theoretical Minimum Examples
Short worked examples for the Susskind classical mechanics notes. For the formula sheet, use Susskind The Theoretical Minimum Equations and definitions.
Example: why position is not enough
Imagine two identical balls at the same height. One is moving upward; one is falling downward. Their positions match, but their futures do not.
same x, different p ⇒ different futureThat is the intuition behind Susskind Classical Mechanics - Phase space.
Example: constant acceleration
Choose z upward and let gravity point downward:
z(t) = z_0 + v_0 t - (1/2)g t²Differentiate:
v_z(t) = v_0 - gt
a_z(t) = -gThe negative sign is not a detail; it says acceleration points downward in the chosen coordinate convention.
Example: harmonic oscillator from a restoring force
Start with:
F = -kx
F = maThen:
m x_ddot = -kx
x_ddot = -(k/m)xDefine ω² = k/m:
x_ddot = -ω² xA standard solution is x(t) = A cos(ωt) + B sin(ωt). The constants are fixed by initial position and velocity.
Example: free particle from stationary action
For a free particle, V = 0:
L = (1/2)m x_dot²Euler-Lagrange gives:
d/dt(m x_dot) = 0Therefore p = m x_dot is constant. In plain English: without a force, the particle moves with constant velocity.
Example: energy conservation from a potential
If:
F_i = -∂V/∂x_iand:
m_i x_ddot_i = F_ithen multiplying each equation by x_dot_i and summing gives:
dT/dt = -dV/dtSo:
d(T + V)/dt = 0This is the compact proof behind Susskind Lecture 5 - Energy.
Example: cyclic coordinate means conserved momentum
Suppose q_2 does not appear in L, though q_dot_2 does. Then:
∂L/∂q_2 = 0Euler-Lagrange says:
d/dt(∂L/∂q_dot_2) = 0So p_2 is conserved. This is the simple form of the symmetry principle in Susskind Lecture 7 - Symmetries and Conservation Laws.
Example: oscillator as a phase-space circle
For a normalized harmonic oscillator:
H = (ω/2)(p² + q²)If H is conserved, then p² + q² is constant. The phase-space point moves around a circle. Project that circle onto q and you see ordinary back-and-forth oscillation.
Example: magnetic field and two momenta
For a charged particle in a vector potential A:
p_canonical = m v + (e/c)A
p_mechanical = m vThe canonical momentum is what goes into Hamilton’s equations. The mechanical momentum is what you infer from the particle’s velocity. Gauge transformations change A and canonical p, but not the observable motion.