Susskind The Theoretical Minimum Equations and definitions

A compact formula sheet for the Susskind classical mechanics notes. For narrative context, start with Susskind Classical Mechanics MOC.

State and phase space

A classical state is the information required to determine future evolution.

one particle in 1D: state = (x, p)
N particles in 3D: phase-space dimension = 6N

Configuration space contains positions q. Phase space contains positions and conjugate momenta (q, p).

Kinematics

v = dr/dt
a = dv/dt = d²r/dt²
speed = |v|

Uniform circular motion:

|v| = Rω
|a| = Rω²
a = -ω² r
period = 2π/ω

Simple harmonic motion:

x_ddot = -ω² x
x(t) = A cos(ωt) + B sin(ωt)

Newtonian dynamics

F = m a
p = m v
p_dot = F

For many particles:

q_dot_i = p_i / m_i
p_dot_i = F_i(q)

Total momentum for an isolated system with pairwise equal-and-opposite internal forces:

P = Σ_i p_i
dP/dt = 0

Potential energy and total energy

One dimension:

F = -dV/dx

Many coordinates:

F_i = -∂V/∂x_i

Kinetic and total energy:

T = (1/2) Σ_i m_i x_dot_i²
E = T + V

For closed conservative systems:

dE/dt = 0

Lagrangian, action, and Euler-Lagrange

L(q, q_dot, t) = T - V        [ordinary mechanical systems]
S[q] = ∫ L dt
δS = 0                       [fixed endpoints]

Euler-Lagrange equation:

d/dt(∂L/∂q_dot_i) - ∂L/∂q_i = 0

Conjugate momentum:

p_i = ∂L/∂q_dot_i

Cyclic coordinate:

if ∂L/∂q_i = 0, then dp_i/dt = 0

Hamiltonian mechanics

Hamiltonian definition:

H = Σ_i p_i q_dot_i - L

For ordinary L = T - V systems:

H = T + V

Hamilton’s equations:

q_dot_i = ∂H/∂p_i
p_dot_i = -∂H/∂q_i

Time-translation relation:

dH/dt = -∂L/∂t

Symmetry and conservation

Infinitesimal transformation:

q_i → q_i + ε f_i(q)

If δL = 0, then:

Q = Σ_i p_i f_i(q)
dQ/dt = 0

Common correspondences:

Symmetry	Conserved quantity
space translation	linear momentum
rotation	angular momentum
time translation	Hamiltonian/energy

Poisson brackets

{F, G} = Σ_i [(∂F/∂q_i)(∂G/∂p_i) - (∂F/∂p_i)(∂G/∂q_i)]

Canonical relations:

{q_i, p_j} = δ_ij
{q_i, q_j} = 0
{p_i, p_j} = 0

Time evolution:

dF/dt = {F, H}

Angular momentum algebra:

{L_i, L_j} = ε_ijk L_k

Liouville theorem

Phase-space divergence of Hamiltonian flow:

Σ_i [∂q_dot_i/∂q_i + ∂p_dot_i/∂p_i] = 0

So Hamiltonian flow preserves phase-space volume.

Electric and magnetic forces

Static electric field:

E = -∇Φ
potential energy = e Φ

Magnetic field and vector potential:

B = ∇ × A
A → A + ∇s        [gauge transformation]

Lorentz force:

F = eE + (e/c) v × B

Charged-particle Lagrangian and Hamiltonian:

L = (1/2)m v² + (e/c) A · v - e Φ
p_canonical = m v + (e/c) A
H = (1/2m)[p - (e/c)A]² + e Φ

Local links

• Susskind Classical Mechanics MOC
• Susskind Classical Mechanics Common pitfalls
• Susskind The Theoretical Minimum Key concepts
• Susskind The Theoretical Minimum Examples
• Susskind Lecture 2 - Motion
• Susskind Lecture 6 - Principle of Least Action
• Susskind Lecture 8 - Hamiltonian Mechanics
• Susskind Lecture 11 - Electric and Magnetic Forces