Knowledge Garden

Susskind Lecture 4 - Systems of More Than One Particle

3 min read

Susskind Lecture 4 - Systems of More Than One Particle

Lecture 4 scales classical mechanics from one particle to many. The lesson is simple but deep: an N-particle system has a state, and that state is a single point in a very high-dimensional space.

Forces depend on the whole configuration

For a system of N particles, the force on particle i can depend on the locations of all particles:

F_i = F_i(r_1, r_2, ..., r_N)

For fundamental forces such as gravity and electrostatics, particle properties such as mass and charge are part of the setup, while positions determine distances and directions.

Newton’s second law applies particle by particle:

m_i a_i = F_i({r})

There are three coordinate equations per particle, so the Newtonian equations are 3N second-order differential equations.

Why the state needs velocities too

Knowing positions is not enough. Newton’s law gives acceleration, not velocity. To predict the next instant you need both:

state = all positions + all velocities

For one particle in 3D this is six numbers. For N particles it is 6N numbers.

This is the continuum version of the Lecture 1 idea: a state is whatever information is needed to determine future evolution.

From state space to phase space

Lecture 4 then replaces velocities with momenta:

p_i = m_i v_i

The state is now described by positions and momenta. This special state-space is phase space.

configuration space = all positions
momentum space      = all momenta
phase space         = configuration space + momentum space

For N particles in 3D, phase space has 6N dimensions.

First-order form of Newtonian mechanics

Using momenta, the motion can be written as first-order equations for each phase-space coordinate:

x_dot_i = p_i / m_i
p_dot_i = F_i({x})

This format is a bridge to Susskind Lecture 8 - Hamiltonian Mechanics, where Hamilton’s equations make the first-order phase-space structure fundamental.

Total momentum conservation

If particles exert pairwise forces on each other and Newton’s third law holds,

F_ij = -F_ji

then internal forces cancel in pairs when summed over the whole isolated system. Therefore the total momentum is conserved:

P_total = Σ_i p_i
P_dot_total = 0

Individual particle momenta can change violently; the total momentum of the closed system does not.

Andrew-friendly mental picture

Imagine a dashboard with one slider for every position component and one slider for every momentum component. The system’s entire state is one point in that dashboard space. Dynamics is not “many particles moving” in the abstract; it is one point tracing one curve through phase space.

Common pitfalls

  • Counting only 3N coordinates. Newtonian state requires positions and velocities/momenta, so the phase-space count is 6N.
  • Treating configuration space and phase space as synonyms. Configuration space has positions only.
  • Assuming total momentum conservation means each particle’s momentum is fixed. Only the sum is conserved in an isolated system.
  • Forgetting that external forces can ruin conservation of the subsystem’s momentum.

See also

Graph View

Backlinks