Classical Mechanics Lecture 1: State, Laws of Motion, and Reversibility

The first surprise in Leonard Susskind’s opening classical-mechanics lecture is that it does not begin with \(F=ma\). It begins with a coin.

The coin strips mechanics down to two questions: what counts as the state, and what rule carries that state to the next moment? Once those questions are clear, position, velocity, vectors, and differential equations have somewhere to live.

A state and a law

Put a coin on a table and call its two possible configurations heads and tails. A simple law might say:

  • heads becomes tails;
  • tails becomes heads.

If the coin starts heads, its history is

\[H \rightarrow T \rightarrow H \rightarrow T \rightarrow \cdots\]

The initial condition tells us where the history begins. The law tells us how to move from one state to the next. Together they determine the sequence.

This toy system also shows why a state must contain enough information for the law to work. Suppose the rule depends not only on the face showing now, but also on the face shown one step earlier. “Heads” alone is no longer a complete state. We must enlarge it to something like \((T,H)\): previous face tails, current face heads.

That is the useful definition of a state in mechanics: the information required to determine the next step, given the law.

Determinism is not the same as reversibility

A deterministic law gives one next state for each present state. A reversible law also lets us recover one unique previous state.

The flip rule above is both. But consider a different rule in which both heads and tails become heads. The future is determined, yet after seeing heads we cannot know which face came before. Information has been lost, so the rule is not reversible.

Many fundamental models in classical mechanics use invertible evolution: a sufficiently complete state can be evolved uniquely forward or backward. Invertibility is not the same as time-reversal symmetry, however, and dissipative or velocity-dependent effective laws require separate care. The irreversibility of everyday macroscopic events is largely statistical: coarse-graining hides the enormous number of microscopic degrees of freedom involved in friction and heat.

Nor does determinism guarantee useful prediction. A tiny error in an initial state can grow. The law may be exact while our measurement of the state is not. “The future follows from the present” is therefore a statement about the mathematical model, not a claim that we can forecast everything indefinitely.

From coins to particles

For a particle moving along a line, position \(x\) alone is not enough. Two particles can pass through the same point while moving in opposite directions. In elementary Newtonian mechanics we add velocity \(v\), so a state can be written

\[(x,v).\]

For a particle moving in three dimensions, position and velocity are vectors:

\[\mathbf r=(x,y,z), \qquad \mathbf v=\frac{d\mathbf r}{dt}.\]

Acceleration is the rate at which velocity changes:

\[\mathbf a=\frac{d\mathbf v}{dt}=\frac{d^2\mathbf r}{dt^2}.\]

The word “change” matters. In circular motion the speed may remain constant while the velocity changes continuously, because its direction changes. If

\[\mathbf r(t)=R(\cos \omega t,\sin \omega t),\]

then

\[\mathbf v(t)=R\omega(-\sin \omega t,\cos \omega t)\]

is tangent to the circle, while

\[\mathbf a(t)=-\omega^2\mathbf r(t)\]

points toward the centre.

For many particles, the state lists the corresponding variables for every particle. The set of all possible states is a state space. In Hamiltonian mechanics, the space built specifically from positions and conjugate momenta is phase space. For a simple Cartesian particle with standard kinetic energy and no velocity-dependent potential, \(\mathbf p=m\mathbf v\). More generally, canonical momentum can differ from mechanical momentum, so phase space is defined using momentum rather than treating velocity and momentum as universally interchangeable.

Three short checks

Try these without looking at the answers first.

  1. A three-state system follows \(A\to B\), \(B\to C\), \(C\to A\). Starting from \(B\), where is it after five steps? Is the law reversible?
  2. A program moves one square left or right according to its current position and the direction it is facing. Is position alone a complete state?
  3. Let \(\mathbf r(t)=(t^2,3t,2)\). Find \(\mathbf v(t)\) and \(\mathbf a(t)\).

Compact checks:

  1. \(B\to C\to A\to B\to C\to A\), so the answer is \(A\). Every state has one predecessor, so the law is reversible.
  2. No. The direction must be included; \((x,\text{direction})\) is a suitable state.
  3. \(\mathbf v(t)=(2t,3,0)\) and \(\mathbf a(t)=(2,0,0)\).

A practical way to study the lecture

Watch Stanford’s official Lecture 1, then close the video and reconstruct the coin argument in your own notation. Explain why determinism and reversibility are different, and derive the circular-motion vectors rather than merely rereading them.

For problem practice, use the MIT OpenCourseWare 8.01 assignments. OpenStax University Physics is a useful reference when vectors or kinematics need a slower second pass.

I also maintain a searchable archive of Susskind transcripts, subtitles, notes, TeX, and PDFs. It is useful for finding a passage after watching the lecture; the generated companions are navigation aids, not a replacement for the official recording.

If you own the lecture, the Bilingual Lecture Pack applies the same source-first workflow to one 20-minute recording, with one target language and a reviewable transcript, pocket companion, subtitles, and preview clip.

The original 2019 version of this post contained only the word “State.” This reconstruction follows Stanford’s official Lecture 1 and preserves that original publication date.

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