Classical Mechanics: Let Energy Write the Equations

A pendulum clock can keep time for years. Add a second joint to the pendulum, which is all a desk-toy "double pendulum" is, and it never repeats itself. Start it twice from positions that differ by the width of an atom and, within seconds, the two swings look nothing alike. The laws are the same Newton wrote down in 1687. Nothing is random in the toy. And yet you cannot predict it.

That strange gap between "fully determined" and "impossible to forecast" is one of the things a classical mechanics course leaves you seeing everywhere. The other is a different way of writing Newton's laws, in which you don't track forces at all. You write down one quantity, kinetic energy minus potential energy, and the motion falls out. It is the same physics, but it works for a bead on a spinning hoop, for a robot arm, for a planet, and for the equations of quantum mechanics.

This is the second time around for mechanics: Physics I (34/06) taught you forces and vectors; this course rebuilds the subject from energy and symmetry, which is the form physicists actually use afterwards.

Pick the coordinates that fit, then write T − V

Newton's method says: find every force on every piece, add them as vectors, write F = ma for each. For a bead sliding on a wire, or a pendulum held by a rod, that is wasteful. The rod's tension is a force you don't care about; it only exists to keep the bob on a circle.

The new method skips constraint forces. First pick the fewest numbers that describe the system's configuration: its generalised coordinates. A pendulum bob has two ordinary coordinates (x and y) but only one that can change, the angle θ. Then write one function, the Lagrangian: kinetic energy minus potential energy, in those coordinates.

The second line is a recipe. It says "the rate of change of the momentum belonging to q equals the force along q". For a pendulum of length l: T = ½ml²θ̇², V = −mgl cos θ, so L = ½ml²θ̇² + mgl cos θ, and the recipe gives θ̈ = −(g/l) sin θ. No tension, no vectors. For small angles sin θ ≈ θ and you get the swing of period 2π√(l/g): 2.006 s for a 1 m pendulum.

The method was built over about a century: Euler and Lagrange (whose Mécanique analytique appeared in 1788) made it precise, and William Rowan Hamilton recast it in the 1830s. Its power is that nothing in it depends on Cartesian axes. A graduate picks the coordinates that make the problem easy, whether angles, distances along a wire or the amplitudes of a vibrating plate.

Nature takes the path of stationary action

Where does the Euler–Lagrange recipe come from? Consider every path a particle could take between two points in a fixed time. For each one, add up L along the way. The total is the action, S = ∫L dt. The real path is the one for which S is stationary: change the path a little and S changes only at second order, the way the bottom of a bowl is flat.

This is the same idea as Fermat's principle for light (see Waves & Optics, 34/80): the real path is picked out from all possible ones by a single global rule. Mechanics got there first; the older name, "least action", is a little too strong because in general the path is only stationary, and over long times it can be a saddle rather than a minimum.

Try it with a ball of mass 1 kg thrown up and caught after 2 seconds. The true path is the parabola reaching 4.9 m. Its action is −32.08 J·s. Add a smooth 1 m bump to that path (still starting and ending at the same points) and the action rises by π²/8 = 1.23 J·s. Add the same bump but with a full wiggle, a second harmonic, and it rises by π²/2 = 4.93 J·s. Bumps up or down both raise it: the parabola sits at the bottom of the bowl.

It is no accident that this ties back to waves. In Richard Feynman's formulation of quantum mechanics (1948), a particle tries all paths, each with a phase set by S/ħ. Away from the stationary path the phases scramble and cancel; near it they add. The classical path is where the quantum waves agree.

Every symmetry hides a conserved quantity

Some quantities never change in an isolated system: energy, momentum, angular momentum. In Newton's physics these look like three lucky accidents. In 1918 Emmy Noether (at Göttingen, in work done for Hilbert and Klein's questions about general relativity) proved that they are not accidents. Each is the shadow of a symmetry of the Lagrangian:

If the physics doesn't change when you…	…this is conserved
shift the clock (time symmetry)	energy
slide everything along an axis	momentum along it
rotate everything about an axis	angular momentum about it

Planets illustrate the third row. The Sun's pull depends only on distance, so the problem looks the same from every direction: rotation symmetry, so angular momentum is conserved, and that is Kepler's second law (a planet sweeps equal areas in equal times). Throw a ball near the ground and gravity points one way: the horizontal sliding symmetry survives, so horizontal momentum is conserved, but not vertical momentum and not angular momentum about any point. Break a symmetry and the quantity is lost.

The theorem extends. Gauge symmetry in electromagnetism gives conservation of electric charge; in particle physics, the Standard Model is built by demanding symmetries and reading off the forces.

Phase space: the whole future in one picture

Newton's second law needs a position and a velocity to start. The two together fix the future. Hamilton's version makes this a picture: draw position q on one axis and momentum p on the other. Every state of the system is one point, and the motion is a curve through phase space which never crosses itself. Two systems can never be in the same state and go different ways.

For a pendulum of mass m and length l, in units where g/l = 1 and ml² = 1, the total energy is H = ½p² + (1 − cos θ). Hamilton's equations say θ̇ = ∂H/∂p and ṗ = −∂H/∂θ. Each curve of constant energy is a possible motion:

• Low energy (H below 2): closed loops, the pendulum swings back and forth.
• Energy exactly 2: the knife-edge curve (the separatrix), which just reaches the upright position in infinite time.
• High energy (above 2): the pendulum goes right over the top and keeps spinning.

Phase space also explains why a "simple" pendulum isn't quite simple. Its swing time depends on how far it swings. Compared with the small-angle period 2π√(l/g): 1.002 at 10°, 1.017 at 30°, 1.073 at 60°, 1.180 at 90°, 1.373 at 120°, 1.762 at 150° and 2.439 at 170°. A pendulum clock is accurate only because its swing is kept small. A theorem of Liouville says the flow in phase space preserves area; counting states by phase-space area is how Statistical Mechanics (34/84) begins.

Chaos: deterministic does not mean predictable

For the single pendulum, phase space is two-dimensional and the motion is orderly. The double pendulum has four dimensions (two angles, two momenta) and almost no tidy loops. Nearby starting points do not stay nearby: the gap between two trajectories grows roughly like e^λt for some rate λ, the Lyapunov exponent. That is the meaning of chaos in this subject: sensitive dependence on initial conditions.

The consequence is a clean fact about prediction. If your measurement is wrong by δ, the forecast is useful until the error has grown to the size you care about, which takes about (1/λ) ln(size/δ). Make the measurement a thousand times better and you gain only (1/λ) ln 1000 more seconds. In the simulator below, starting two copies of the same pendulum 0.001 rad apart gives a visible split after about 6 seconds; a million times closer buys only about 7 more seconds. Predictions don't fail from bad arithmetic, but from the gap between the world and any finite number of digits.

Chaos was found in celestial mechanics first. Henri Poincaré, studying the three-body problem in the late 1880s, saw that its trajectories could be hopelessly tangled. Edward Lorenz found the same sensitivity in a weather model in 1963. The inner solar system is itself weakly chaotic: its Lyapunov time is about 5 million years (Laskar, 1989), which is why detailed predictions of the planets' positions fade out beyond some tens of millions of years, even though the system stays statistically stable far longer. Small angles stay orderly: the 10° preset below never splits.

Every valley is a spring

One more trick, and it is the one engineers lean on every day. Near the bottom of any smooth potential-energy valley, the curve looks like a parabola, V ≈ V₀ + ½k(x − x₀)², where k is the curvature at the bottom. A parabola is a spring. So any system disturbed slightly from a stable equilibrium oscillates at ω = √(k/m), the same equation as in Vibrations (34/17). A chain of atoms, a bridge, a molecule and a skyscraper all become coupled springs.

The standard model of the force between two neutral atoms is the Lennard-Jones potential, V = 4ε[(σ/r)¹² − (σ/r)⁶], first written by John Lennard-Jones in 1924. It has its minimum at r₀ = 2^1/6σ = 1.1225σ and there the effective spring constant is 57.1 ε/σ². But a valley is not exactly a parabola: it is steeper on the near side and softer on the far side. The simulator shows what that does. Give the pair more energy and the swing gets longer than the spring would predict, and the midpoint of the vibration drifts outward. That drift is the origin of thermal expansion: hot solids expand because their atoms vibrate in lopsided wells.

What a graduate just sees

• Pick generalised coordinates to fit the problem; the Lagrangian T − V gives the equations without constraint forces.
• The real path is the one that makes the action stationary: a single global rule behind F = ma, and the same trick as Fermat's for light.
• Every continuous symmetry gives a conserved quantity (Noether): time → energy, position → momentum, rotation → angular momentum.
• If a quantity isn't conserved, ask which symmetry has been broken.
• A state is a point in phase space; the motion is a non-crossing curve, and constant-energy curves tell you the kinds of motion before you solve anything.
• Even a simple pendulum is only isochronous for small swings; a bigger swing is slower.
• Deterministic and predictable are different things: with chaos, extra digits buy only a fixed extra time.
• Near any stable minimum, a system is a spring; the lopsidedness of the valley explains thermal expansion.

Where it shows up next

• Quantum Mechanics (34/83): the Hamiltonian becomes the energy operator, and Poisson brackets become commutators; Feynman's sum over paths uses the action directly.
• Statistical Mechanics (34/84): a gas of N particles is a point in a 6N-dimensional phase space, and entropy counts the volume.
• General Relativity (34/85): free-falling objects follow paths that make a spacetime interval stationary, and Noether's theorem is why energy is subtle in GR.
• Dynamics (34/12) and Control Systems (34/20): robot arms and spacecraft are modelled with Lagrangians; stability of an equilibrium is the spring picture.
• The real world: spacecraft trajectories designed in the three-body regime, molecular dynamics simulation (34/09, 34/78), chaotic mixing in fluids and the limits of weather forecasting (see Chaos Theory and the Butterfly Effect, 12/07).

If you want to go deeper

• 📘 Textbook: John R. Taylor, Classical Mechanics (University Science Books, 2005): the clearest undergraduate treatment of Lagrange, Hamilton and chaos. For a more advanced level, Goldstein, Poole and Safko, Classical Mechanics (3rd edition, Pearson/Addison-Wesley, 2002), the long-standing graduate reference. David Morin's Introduction to Classical Mechanics (Cambridge University Press) is rich in worked problems.
• 📘 Gentler: Leonard Susskind and George Hrabovsky, The Theoretical Minimum: What You Need to Know to Start Doing Physics (Basic Books, 2013), or Richard Feynman's chapter "The Principle of Least Action" in The Feynman Lectures on Physics, Vol. II, ch. 19, free online.
• ▶️ Free course: MIT OpenCourseWare 8.223 Classical Mechanics II (IAP 2017); for the graduate level, 8.09 Classical Mechanics III (Fall 2014).
• 🌐 Easier first reads in this library: Energy: The One Currency (13/02), Orbits: Why the Moon Keeps Falling (13/11), Chaos Theory and the Butterfly Effect (12/07), and in this series Physics I (34/06) and Vibrations (34/17).