Hamilton's principle/Explainers/The Hamiltonian

Technical version: The Hamiltonian and Hamilton's equations

Hamilton's principle: map
Reading the volume How to read this volume Notation Glossary Equation index Figure gallery
Reading by level Start here: Comparing paths The mathematics: Functionals and the space of paths Stationary is not least: The Jacobi condition and conjugate points The applications: The brachistochrone The history: Hamilton 1834 and 1835
Sources Sources Chronology
Related Noether's theorems On Physical Lines of Force

This page explains a reformulation of mechanics in terms of position and momentum. It uses no mathematics. The technical page it goes with is named above.

A different pair of quantities

Phase portrait of a pendulum.

The usual description of a moving object tracks where it is and how fast it is going.

Apply the swap from Reading a curve two ways and you get a description tracking where it is and its momentum instead.

For a simple object momentum is just mass times speed, so this looks like a relabelling. It is not, and the reason is what happens to the equations.

Fewer steps, more equations

Lagrangian and Hamiltonian descriptions compared.

Newton's laws, and everything in this volume so far, involve accelerations — how the speed is changing. Acceleration is a rate of change of a rate of change. Two steps removed from position.

In the new description, that never happens. You get twice as many equations, but each involves only one rate of change.

The trade is real. Equations with two steps of change are harder to handle, harder to solve numerically, and harder to reason about, than twice as many with one step each.

The two forms have the same content. Neither knows anything the other does not.

A space where a point is a whole state

Here is the change that actually matters.

In the usual description, knowing where an object is does not tell you what it will do. A ball at a given place might be rising or falling. Position alone is not the state.

In the new description, position and momentum together are the state, and they are treated as coordinates of equal standing. Every possible state of the system is a single point in a space with both.

That space is called phase space.

The consequence is the useful one: a single point in phase space determines the entire future. There is no ambiguity left over, because the information that was missing is now part of the location.

Paths that never cross

From that follows a fact with real force.

Trajectories in phase space never cross.

If two trajectories met at a point, that point would be one state with two futures. Impossible, because the state fixes the future completely.

So the motion of a system becomes a flow, like a fluid, with streamlines that never meet. Every point moves, nothing collides, nothing merges.

This gives you a way to reason about a system without solving anything. Trajectories are hemmed in by their neighbours; regions of phase space get carried along and deformed but cannot be torn.

Reading off the answer

For a system where energy is conserved, the payoff is immediate.

Energy has a fixed value along any trajectory. So each trajectory stays on a single surface of constant energy.

Draw those surfaces and you have the shape of the motion, without solving anything.

For a pendulum swinging gently, the surfaces are closed loops. A trajectory going round a closed loop returns to where it started: the motion repeats. That is the swing.

For a pendulum with enough energy to go over the top, the surfaces are not closed loops but curves running off in one direction. The pendulum keeps rotating the same way and never comes back.

Between the two lies a special curve separating the swinging region from the spinning one. On it, the pendulum takes infinitely long to creep up to the vertical and balance.

All of that came from looking at the shape of the energy surfaces. No equation was solved. That is the characteristic move of this description, and it is why phase space is the right picture for asking what a system does in general rather than what it does in one particular case.

A caution about energy

The quantity that plays the leading role here is usually the energy, and it is worth knowing when it is not.

For ordinary systems — masses, springs, gravity, fixed constraints — it is the familiar energy, kinetic plus potential.

For systems where the constraints themselves are moving, it can differ. Something is still conserved, and it is still the quantity that runs the equations, but calling it the energy will mislead you.

Conserved and equal to the energy are two separate claims. Most treatments merge them because for the standard examples they coincide.

Where this leads

Phase space is not just tidier. It is the setting for most of what came next.

The idea that a region of phase space keeps its volume as it flows became the foundation of statistical mechanics — the physics of systems with too many parts to track individually.

The pairing of position with momentum became, in quantum mechanics, the pairing that cannot be measured simultaneously to arbitrary precision.

And the search for descriptions in which the flow is as simple as possible leads directly to the next page.

Where to go next

Pushed to its conclusion, that search produces a single equation replacing all the others, and it turns out to describe waves. It is at Waves and rays.