Hamilton's principle/Explainers/What the Euler-Lagrange equation says

Technical version: The fundamental lemma· The Euler-Lagrange equation

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 the central equation of the subject. It uses no mathematics. The technical pages it goes with are named above.

The problem so far

A localised test function.

Comparing paths set things up. You have a rule that gives a number for any whole path. You want the path where that number stops changing when you nudge it.

Turning the dial and watching the number tells you something, but it tells you something about the whole path at once. It says the total stops changing.

That is not yet useful. What you want is a rule the path obeys at each point, so you can work the path out step by step. Getting from the first kind of statement to the second is the job of this page.

Adding up to zero

The two rates the Euler-Lagrange equation balances.

Here is the difficulty, in everyday terms.

Suppose a shop's takings for the day come to exactly zero. There are two possible reasons.

Every single sale might have been for nothing. Or there might have been large sales and large refunds that happened to cancel out.

The first tells you a lot. The second tells you almost nothing.

When physics says the total change in the number is zero, it faces the same ambiguity. Perhaps nothing is changing anywhere along the path. Or perhaps changes in one stretch are being cancelled by opposite changes elsewhere.

To get a rule at each point, the second possibility has to be ruled out.

Ruling out the cancellation

The way to rule it out is to be greedy about the comparisons.

Remember that you get to choose the nudge. Any smooth nudge is allowed, as long as it is zero at both ends.

So choose a nudge that is zero everywhere except in one small stretch. This is like poking the path with a finger in one place and leaving the rest alone.

If something were wrong in that stretch, this nudge would change the total. There is no other stretch involved, so nothing can cancel it.

Now do the same thing everywhere. Poke every stretch of the path in turn. If the total refuses to change no matter where you poke, then nothing is wrong anywhere.

That argument, done properly, is what mathematicians call the fundamental lemma. It has a long name and a simple idea: if you can test anywhere, and every test comes back clean, then everything is clean.

The greediness is essential. If you were only allowed a few fixed nudges, the cancellation dodge would still be open.

The rule at each point

Once the cancellation is ruled out, a rule at each point drops out. It is called the Euler-Lagrange equation, after the two people who found it in the 1740s and 1750s.

In words, and with no symbols, it says this.

At every moment, the rate at which the path's cost changes because of where you are must be exactly balanced by the rate at which it changes because of how fast you are going.

For a moving object that turns out to be Newton's second law: force equals mass times acceleration. The two statements say the same thing.

That is worth pausing on. A rule about total cost over a whole journey, and a rule about the push on an object at one instant, are the same rule. One is expressed as a statement about the journey. The other as a statement about the moment.

What you get for free

The equation has a feature that saves an enormous amount of work.

If the cost rule does not care where you are, something is conserved. A skater on flat ice feels the same conditions everywhere. Nothing about the ice depends on position. As a result the skater's momentum does not change.

If the cost rule does not care what time it is, something else is conserved. The laws of physics were the same yesterday. As a result energy does not change.

You can read these off without solving anything. Look at the rule, see what it leaves out, and you know what stays constant.

This is one of the most useful habits in physics. Later in the volume it turns out to be an instance of a general theorem — every symmetry gives a conserved quantity — and that is explained at Symmetry and conservation.

What this does not tell you

One warning, and it runs through the whole volume.

The equation finds paths where the number stops changing. It does not tell you the number is smallest.

The straight line on paper really is the shortest. But that is an extra fact, proved separately. The equation on its own does not establish it.

There are cases where the equation is satisfied and the path is not shortest at all. The clearest is the long way round a globe: it obeys the rule at every point and is obviously not the short route.

Paths found by this equation are called extremals. The name records that the rule is satisfied, and nothing more. Whether an extremal is best is a separate question with its own answer, and that answer is at Is it really a minimum.

What comes next

Two things are still missing.

Everything so far assumed both ends of the path are pinned down. What if one end is free to move? That is at Free ends.

And everything so far assumed one thing is moving. Real systems have many parts. That is at More than one variable.