Reading a curve two ways
This page explains a change of viewpoint that reorganises the whole subject. It uses no mathematics. The technical page it goes with is named above.
Describing a curve without listing its points
Draw a curve on paper. How would you describe it to someone who cannot see it?
The obvious way is to list its points. At this position it is this high, and so on.
There is a second way, and it is not obvious. Instead of listing points, list the straight lines that just touch it.
For each possible slope, there is one line of that slope that touches the curve and does not cross it. Say where that line sits. Do this for every slope.
A curve is completely determined by the family of lines that graze it. So the two descriptions carry the same information, and either can be recovered from the other.
Swapping between them is called the Legendre transform.
Why bother
Because sometimes the second description is the useful one.
Think about what you know and what you want. Sometimes you know where you are and want to know how steeply things are changing. Sometimes you know how steeply things are changing and want to know where you are.
The transform swaps which of these is the input.
In mechanics the swap is between speed and momentum. The usual description takes speed as the basic quantity. The transform gives one where momentum is basic instead.
That sounds like relabelling. It is not, and the next page shows what it buys.
The condition that makes it work
There is one requirement, and it is easy to state.
The curve must bend one way only.
If it bends consistently — always curving upward, say — then each slope touches it at exactly one place, and the description by tangent lines is unambiguous.
If the curve wiggles, several places have the same slope. Asked to name the point with that slope, you cannot answer. The second description fails.
This turns out to be the same condition that appeared on Is it really a minimum as the local test: wobbling must not pay. The same requirement shows up twice, in two subjects, for two apparently different reasons.
That is worth noticing. When one condition keeps reappearing, it is usually telling you something structural rather than technical.
Where it fails, and why that matters
The requirement is not always met, and the failures are not obscure corner cases. They are some of the most important theories in physics.
Electromagnetism fails it. So does general relativity.
In both cases the reason is the same, and it is respectable: the description contains more quantities than the physics needs. Different descriptions represent the same physical situation. So knowing the physics does not pin down the description, and the transform has nothing unique to return.
That redundancy is not a defect. It is what physicists call gauge freedom, and it is treated at When symmetry is local.
For now the point is that the failure of a mathematical transform and the presence of a physical redundancy are the same fact. Meeting the failure tells you something true about the theory.
The same idea in other places
The transform is not confined to mechanics. Once you know its shape you see it in several subjects.
Thermodynamics uses it constantly. The different energy quantities in a textbook — internal energy, free energy, enthalpy — are transforms of one another, swapping which quantity is treated as the input. An engineer working at fixed temperature wants a different one from an engineer working at fixed heat content, and the transform gets between them.
Economics uses it, though rarely by name. Describing a producer by what they make, or by the prices they face, are two descriptions of one situation. The transform swaps between quantity and price.
The recurring pattern is: when two quantities pair up so that knowing one determines the other, you can build your description on either. Which you pick should depend on which you actually measure.
A picture to keep
Here is the version worth remembering.
Think of a hill. You can describe it by giving the height at every location. Or you can describe it by giving, for each possible slope, the flattest position where the ground tilts that steeply.
Both descriptions contain the hill. Neither is more fundamental.
The first is natural if you are standing on it. The second is natural if you are choosing a route by how steep you are willing to walk.
Physics has both descriptions of mechanics, for the same reason. Which is better depends on the question.
Where to go next
Applying the swap to mechanics produces a reformulation with real advantages, and it is at the Hamiltonian.