Legendre Transformation

Regular change from generalized velocities to conjugate momenta. Theory, formulas, worked examples and interactive graphs.

Conceptual context

The Legendre transformation appears when a physical description written with a variable and its velocity must be rewritten with the variable and its conjugate momentum. In analytical mechanics, the Lagrangian $L$ naturally lives in configuration space: it uses the generalized coordinate $q$, the generalized velocity $\dot{q}$, and, when needed, time. The Hamiltonian $H$ lives in phase space: it uses $q$ and the conjugate momentum $p$.

This change is not a cosmetic translation. It changes which information is treated as primary. In the Lagrangian formalism one asks which path makes the action stationary; in the Hamiltonian formalism one asks how a phase-space point evolves. The Legendre transformation is the technical bridge between both readings when the relation between velocity and momentum is invertible.

🟢 Essential level

The essential idea is simple: if you know how the Lagrangian changes when generalized velocity varies, you can define a new dynamical variable. That variable measures a physical slope, not necessarily a Cartesian linear momentum. For an ordinary free particle it matches the intuition of mass and motion; in generalized coordinates or in systems with fields, it may have a different form.

Then a new dynamical function is built that no longer takes velocity as the primary variable. But the result is truly Hamiltonian only if velocity can be eliminated and written from conjugate momentum. That is why a Legendre transformation always includes a physical condition: writing symbols is not enough; the velocity-momentum map must be invertible and must preserve state information.

A practical way to consolidate Legendre Transformation is to separate the physical system, the independent variables, and the mathematical representation before calculating. Regular change from generalized velocities to conjugate momenta. The student should be able to explain what information the analytical description preserves, what it replaces from the Newtonian treatment, and which observable consequence would reveal an inconsistent choice.

🔵 Formal level

Basic formulas

This first formal layer presents the reference relation used to recognize the model and read its variables before entering more abstract derivations. Identify what each term represents, which assumptions make it valid, and how the chosen coordinates connect to the physical motion that the model is intended to describe.

To fix the simplest mechanical case, start from a Lagrangian with quadratic kinetic energy and potential energy $V$:

Quadratic Lagrangian

Real student questions

A common question is why the Hamiltonian is not simply defined as total energy. The reason is that the formalism needs a construction that works from $L$. In many systems the result equals energy, but that equality is a consequence of physical hypotheses, not a universal definition.

Another doubt is why $p$ is called momentum if it is obtained by differentiating. In ordinary Cartesian coordinates that derivative reproduces linear momentum. In generalized coordinates, the name is kept because the variable plays the dynamical role conjugate to the coordinate, although its unit and interpretation depend on the system.

Final synthesis

The Legendre transformation converts a velocity-based description into a conjugate-momentum description. Its core is to compute $p$, check that $\dot{q}$ can be recovered, and build $H$ without hidden dependencies. By the end of this node, you should be able to decide when the Hamiltonian represents mechanical energy and when it is only the phase-space function organizing the dynamics.