Legendre transform

Intuition [Beginner]

The Legendre transform is a way to swap which variable you treat as fundamental. In mechanics you start with a Lagrangian $L(q,\dot{q})$, where the basic data is position $q$ and velocity $\dot{q}$. The Legendre transform replaces $\dot{q}$ by a new variable $p$ called the conjugate momentum, defined as the slope of $L$ in the $\dot{q}$ direction. The output is the Hamiltonian $H(q,p)$, which encodes the same physics but lives on a new space — phase space — where positions and momenta sit as independent coordinates.

Here is the analogy. A convex curve can be described by listing the height of the curve above each horizontal position, or by listing the slope of each supporting tangent line and where that line cuts the vertical axis. Both descriptions carry the same information. The Legendre transform is the dictionary between them.

The geometric payoff is that phase space — the space where $H$ lives — has more structure than velocity space. It has a natural symplectic form, an extra geometric object that encodes time evolution as a rotation in pairs of coordinates. The Lagrangian story takes place on the tangent bundle $TQ$; the Hamiltonian story takes place on the cotangent bundle $T^{\ast }Q$. The Legendre transform is the bridge.

Visual [Beginner]

A schematic showing two graphs side by side. On the left, a convex curve with a horizontal axis labelled $q$ and a vertical axis labelled $L$, with a tangent line at one point and its slope marked $p$. On the right, a curve with horizontal axis labelled $p$ and vertical axis labelled $H$, with the height of the curve at each $p$ equal to the negative of the $y$-intercept of the corresponding tangent line on the left. An arrow labelled "Legendre transform" connects the two.

The key picture is that each tangent line on the left becomes a single point on the right, and vice versa. The transform turns slopes into positions and intercepts into heights.

Worked example [Beginner]

Take the Lagrangian for a free particle of mass $m$ in one dimension: $L(q,\dot{q})=\frac{1}{2}m{\dot{q}}^2$.

The conjugate momentum is the slope of $L$ in the $\dot{q}$ direction. Differentiating $\frac{1}{2}m{\dot{q}}^2$ with respect to $\dot{q}$ gives $p=m\dot{q}$. This matches the elementary physics formula "momentum equals mass times velocity." Solve for the velocity: $\dot{q}=p/m$.

The Hamiltonian is built by the rule $H=p\dot{q}-L$, with $\dot{q}$ rewritten in terms of $p$. So $H=p\cdot (p/m)-\frac{1}{2}m(p/m{)}^2=p^2/m-p^2/(2m)=p^2/(2m)$.

Now add a potential energy $V(q)$, so $L=\frac{1}{2}m{\dot{q}}^2-V(q)$. The momentum is still $p=m\dot{q}$, and the Hamiltonian becomes $H=p^2/(2m)+V(q)$. What this tells us: the Lagrangian was kinetic minus potential, the Hamiltonian is kinetic plus potential, and the Legendre transform converted between them.

Check your understanding [Beginner]

Formal definition [Intermediate+]

There are two complementary settings — one from convex analysis on ${\mathbb{R}}^n$, one from differential geometry on a smooth manifold. The mechanics application is the second; the conceptual content is in the first.

Convex form. Let $f:{\mathbb{R}}^n\to \mathbb{R}$ be a smooth strictly convex function with superlinear growth ($f(q)/|q|\to \infty$ as $|q|\to \infty$). The Legendre transform of $f$ is the function $f^{\ast }:{\mathbb{R}}^n\to \mathbb{R}$ defined by

$$f^{\ast }(p)=\underset{q\in {\mathbb{R}}^n}{\sup }(⟨p,q⟩-f(q)).$$

Under the strict-convexity-plus-superlinear hypothesis the supremum is attained at a unique $q=q(p)$ characterised by the equation $p=\nabla f(q)$. The map $p\mapsto q(p)$ is the inverse of the gradient map $\nabla f$, and one has the symmetric relation $\nabla f^{\ast }(p)=q(p)$. The transform is involutive: $f^{\ast \ast }=f$ (Fenchel-Moreau theorem ^{[Rockafellar §12]}). Geometrically, $f^{\ast }(p)$ equals minus the $y$-intercept of the supporting hyperplane to the graph of $f$ with slope $p$.

Fibre form on a manifold. Let $Q$ be a smooth manifold and $L:TQ\to \mathbb{R}$ a smooth Lagrangian. The fibre Legendre transform (or fibre derivative) is the smooth bundle map

$$\mathbb{F}L:TQ\to T^{\ast }Q,\mathbb{F}L(q,\dot{q})(\xi )=\frac{d}{ds}{|}_{s=0}L(q,\dot{q}+s\xi ),$$

for $(q,\dot{q})\in TQ$ and $\xi \in T_{q}Q$. In coordinates this is $\mathbb{F}L(q,\dot{q})=(q,\partial L/\partial {\dot{q}}^{i}d{q}^i)$, recovering the conjugate-momentum prescription $p_i=\partial L/\partial {\dot{q}}^i$.

The Lagrangian $L$ is regular if $\mathbb{F}L$ is a local diffeomorphism — equivalently, the vertical Hessian matrix $H_{ij}(q,\dot{q})={\partial }^{2}L/\partial {\dot{q}}^i\partial {\dot{q}}^j$ is invertible at every $(q,\dot{q})$. It is hyper-regular if $\mathbb{F}L$ is a global diffeomorphism $TQ\to T^{\ast }Q$. Hyper-regularity is the hypothesis under which Hamiltonian mechanics on $T^{\ast }Q$ is fully equivalent to Lagrangian mechanics on $TQ$.

Hamiltonian. Given a hyper-regular $L$, the Hamiltonian $H:T^{\ast }Q\to \mathbb{R}$ is

$$H(q,p)=⟨p,\dot{q}(q,p)⟩-L(q,\dot{q}(q,p)),$$

where $\dot{q}(q,p)=(\mathbb{F}L{)}^{-1}(q,p)$. The phase space $T^{\ast }Q$ carries the canonical Liouville one-form $\lambda =p_{i}d{q}^i$ and the symplectic form $\omega =-d\lambda =d{q}^i\wedge d{p}_i$, providing the geometric setting for Hamilton's equations.

Counterexamples to common slips

• The Legendre transform is not the same as a Fourier transform, despite both being involutive integral-style transforms; the Legendre transform is purely pointwise (a supremum at each $p$) and requires convexity, not periodicity or integrability.
• Regularity of $L$ is not the same as hyper-regularity. Regularity is a pointwise condition (the Hessian is invertible everywhere); hyper-regularity adds global injectivity and surjectivity of $\mathbb{F}L$. A free particle on $\mathbb{R}$ is hyper-regular; a free particle on $S^1$ with a multivalued Lagrangian construction can fail global injectivity.
• For non-hyper-regular $L$, the formula $H(q,p)=⟨p,\dot{q}⟩-L$ is not well-defined as a function on $T^{\ast }Q$ — one cannot solve $p=\partial L/\partial \dot{q}$ uniquely for $\dot{q}$. The Dirac-Bergmann algorithm is the systematic substitute.

Key theorem with proof [Intermediate+]

Theorem (Lagrange-Hamilton equivalence). *Let $Q$ be a smooth manifold, $L:TQ\to \mathbb{R}$ a hyper-regular Lagrangian, and $H:T^{\ast }Q\to \mathbb{R}$ the corresponding Hamiltonian. A smooth curve $\gamma :I\to Q$ satisfies the Euler-Lagrange equations on $TQ$ if and only if the curve $(\gamma (t),p(t))\in T^{\ast }Q$ defined by $p(t)=\mathbb{F}L(\gamma (t),\dot{\gamma }(t))$ satisfies Hamilton's equations*

$${\dot{q}}^i=\frac{\partial H}{\partial p_i},{\dot{p}}_i=-\frac{\partial H}{\partial q^i}.$$

Proof. Work in a chart on $Q$ with coordinates $q^i$, induced coordinates $(q^i,{\dot{q}}^i)$ on $TQ$ and $(q^i,p_i)$ on $T^{\ast }Q$. Hyper-regularity says $p_i=\partial L/\partial {\dot{q}}^i$ defines a diffeomorphism, so ${\dot{q}}^i$ is a smooth function of $(q,p)$.

Differentiate the defining relation $H(q,p)=p_j{\dot{q}}^j(q,p)-L(q,\dot{q}(q,p))$ with respect to $p_i$ at fixed $q$:

$$\frac{\partial H}{\partial p_i}={\dot{q}}^i+p_j\frac{\partial {\dot{q}}^j}{\partial p_i}-\frac{\partial L}{\partial {\dot{q}}^j}\frac{\partial {\dot{q}}^j}{\partial p_i}.$$

Since $p_j=\partial L/\partial {\dot{q}}^j$, the last two terms cancel and one obtains $\partial H/\partial p_i={\dot{q}}^i$. This is the first Hamilton equation, identical in content to the inverse fibre derivative.

Differentiate the same relation with respect to $q^i$ at fixed $p$:

$$\frac{\partial H}{\partial q^i}=p_j\frac{\partial {\dot{q}}^j}{\partial q^i}-\frac{\partial L}{\partial q^i}-\frac{\partial L}{\partial {\dot{q}}^j}\frac{\partial {\dot{q}}^j}{\partial q^i}.$$

Again $p_j=\partial L/\partial {\dot{q}}^j$ cancels the first and third terms, leaving $\partial H/\partial q^i=-\partial L/\partial q^i$.

Now if $\gamma$ satisfies Euler-Lagrange, $\frac{d}{dt}(\partial L/\partial {\dot{q}}^i)=\partial L/\partial q^i$. The left side is ${\dot{p}}_i$ along the curve. The right side equals $-\partial H/\partial q^i$ by the previous calculation. Combining with the first equation gives Hamilton's system. The reverse direction follows by reading the same equalities backward. $\square$

Bridge. The Legendre transform builds toward 05.01.02 (symplectic manifold) and 05.02.05 (cotangent bundle as canonical symplectic manifold): the same fibre derivative $\mathbb{F}L:TQ\to T^{\ast }Q$ appears again in the construction of canonical phase-space coordinates, where the Liouville one-form $\lambda =p_{i}d{q}^i$ on $T^{\ast }Q$ and the symplectic form $\omega =-d\lambda$ supply the geometric scaffolding. Putting these together, the equivalence of Euler-Lagrange and Hamilton equations is exactly the statement that the fibre derivative identifies Lagrangian dynamics with symplectic dynamics; the foundational reason the symplectic side is preferred for further development is that $T^{\ast }Q$ carries an intrinsic geometric structure ($\omega$) that $TQ$ does not. The bridge between variational mechanics and symplectic geometry is precisely the Legendre transform.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

Mathlib does not yet contain the integrated layer described in the lean_mathlib_gap field. A skeleton statement of what the formalisation would look like, paralleling the key theorem above:

[object Promise]

The roadmap items (1)-(5) in lean_mathlib_gap describe the supporting infrastructure each sorry would require. The Fenchel-Moreau involution and the smooth-bundle-map structure of $\mathbb{F}L$ are independent contributions to convex analysis and to the differential-geometry library respectively.

Advanced results [Master]

The Legendre transform is the conceptual hinge of analytical mechanics and a recurring template across mathematical physics. The advanced material falls into three groups: convex-analytic generalisations, geometric refinements on the manifold side, and the singular case where hyper-regularity fails.

Convex-analytic generalisations. The pointwise definition $f^{\ast }(p)={\sup }_q(⟨p,q⟩-f(q))$ extends without modification to convex functions on a topological vector space, with values in $\mathbb{R}\cup \{+\infty \}$. The Fenchel-Moreau theorem then reads $f^{\ast \ast }=f$ on the class of lower-semicontinuous convex functions ^{[Rockafellar §12]}. Strict convexity is what guarantees a unique attaining $q$ at each $p$ and hence a single-valued gradient relation; convexity without strictness yields a multivalued subdifferential. The Legendre transform is dual to the Fenchel conjugacy of convex sets via the indicator-function correspondence: the polar of a convex set $C$ is the Legendre transform of its indicator function.

Geometric refinements. The fibre derivative $\mathbb{F}L:TQ\to T^{\ast }Q$ is the vertical-derivative map for the bundle $TQ\to Q$. When $L$ is hyper-regular, the pullback $(\mathbb{F}L{)}^{\ast }{\omega }_{T^{\ast }Q}$ of the canonical symplectic form on $T^{\ast }Q$ is the Lagrangian symplectic form ${\omega }_L=-d{\theta }_L$, where ${\theta }_L=(\mathbb{F}L{)}^{\ast }\lambda$ is the Lagrangian one-form. In coordinates ${\theta }_L=(\partial L/\partial {\dot{q}}^i)d{q}^i$ and ${\omega }_L=-d{\theta }_L$. Hamilton's equations on $T^{\ast }Q$ pull back along $\mathbb{F}L$ to the Euler-Lagrange flow on $TQ$. The whole symplectic geometry of phase space is thus implicit on the Lagrangian side via the fibre derivative, and explicit on the Hamiltonian side via the canonical structure of $T^{\ast }Q$.

Singular Lagrangians and the Dirac-Bergmann algorithm. Many physical Lagrangians fail to be regular: gauge theories (Maxwell, Yang-Mills), general relativity in the ADM formulation, and reparametrisation-invariant systems all have $\det ({\partial }^{2}L/\partial {\dot{q}}^i\partial {\dot{q}}^j)=0$ identically. The fibre derivative is then not a local diffeomorphism. Dirac and Bergmann developed a constraint algorithm that handles this case ^{[Marsden-Ratiu Ch. 7]}.

The image $\mathbb{F}L(TQ)\subset T^{\ast }Q$ is locally cut out by primary constraints ${\varphi }_a(q,p)=0$, the relations among the $p_i$ that hold automatically because the fibre derivative is not surjective. Consistency of the dynamics (preservation of these constraints under Hamiltonian flow) generates secondary constraints. Iterating, the constraint manifold is the maximal subset of $T^{\ast }Q$ on which the dynamics is consistent. Constraints split into first-class (their Poisson brackets vanish on the constraint manifold; these generate gauge transformations) and second-class (those whose brackets are non-degenerate on the constraint manifold; these reduce phase-space dimension). The reduced phase space, obtained by symplectic reduction with respect to the first-class constraints and the symplectic submanifold structure inherited via the second-class constraints, recovers a symplectic manifold on which the gauge-fixed dynamics is genuinely Hamiltonian.

Synthesis. The Legendre transform is precisely the bridge between two formulations of mechanics that look different but encode the same data: $L:TQ\to \mathbb{R}$ on velocity space and $H:T^{\ast }Q\to \mathbb{R}$ on phase space. The fibre derivative identifies $TQ$-mechanics with $T^{\ast }Q$-symplectic-geometry, and this identification is exactly the foundational reason every classical mechanical system has both a variational ("least action") and a flow-on-symplectic-manifold formulation. Putting these together, the Legendre transform is an instance of a more general duality: convex Fenchel duality on a vector space, fibre-wise on a bundle, and reduction-modulo-gauge in the singular setting all generalise the same template — the gradient of a convex function gives a diffeomorphism between a space and its dual, and the dual function is the conjugate. The bridge between Lagrangian variational mechanics and Hamiltonian symplectic geometry is exactly this gradient-as-diffeomorphism construction, fibred over the configuration manifold $Q$.

Full proof set [Master]

Theorem (Fenchel-Moreau involution). Let $f:{\mathbb{R}}^n\to \mathbb{R}$ be a smooth strictly convex function with superlinear growth. Then $f^{\ast \ast }=f$.

Proof. Let $g=f^{\ast }$. Since $f$ is strictly convex with $\nabla f$ a diffeomorphism ${\mathbb{R}}^n\to {\mathbb{R}}^n$ (by superlinear growth and strict convexity), the supremum in $g(p)={\sup }_q(⟨p,q⟩-f(q))$ is attained at the unique $q$ satisfying $p=\nabla f(q)$. Differentiate $g$: $\nabla g(p)=q(p)=(\nabla f{)}^{-1}(p)$. Hence $\nabla g$ is a diffeomorphism (the inverse of $\nabla f$), and $g$ is strictly convex with superlinear growth.

Now compute $g^{\ast }(q)={\sup }_p(⟨p,q⟩-g(p))$, with the supremum attained at $p$ satisfying $q=\nabla g(p)=(\nabla f{)}^{-1}(p)$, which inverts to $p=\nabla f(q)$. Substituting, $$g^{\ast }(q)=⟨\nabla f(q),q⟩-g(\nabla f(q))=⟨\nabla f(q),q⟩-(⟨\nabla f(q),q⟩-f(q))=f(q).$$ $\square$

Theorem (fibre derivative is a smooth bundle map). *For $L:TQ\to \mathbb{R}$ smooth, the fibre derivative $\mathbb{F}L:TQ\to T^{\ast }Q$ is a smooth map of fibre bundles over $Q$.*

Proof. In a coordinate chart $(q^i,{\dot{q}}^j)$ on $TQ$ and $(q^i,p_j)$ on $T^{\ast }Q$, the fibre derivative is given by $\mathbb{F}L(q^i,{\dot{q}}^j)=(q^i,\partial L/\partial {\dot{q}}^j)$. Each component is smooth as a partial derivative of a smooth function, so $\mathbb{F}L$ is smooth in coordinates. The expression respects the bundle projections: the $q^i$-coordinate is preserved, so ${\pi }_{T^{\ast }Q}\circ \mathbb{F}L={\pi }_{TQ}$, where $\pi$ denotes the bundle projection to $Q$. Thus $\mathbb{F}L$ is a smooth map over $Q$. The coordinate-free definition $\mathbb{F}L(q,\dot{q})(\xi )=\frac{d}{ds}{|}_{s=0}L(q,\dot{q}+s\xi )$ verifies that the construction is intrinsic, independent of the chart. $\square$

Theorem (Lagrangian-Hamiltonian equivalence; restated). See the key theorem above; the proof there is the full proof. $\square$

Theorem (regular implies local-diffeomorphism). Let $L:TQ\to \mathbb{R}$ be regular at $(q_0,{\dot{q}}_0)$. Then $\mathbb{F}L$ is a local diffeomorphism near $(q_0,{\dot{q}}_0)$.

Proof. By Exercise 4 above, the Jacobian of $\mathbb{F}L$ in coordinates is block-triangular with diagonal blocks the identity and the vertical Hessian. The total determinant equals the determinant of the vertical Hessian, which is nonzero by regularity. The inverse function theorem then yields the local diffeomorphism. $\square$

Connections [Master]

• Lagrangian on the tangent bundle 05.00.01. The starting point: the Legendre transform takes a Lagrangian $L:TQ\to \mathbb{R}$ as input. Hyper-regularity of $L$ is the hypothesis under which the transform yields a globally well-defined Hamiltonian description.

• Symplectic manifold 05.01.02. The output of the Legendre transform — the Hamiltonian $H:T^{\ast }Q\to \mathbb{R}$ — lives on a symplectic manifold. The cotangent bundle's canonical symplectic structure is what makes the Hamiltonian formulation geometric rather than merely algebraic.

• Cotangent bundle as canonical symplectic manifold 05.02.05. $T^{\ast }Q$ is the canonical example of a symplectic manifold; the Legendre transform is exactly what justifies this canonicity for mechanics. The Liouville one-form $\lambda =p_{i}d{q}^i$ has its mechanical meaning fixed by the Legendre construction.

• Hamiltonian vector field 05.02.01. The Hamilton equations $\dot{q}=\partial H/\partial p$, $\dot{p}=-\partial H/\partial q$ are exactly the integral curves of the Hamiltonian vector field $X_H$ defined by ${\iota }_{X_H}\omega =dH$. Without the Legendre transform there is no canonical $H$ to plug in; with it, the Hamiltonian flow on $T^{\ast }Q$ is the symplectic-geometric image of the Euler-Lagrange flow on $TQ$.

• Noether's theorem 05.00.04. Noether's theorem on the Lagrangian side has its Hamiltonian counterpart via the moment map. The Legendre transform connects the two: a one-parameter symmetry of $L$ becomes a one-parameter symmetry of $H$, and the conserved quantity transforms accordingly.

• *Geodesic flow as Hamiltonian flow on $T^{\ast }M$ 05.02.06.* The free-particle Lagrangian on a Riemannian manifold is $L=\frac{1}{2}{g}_{ij}{\dot{q}}^i{\dot{q}}^j$; Legendre transform gives $H=\frac{1}{2}{g}^{ij}{p}_i{p}_j$, the geodesic Hamiltonian. The geodesic flow on $T^{\ast }M$ is a special case of Hamiltonian flow, accessible via the Legendre transform.

• Hamilton-Jacobi equation 05.02.07. Hamilton-Jacobi theory expresses the action $S(q,t)$ as a generating function for the Hamiltonian flow; the Legendre relation $p=\partial L/\partial \dot{q}=\partial S/\partial q$ is the variational form of the Legendre transform applied to the action.

Historical & philosophical context [Master]

Adrien-Marie Legendre introduced the transform in 1787 in Mémoire sur l'intégration de quelques équations aux différences partielles (Mém. Acad. Sci. Paris) ^{[Legendre 1787]}, in the context of integrating partial differential equations. The transform was a tool for swapping dependent and independent variables, exploiting the duality between a convex curve and the family of its supporting tangent lines. Legendre's original setting was analytic, not mechanical.

William Rowan Hamilton's 1834 essay On a General Method in Dynamics and its 1835 sequel introduced what is now called the Hamiltonian formulation of mechanics. Hamilton arrived at the equations $\dot{q}=\partial H/\partial p$, $\dot{p}=-\partial H/\partial q$ via the principal function (now called the action), without explicit reference to Legendre's transform; the connection between Hamilton's $H$ and Lagrange's $L$ via $H=p\dot{q}-L$ was understood by Hamilton but presented as part of the variational machinery rather than as a transform.

The recognition that Hamilton's substitution is exactly Legendre's 1787 transform — applied fibre-wise on $TQ\to T^{\ast }Q$ — was developed in the 19th century by Jacobi and others, and given its modern coordinate-free formulation in the 20th century. Vladimir Arnold's Mathematical Methods of Classical Mechanics ^{[Arnold §15]} gives the geometric statement: $\mathbb{F}L:TQ\to T^{\ast }Q$ as a fibre derivative, with hyper-regularity as the condition for the Hamiltonian formulation to be globally equivalent. Ralph Abraham and Jerrold Marsden's Foundations of Mechanics (1978) ^{[Abraham-Marsden §3.6]} built the fully geometric Lagrangian-Hamiltonian theory on this footing, including the Lagrangian symplectic form ${\omega }_L$ and the systematic treatment of regularity.

The convex-analytic side was developed independently by Werner Fenchel in the 1940s and systematised by R. Tyrrell Rockafellar in Convex Analysis (1970) ^{[Rockafellar §12]}; the Fenchel-Moreau involution theorem $f^{\ast \ast }=f$ on lower-semicontinuous convex functions is the precise general statement of which Legendre's smooth-strictly-convex case is a special case. The singular-Lagrangian extension via the Dirac-Bergmann constraint algorithm grew out of P.A.M. Dirac's 1950s work on canonical quantisation of gauge theories.

Bibliography [Master]

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