nLab Legendre transformation

The Legendre transformation


Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)


physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

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The Legendre transformation


The Legendre transformation is an operation on convex functions from a real normed vector space to the real line; it is one of the cornerstones of convex analysis. The space of arguments changes accordingly.


Two differentiable functions f,f˜:f, \tilde f \;\colon\; \mathbb{R} \to \mathbb{R} on the real line are said to be Legendre transforms of each other, if their derivatives Df,Df˜:D f, D\tilde f \;\colon\; \mathbb{R} \to \mathbb{R} are inverse functions of each other:

(1)DfDf˜=idAAADf˜Df=id D f \circ D \tilde f = id \phantom{AAA} D \tilde f \circ D f = id

In classical mechanics – Hamiltonians and Lagrangians

The main application of and the historical root of the notion of Legendre transform (in differential geometry) is in classical physics and its formalization by symplectic geometry. In classical mechanics, the Hamiltonian function HH is a Legendre transform of the Lagrangian LL and vice versa.

When one formalizes classical mechanics as the local prequantum field theory given by prequantized Lagrangian correspondences, then the Legendre transform is exhibited by the lift from a Lagrangian correspondence to a prequantized Lagrangian correspondence. For more on this see at The classical action, the Legendre transform and Prequantized Lagrangian correspondences.

In many dimensions, hybrid versions are possible. When the physics of the system is given by the variational principle, then the Legendre transform of an extremal quantity is a conserved quantity. In thermodynamics, we can have some quantities set to be fixed (some candidates: entropy SS, temperature TT, pressure PP, volume VV, magnetization MM); this dictates the choice of variables and quantity which is extremized as well as which one takes the role of conserved energy. Some of the standard choices are enthalpy HH, Helmholtz free energy FF, Gibbs free energy GG, internal energy UU, etc.

See also wikipedia:Legendre transformation and wikipedia:Legendre-Fenchel transformation; the two wikipedia articles have much detail in certain specific approaches and cases, but also miss some of the basic ones to be balanced.

Via prequantized Lagrangian correspondences

See at prequantized Lagrangian correspondence.

In multisymplectic geometry

See at multisymplectic geometry – de Donder-Weyl-hamilton equations of motion.

Hamiltonian\leftarrow Legendre transform \rightarrowLagrangian
Lagrangian correspondenceprequantizationprequantized Lagrangian correspondence
  • Relation with Fourier transform?

… tropical


The concept is named after Adrien-Marie Legendre.

Reviews include

See also

Discussion of Legendre transformation in the context of Lie algebroids is in:

  • Paulette Liberman, Lie algebroids and mechanics (ps)

  • Jorge Cortes et al, A survey of Lagrangian mechanics and control on Lie algebroids and Lie groupoids (arxiv)

  • Juan Carlos Marrero, Nonholonomic mechanics: a Lie algebroid perspective (pdf talk notes)

category: physics

Last revised on July 2, 2023 at 19:09:44. See the history of this page for a list of all contributions to it.