# nLab Legendre transformation

The Legendre transformation

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# The Legendre transformation

## Idea

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.

## Definition

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

(1)$D f \circ D \tilde f = id \phantom{AAA} D \tilde f \circ D f = id$

## 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 $H$ is a Legendre transform of the Lagrangian $L$ 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 $S$, temperature $T$, pressure $P$, volume $V$, magnetization $M$); 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 $H$, Helmholtz free energy $F$, Gibbs free energy $G$, internal energy $U$, 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.

### In multisymplectic geometry

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

… tropical

## References

The concept is named after Adrien-Marie Legendre.

Reviews include