nLab 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.

Via prequantized Lagrangian correspondences

See at prequantized Lagrangian correspondence.

In multisymplectic geometry

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

Related concepts

• canonical coordinates, canonical momenta

Hamiltonian	$\leftarrow$ Legendre transform $\rightarrow$	Lagrangian
Lagrangian correspondence	prequantization	prequantized Lagrangian correspondence

• Relation with Fourier transform?

… tropical

References

The concept is named after Adrien-Marie Legendre.

Reviews include

• Torontowiki: Legendre Transformation

See also

• Wikipedia, Legendre transformation

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