##  [Legendre Transform](/legendre-transform-0) 

 Definition

An operation that maps a (convex, proper, lower semicontinuous) function f(x) of a variable x to its convex conjugate f*(p) = sup_x (p·x − f(x)), exchanging variable and slope and producing a dual representation.

 

 

 

 

 

 





## Principle

Principle

Realize a duality between 'coordinates' and 'conjugate momenta' (or slopes) by taking a supremal linearization; the Legendre transform converts variational descriptions and often turns constrained optimization into an unconstrained dual problem.

 

 

 

 

 





## Demonstration

Demonstration

For f(x)=½ax^2 (a>0), the Legendre transform yields f*(p)=½ p^2 / a; in mechanics, the Legendre transform of a regular Lagrangian with respect to velocities produces the Hamiltonian in momentum coordinates.

 

 

 

 

## Misapplication

Misapplication

Applying the classical Legendre transform to nonconvex functions without accounting for multiple local suprema; ignoring lower semicontinuity can lead to transforms that correspond to convex envelopes rather than an involutive inverse.

 

 

 

 

 





## Consequence

Consequence

Provides a bridge between primal and dual formulations: solves optimization by passing to the convex conjugate, yields thermodynamic potentials from free energies, and produces Hamiltonian from Lagrangian under regularity.

 

 

 

 

## Reversal

Reversal

The inverse Legendre transform recovers the original function when f is proper, convex and lower semicontinuous; for nonconvex f the inverse may produce the convex hull rather than the original function.

 

 

 

 

 





## Boundary

Boundary

Standard Legendre transform requires convexity and appropriate growth (properness, lsc) for involutivity; for nondifferentiable points replace gradient by subgradient and for nonconvex settings use generalized convexification.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Confused with Fourier or Laplace transforms; Legendre transform is a convex duality via supremum of linear functionals, not an integral transform — it exchanges geometrical slope information rather than frequency content.

 

 

 

 

 





## Synthesis

Synthesis

The Legendre transform assigns to a function its convex conjugate by supremizing linear forms p·x−f(x), effecting a coordinate‑to‑slope duality that underpins dual optimization, thermodynamic potentials and passage from Lagrangian to Hamiltonian formalisms.