That is why in such cases a more generalized Legendre transformation should be considered.

This change of variables in the differentials is the Legendre transformation.

The various expressions for chemical potential are related to each other by Legendre transformations.

The Legendre transformation is an application of the duality relationship between points and lines.

When restricted to a particular point, the fiber derivative is a Legendre transformation.

Free energy functions are Legendre transformations of the internal energy.

Other thermodynamic potentials can also be obtained through Legendre transformation.

In particular, for and we obtain the total Legendre transformation and the identity, respectively.

In mathematics, convex conjugation is a generalization of the Legendre transformation.

The advantage of the Legendre transformation is that the relation between the new and old functions and their variables are obtained in the process.

The result is compared with that obtained from an ad hoc method with the Legendre transformation.

The symmetry of this expression underscores that the Legendre transformation is its own inverse (involutive).

See also Legendre transformation.

The previous formulation has been recast using Legendre transformations to allow expression of the necessary functions in plasticity theory.

The Legendre transformation can be generalized to the Legendre-Fenchel transformation.

The Hamiltonian is calculated using the usual definition of as the Legendre transformation of :

For dissipation-free systems this adjustment can be achieved for each case by means of an ad hoc Legendre transformation on the internal energy density.

Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named after him.

Another result from the Legendre transformation relates the time derivatives of the Lagrangian and Hamiltonian:

Legendre transformation is necessary because mere substitutive replacement of extensive variables by intensive variables does not lead to thermodynamic potentials.

He is known for the Legendre transformation, which is used to go from the Lagrangian to the Hamiltonian formulation of classical mechanics.

The construction of the dual curve is the geometrical underpinning for the Legendre transformation in the context of Hamiltonian mechanics.

In large deviations theory, the rate function is defined as the Legendre transformation of the logarithm of the moment generating function of a random variable.

Since the Gibbs free energy is the Legendre transformation of the internal energy, the derivatives can be replaced by its definitions transforming the above equation into:

The fLT should not be confused with the Legendre transform or Legendre transformation used in thermodynamics and quantum physics.