Types of quantum field thories
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.
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 is a Legendre transform of the Lagrangean 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 , temperature , pressure , volume , magnetization ); 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 , Helmholtz free energy , Gibbs free energy , internal energy , 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.
|Lagrangian correspondence||prequantization||prequantized Lagrangian correspondence|
Discussion of Legendre transformation in the context of Lie algebroids is in:
Paulette Liberman, Lie algebroids and mechanics (ps)
Juan Carlos Marrero et al, A survey of Lagrangian mechanics and control on Lie algebroids and Lie groupoids (pdf)
Juan Carlos Marrero, Nonholonomic mechanics: a Lie algebroid perspective (pdf talk notes)