Intuitive

Concrete

• See: Making sense of the Legendre transform by R. K. P. ZiaEdward, F. RedishSusan, R. McKay
• and Ryder - Quantum Field Theory page 260
• see also https://blog.jessriedel.com/2017/06/28/legendre-transform/
• http://blog.sigfpe.com/2005/10/quantum-mechanics-and-fourier-legendre.html
• https://www.andrew.cmu.edu/course/33-765/pdf/Legendre.pdf
• http://mathforum.org/kb/message.jspa?messageID=3868685 (Legendre transformation is "zero temperature limit" of the Laplace Transformation)
• http://blog.sigfpe.com/2012/01/some-parallels-between-classical-and.html

Abstract

The Fourier transform and the Legendre transform may be interpreted as the same thing, just over different semirings.http://blog.sigfpe.com/2005/10/quantum-mechanics-and-fourier-legendre.html

Why is it interesting?

The Legendre transform shows up whenever we minimize or maximize something subject to constraints. That happens a lot.

https://johncarlosbaez.wordpress.com/2012/01/19/classical-mechanics-versus-thermodynamics-part-1/

The Legendre transformation is a useful mathematical tool that is used in thermodynamics, classical mechanics and quantum field theory.

Maybe the most famous application is that in classical mechanics, quantum mechanics and quantum field theory the Hamiltonian and the Lagrangian are connected by a Legendre transformation.

Moreover, the Legendre transformation is used in thermodynamics to motivate the connection between the internal energy, enthalpy, and Gibbs and Helmholtz free energies.

Both uses can be compactly motivated if the Legendre transform is properly understood. Unfortunately, that transform is often relegated to a footnote in a textbook, or worse is presented as a complicated mathematical procedure. […] In a nutshell, a Legendre transform simply changes the independent variables in a function of two variables by application of the product rule.

https://www.aapt.org/docdirectory/meetingpresentations/SM14/Mungan-Poster.pdf