Most physicists teach partial differential equations poorly, especially when it comes to topics like thermodynamics and analytical mechanics. The way the Legendre transform is traditionally taught in textbooks lacks clarity and elegance. The typical explanation is often muddled, resulting in confusion regarding the connection between Lagrangian and Hamiltonian mechanics. However, a clear and concise definition of the Legendre transform is crucial for understanding the essence of this transformation, which serves as the bridge between key formulations in classical and quantum physics. The blog post suggests a more straightforward definition to grasp the concept, emphasizing the importance of understanding the fundamentals of physics.
https://blog.jessriedel.com/2017/06/28/legendre-transform/