The discussion centers on the relationship between Hamiltonian and Lagrangian formulations in physical theories, specifically addressing the application of the Legendre transform. It is established that the Legendre transform is essential for deriving the Hamiltonian from the Lagrangian, but it requires the Lagrangian to be a convex function of the time derivative of generalized coordinates. A specific example illustrating this relationship is requested, highlighting the necessity of the Legendre transform's existence for successful conversion between the two formulations.
PREREQUISITESPhysicists, students of theoretical mechanics, and anyone interested in the mathematical foundations of classical mechanics will benefit from this discussion.
Ah, sorry. Legendre transformation. That was what I meant. But doesn't it always exist?Orodruin said:Can you be more specific and give a specific example?
In general, you can get to the Hamiltonian formulation of the same theory by applying the Legendre transform (which is what I assume that you mean) to your Lagrangian. In order for this to work, the Legendre transform needs to exist, which in turn requires the Lagrangian to be a convex function of the time derivative of your generalised coordinates.
No. See the wikipedia page.Narasoma said:Ah, sorry. Legendre transformation. That was what I meant. But doesn't it always exist?