The discussion revolves around the relationship between Hamiltonian and Lagrangian formulations of physical theories, specifically addressing the process of obtaining a Hamiltonian from a Lagrangian through the Legendre transform. Participants explore the conditions under which this transformation is valid and the implications of those conditions.
Participants do not reach a consensus on whether the Legendre transform always exists, indicating a disagreement on this point. There is also uncertainty regarding the implications of the Lagrangian's properties for the transformation process.
The discussion highlights the dependency of the Legendre transform's existence on the convexity of the Lagrangian, which remains an unresolved aspect of the conversation.
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?