Hamiltonian of a Physical Theory: Lagrangian vs Transformation

[Narasoma]

What does it means for a physical theory to have hamiltonian, if it is formulated in lagrangian form? Why doesn't someone just apply the lagrangian transformation to the theory, and therefore its hamiltonian is automatically gotten?

 

[Orodruin]

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.

 

[Narasoma]

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.

Ah, sorry. Legendre transformation. That was what I meant. But doesn't it always exist?

 

[Orodruin]

Ah, sorry. Legendre transformation. That was what I meant. But doesn't it always exist?

No. See the wikipedia page.