What is exactly a Hamiltonian or Lagrangian?

Question

I wasn't able to find anywhere a definition of a Lagrangian, is it a (dynamical) system (and what does that actually mean), is it an interpretation of other equations or what? What is the different *vision° or rationale of a Hamiltonian and a Lagrangian? How would you describe them in the most simple and profound way?

Answer 1

Imagine you have a system whose system is described by $N$ parameters $\{ q^\alpha\}_{\alpha = 1}^N$. In mechanics you can think of these numbers as coordinates, but this formalism can be extended to many other areas that do not involve particles moving.

Hamilton's principle states said system is going to move in such a way that the functional

$$ S[q,\dot{q}] = \int_{t_i}^{t_f}{\rm d}t\; L(q, \dot{q}) $$

has an extreme value. $S$ is called the action and $L$ the Lagrangian of the system. You can then think of the Lagrangian as the quantity that allows you to distinguish, among all possible paths that the system can follow connecting the times $t_i$ and $t_f$, which is physical and which are not.

Turns out that for a mechanical problem $L$ has a simple expression $L = T- V$, because doing that way you get back Newton's second law. But again, this is much more general than that.

A Hamiltonian will also contain information about the possible paths that the system can follow, but instead of using $\{ q^\alpha, \dot{q}^\alpha\}_{\alpha = 1}^N$ as the dynamical variables, it uses $\{ q^\alpha, p_\alpha\}_{\alpha = 1}^N$, which are related via a Legendre transformation.

I deliberately used this notation with the indices, because when you think of $q$ as being coordinates of a manifold and $p$ elements of the tangent vector, a lot of interesting results about the interpretation of $H$ and $L$ naturally emerge. For that I recommend you to look at this book.