The Hamiltonian of a system is defined as the Legendre transform of the Lagrangian and I was wondering if there is a way to construct the Hamiltonian formalism without ever using a Lagrangian. This question is motivated by section 8.5 of Goldstein classical mechanics. There Goldstein derives Hamilton's equations of motion from Hamilton's principle. He states that if we require both $\delta q_i = 0$ and $\delta p_i = 0$ at the endpoints, then one can consider actions of the form: $$ S = \int_{t_1}^{t_2} (\dot{q}_ip_i - H(q,p,t) + \frac{dF(q,p,t)}{dt})dt\tag{8.65} $$ without affecting the validity of Hamilton's principle. I imagine that this is simply because under those requierements: $$ \delta\int_{t_1}^{t_2}\frac{dF(q,p,t)}{dt} = 0\tag{1} $$ Then, below eq. (8.71) he says that the conditions $$\delta q_i = 0\qquad\text{and}\qquad\delta p_i = 0\tag{2} $$ at the endpoints provide an independent and general way of setting up Hamilton’s equations of motion without a prior Lagrangian formulation. That is what I do not understand. How is the Hamiltonian defined under those assumptions?
2 Answers
Hamiltonian and (variational) Lagrangian formulations are intimately related, cf. this Phys.SE post, so even if one starts with the Hamiltonian formulation, there is implicitly also a (variational) Lagrangian formulation.
By imposing $4n$ boundary conditions (2), Goldstein is overconstraining the Hamiltonian system, cf. this Phys.SE post.
I want to comment on how you phrased the question.
Path-of-construction and independent-versus-dependent are distinct properties.
While it is possible to construct the Hamiltonian formalism without using the Lagrangian formalism as intermediate, that does not alter the fact that the Hamiltonian formalism and the Lagrangian formalism are related by Legendre transform.
The are-related-by-Legendre-transform property is valid anyway, independent of which path you traverse to get to your destination.
I prefer to think of the relations in this case as the relations between the corners of a triangle.
There is a mathematical relation between the Lagrangian formulation of mechanics and Hamilton's stationary action, allowing construction.
There is a mathematical relation between Hamiltonian formulation and Lagrangian formulation, allowing bi-directional interconversion: Legendre transformation.
It follows logically that there must be a way to construct Hamiltonian formulation from Hamilton's stationary action directly. The method of construction may be convoluted, and in that sense unpractical, but logic tells us that it must exist.
I vaguely remember seeing that very construction somewhere: Hamiltonian formulation from stationary action directly, but my memory is blank on where that was, and how it was accomplished. But I argue that doing that actual construction doesn't bring additional value; what it will tell you is something that you already know.
