Hello, I am trying to "integrate into my understanding" the difference between Hamiltonian and Lagrangian mechanics. In a nutshell: If Lagrange did all the work and formulated L = T - V, they why is Hamilton's name attached to the minimization principle? YES; I KNOW about Hamilton's Second Principle with phase space and momenta, but I ask you to ignore that in this. For example, I understand this: "Two dominant branches of analytical mechanics are Lagrangian mechanics (using generalized coordinates and corresponding generalized velocities in configuration space) and Hamiltonian mechanics (using coordinates and corresponding momenta in phase space). Both formulations are equivalent by a Lengendre transformation on the generalized coordinates, velocities and momenta, therefore both contain the same information for describing the dynamics of a system." HOWEVER! I am also learning about The Calculus of Variations. And I am learning it is called Hamiltonian Dynamics (so I assume Hamilton's First principle involves this, BEFORE he gets to phase space with his other principle), but uses a Lagranian And there you have my confusion: a mish-mosh soup of THREE terms and I am unable to state, concisely, the difference in these views with regard to using variational methods to extract the equations of motion for simple dynamical systems. Can someone please distinguish these three words and the contributions of Hamilton vs Lagrange, simply, concisely (but under the aegis of ONLY variational methods, and not involving momentum/phase space)?