Lagrange vs Hamilton: What's the Difference?

[Sami Ullah]

if lagrangian mechanics is available the why we needed hamiltonian mechanics. where they differ from each other?

 

[Joey21]

If natural sciences only studied what was neccesary I believe two things would happen:

- Things would get boring,
- Human knowledge would lose a hell of a load of insight.

I reckon if you're going to ask that question, there is one that should go before it: why do we need Lagrangian mechanics when Newtonian mechanics is available? Lagrangian mechanics, for all the power it has, is really limited to a handful of situations: non-holonomic constraints blow Lagrangian mechanics put of the water ¹. Newtonian vector mechanics, however has not problem with them. So why do we use Lagrangian mechanics? It turns out that the equations of motion both methods supply are identical, but arriving at those equations, while relativly easy in Lagrangian mechanics, can be quite a pain in the backside if one uses vector mechanics ( forces, reactions, etc). And, apart from the practical issues, which I believe are what would interest an engineer when he or she has to finish a project, from a theoretical point of view Lagrangian mechanics offers a lot of insight into the structure of mechanics as a theory, gives a neat explanation for conservation laws (based of symmetries in the Lagrangian), and builds up concepts which are used all throughout physics.

Back to your question, why use hamiltonian mechanics if Lagrangian mechanics is available? Like in the Lagrangian vs Newtonian case, Hamiltonian mechanics has a characteristic that from a practical point of view : while Lagrangian Mechanics spits out a second order ordinary differential equations (ODE), Hamiltonian mechanics gives us two first order ODE's, which in a lot of cases are easier to solve, and offers new ways to calculate conserved quantities of motion. Also, Hamiltonian mechanics presents new theoretical concepts, such as Poisson Brackets, which is used in any branches of physics to study the time evolution of a system, canonical transforations, phase space, Liouville's theorem, etc. Even the Hamiltonian, which a lot of the time can be thought of as the mechanical energy of the system ( if certain conditions are met) becomes an important concept in Modern Physics ( I believe, I still have to trust my prof on that one).

During my still short time in college I have studied Newtonian Mechanics, Lagrangian Mechanics and Hamltonian Mechanics. We learned about Lagrangian Mechanics from the point of view of virtual work and D'Alembert's principle, and from the point of view of the principle of least action. In the first, you use Newton's laws as a starting point. In the later, you can recover Newton's laws very quickly. Hamiltonian mechanics can be formulated by using a partial legendre transform on the Lagrangian ( changing the generalized velocities for the canconical momenta) or from a modified action principle. And there are more formulations, such as the Hamilton-Jacobi equation.

What I am trying to say with this overgrown response is that looking at things from different points of view often reveals information which was hidden behind the curtains. And the fact is that if we want to understand the way the universe works, we have to look behind all of them. Especially when it comes to the basics, like classical mechanics. Because really, if you come to think about it, a lot of the results of physics are concepts that you can later plug into equations of motion from mechanics to learn how things move and work, so the better we understand mechanics (classical, quantum, statistical...) the better the basis we will have to study branches like EM, optics, thermodynamics, solid state, etc.





¹http://physics.clarku.edu/courses/201/sreading/AJP73_March2005_265-272.pdf