# What exactly are Hamiltonian Mechanics (and Lagrangian mechanics)

I want to self-study QM, and I've heard from most people that Hamiltonian mechanics is a prereq. So I wikipedia'd it and the entry only confused me more. I don't know any differential equations yet, so maybe that's why.

• But what's the difference between Hamiltonian (& Lagrangian mechanics) and Newtonian mechanics?

• And why is Hamiltonian mechanics used for QM instead of Newtonian?

• Also, what would the prereqs for studying Hamiltonian mechanics be?

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I have the feeling I answered this before: physics.stackexchange.com/q/39677 - regarding the prerequisites for Hamiltonian mechanics: if you know how to solve differential equations (both PDE and ODE), you should be fine. –  Claudius Nov 20 '12 at 21:22
@Claudius Are there no physics prereqs? Can I jump straight into Hamiltonian mechanics from Newtonian if I understand the math behind it, or would you recommend studyinh langrangian first? –  user14445 Nov 20 '12 at 21:26
You should do Lagrangian first, then Hamiltonian. You don’t necessarily need Newtonian for that, but it is nice to derive Newtonian mechanics from Lagrangian/Hamiltonian mechanics, so it might help knowing $F = m a$ (it also helps in recalling the Hamiltonian equations of motion, $F = - \nabla U \equiv \dot p = - \frac{\partial H}{\partial q}$ and $v = \frac{1}{2} \frac{\partial m v^2}{\partial p} = \frac{\partial H}{\partial p}$). –  Claudius Nov 20 '12 at 21:31
–  Qmechanic Nov 20 '12 at 21:40
Watch first Leonard Susskind courses in youtube. About ten lessons each, one called Classical Mechanics, and then Quantum Mechanics. They are an excellent introduction with extremely simplified, yet serious maths –  Eduardo Guerras Valera Nov 20 '12 at 21:44

I'd say there were almost no prerequisites for learning Langrangian and Hamiltonian mechanics.

First thing to say is that there's almost no difference between them. They're both part of the same overarching framework. Basically it's a convenient way to write down general laws of physics. There's nothing too difficult or scary about it, and it's a lot more elegant than Newtonian theory.

If you have a rough grasp of basic physics, I don't think you need to formally learn Newtonian theory first. I had to as an undergraduate and it was a horrible mess. I've never needed to do anything using purely Newtonian theory since.

You might need to know how to solve differential equations, both ordinary and partial, but it's possible to pick this up as you go along. There's almost no linear algebra needed, so don't worry about that.

If you're looking for a book, the best one is Landau and Lifschitz, Volume I. Their exposition is very clear and concise, ideal in a textbook! Good luck!

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Definite vote for Landau and Lifschitz! –  Dylan Sabulsky Nov 20 '12 at 22:40
But one definitely needs some linear algebra once one has more than a few degrees of freedom. –  Arnold Neumaier Nov 21 '12 at 10:05

In university, here is the way the material was presented to me:

1. Newtonian Mechanics
2. Learn solutions to ODE and PDE
3. Lagrangian Mechanics
4. Hamiltonian Mechanics

This was over two course, back to back. Hamiltonian and Lagrangian Mechanics provide a formalism for looking at problems using a generalized coordinate system with generalized momenta. Hamiltonians and Lagrangians are written in terms of energy, a departure somewhat from Newtonian mechanics, if I recall properly.

Hamiltonian Mechanics is suitable for quantum mechanics in that one can describe a system's energy in terms of generalized position and momentum. Newtonian mechanics is for macro scale systems, like throwing a baseball. Quantum mechanics is on a much smaller scale. It is the only way I have been taught QM and it is the only way in which I've seen it taught.

Prerequisites for Hamiltonian mechanics would be solving ODEs and PDEs, familiarity with matrix operations and some linear algebra. This would do well for Hamiltonian mechanics through beginners Quantum Mechanics. Books for starting; I would recommend Boas (Mathematical Methods of the Physical Sciences) and Arfkan (Mathematical Methods for Physicists, Sixth Edition; another math methods book). For classical mechanics, Taylor Classical Mechanics. For intro QM, Giffiths Introduction to Quantum Mechanics (2nd Edition).

Best of luck!

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Please, anyone correct me if I'm wrong or misleading! –  Dylan Sabulsky Nov 20 '12 at 21:42
What topics in Linear Algebra are necessary for Hamiltonian Mechanics? I'm currently self-studying LA, so I'd like to know what I actually need. –  user14445 Nov 20 '12 at 21:48
You will need LA for QM, but I was introduced to LA before I took CM and it helped. I think matrix operations and such are important. Learning about vector spaces is key. Perhaps I did not say this right, because I think it is worth studying specifically for QM. If you go deep into CM you will use LA, but you can get through Hamiltonian and Lagrangian Mech without it. –  Dylan Sabulsky Nov 20 '12 at 22:39

I always felt like I learned Hamiltonian and Lagrangian mechanics without meaning to. The moment you can comprehend the implications of a differential equation (and by that i dont mean just being able to solve it) you kind of grasp Hamiltonian mechanics if you have a solid background in physics. As for that background you definitely need to know Newtonian mechanics and calculus (you should be able to solve ordinary and partial differentials) obviously. You can start to self study QM by the way. I learned quite a bit of QM way before I even heard of Hamiltonian mechanics. However it is always better to know HM and LM before QM to have the intuition and the math behind most of the basic concepts of QM.

Hamiltonian and Lagrangian mechanics are generally regarded as very similar to one another if not synonymous. They are both reformulations of classical mechanics and if I remember correctly Hamiltonian mechanics is derived from Lagrangian mechanics but it could be the other way around.

As learning material I would recommend Classical Mechanics: Hamiltonian and Lagrangian Formalism by A. Deriglazov. It isn't a very good source if you do not know much physics and differential equations however if you do it's a pretty good textbook.

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aren't they all equivalent? I think there is a transformation between the Hamiltonian and Lagrangian formulations. –  Brady Trainor Oct 26 '13 at 21:56

## protected by Qmechanic♦Dec 1 '13 at 13:59

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