# Book for Hamiltonian and Lagrangian mechanics

Avichal
I am learning Hamiltonian and Lagrangian mechanics and looking for a book that starts with Newtonian mechanics and then onto Lagrangian & Hamiltonian mechanics.
It should have some historical context explaining the need to change the approaches for solving equation of motions. Also it should explain how all approaches are equivalent to each other intuitively.

Also since it requires lot of calculus, recommend a book that has the required math for it too.

Thank you!

## Answers and Replies

tia89
Well I am not aware of books treating mechanics as you want... they either are books on Newtonian mechanics (usual way) or on Lagrangian and Hamiltonian mechanics... I don't think anyone does both on the same book. And I don't understand why you want such thing, for Newtonian mechanics there are a lot of good books. Study them and then go to the others.

As for Lagrangian and Hamiltonian mechanics, I can recommend the first book of the Landau-Lifshitz series, even if it is quite difficult, better read it when you have already some knowledge of the topic. Also there is Arnold's "Mathematical methods for classical mechanics" and Goldstein's "Classical mechanics". Arnold has also (and mainly) the mathematics you need, as for Goldstein I don't know, probably something yes but don't know how much, never used it.

Gold Member
2021 Award
My favorite for everything concerning classical physics is

A. Sommerfeld, Lectures on Theoretical Physics, 6 Vols.

For mechanics it's, of course, Vol. 1. It's mostly Newtonian mechanics but has very good final chapters on the various integral and differential principles of mechanics.

The book by Arnold is very mathematical. I doubt that you can learn the physics out of it, although it's a master piece for the mathematical foundations. I'd recommend to read it only after having a good grasp from physics books. Landau/Lifshitz treats everything from the very beginning with Hamilton's principle of least action. That's a very systematic and modern approach, but it's prehaps a bit tough for a beginner in theoretical physics.

Poley
www.amazon.com/Classical-Mechanics-John-R-Taylor/dp/189138922X[/URL]

The first 5 chapters are a review of mechanics in a Newtonian setting, though probably with more rigour and vector calculus than a typical introductory class. Then, the book moves onto variational calculus, Lagrangian mechanics, and Hamiltonian mechanics, along with several other more advanced topics.

The level is certainly beneath other books suggested, like Arnold or Landau/Lifshitz. Nevertheless, I still think Taylor is a good introduction if you haven't seen all the math before, but keep in mind that it's only the beginning.

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Avichal
Thank you for replies.

I'm particularly looking for a book that connects Newtonian mechanics with principle of least action, Hamiltonian and Lagrangian mechanics. I'm want to know how all three are equivalent.
Anyways I'll check out the books.

I heard that Marion and Thornton is a "standard". How is it?

smodak
I am learning Hamiltonian and Lagrangian mechanics and looking for a book that starts with Newtonian mechanics and then onto Lagrangian & Hamiltonian mechanics.
It should have some historical context explaining the need to change the approaches for solving equation of motions. Also it should explain how all approaches are equivalent to each other intuitively.

Also since it requires lot of calculus, recommend a book that has the required math for it too.

Thank you!

https://www.amazon.com/dp/046502811X/?tag=pfamazon01-20

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Daverz
Many undergraduate mechanics books cover Lagrangian and Hamiltonian mechanics, e.g.

Symon, Mechanics
Fowles, Analytical Mechanics

You'll just need vector calc for these.

For a more historical development, the book by Lanczos might be what you want.

tia89
I'm particularly looking for a book that connects Newtonian mechanics with principle of least action, Hamiltonian and Lagrangian mechanics. I'm want to know how all three are equivalent.

Well, for the connection between least action principle and Lagrangian mechanics it is easy, you don't even need a book... just do the math (it is variation calculus though) and from least principle you recover Lagrange equations... and you are done. As well as for the connection between Lagrangian and Hamiltonian mechanics, there is a Legendre transformation in the middle, again math.

These things are found in any good book on Lagrangian and Hamiltonian mechanics. What I am perplexed about is that I am not aware of any book drawing a connection between these things and Newtonian mechanics done as it is usually done...

Avichal
Well, for the connection between least action principle and Lagrangian mechanics it is easy, you don't even need a book... just do the math (it is variation calculus though) and from least principle you recover Lagrange equations... and you are done. As well as for the connection between Lagrangian and Hamiltonian mechanics, there is a Legendre transformation in the middle, again math.

These things are found in any good book on Lagrangian and Hamiltonian mechanics. What I am perplexed about is that I am not aware of any book drawing a connection between these things and Newtonian mechanics done as it is usually done...

Yes, the math is complicated. I'll look for a book on variation calculus.
So there is no book that tells you how principle of least action and newton's equations tell the same thing from a different perspective. The connection is what I want to know.

tia89
So there is no book that tells you how principle of least action and newton's equations tell the same thing from a different perspective. The connection is what I want to know.

I am notaware of any book doing so, I'm sorry... anyway I think the answer should go through th Lagrange equations... I mean the Newton equations are given by the Lagrange equations (not sure in general of the opposite though) and Lagrange equations are derived from least action... therefore for transitivity Newton equations come from least action... becareful though because least action and the lagrangian formalism is quite more general than Newton itself... proof of this in quantum field theory you still extremize action to get eoms

Gold Member
Try "The Variational Principles of Mechanics" by Lanczos - not normally used as a text book, it is great for self study. I relied upon it while preparing a lengthy seminar series for some colleagues; you can find my free "Notes on Analytical Mechanics" via my profile.

zurdo
Perhaps it's late to tell... I think you will find "Classical Mechanics" from John R. Taylor useful, alhtough it's quite expensive. Chapters 1 - 5 are Newtonian mechanics, chapter 6 is a "link" betwen Lagrange and Newton, 7 is Lagrange mechanics and 13 deals with Hamiltonian mechanics.

Regards.

Bene Chen
I am learning Hamiltonian and Lagrangian mechanics and looking for a book that starts with Newtonian mechanics and then onto Lagrangian & Hamiltonian mechanics.
It should have some historical context explaining the need to change the approaches for solving equation of motions. Also it should explain how all approaches are equivalent to each other intuitively.

Also since it requires lot of calculus, recommend a book that has the required math for it too.

Thank you!

Take a look at The Variational Principles of Mechanics by Cornelius Lanczos. It has some comparisons between these two approaches, together with some historical account.