Book for Hamiltonian and Lagrangian mechanics

In summary, there is no book that tells you explicitly how all three approaches are equivalent. However, any good book on Lagrangian and Hamiltonian mechanics will have the necessary math for it.
  • #1
Avichal
295
0
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!
 
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  • #2
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.
 
  • #3
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.
 
  • #4
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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  • #5
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?
 
  • #6
Avichal said:
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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  • #7
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.
 
  • #8
Avichal said:
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...
 
  • #9
tia89 said:
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.
 
  • #10
Avichal said:
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
 
  • #11
Try "The Variational Principles of Mechanics" by Lanczos - not normally used as a textbook, 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.
 
  • #12
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.
 
  • #13
Avichal said:
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.
 

1. What is the difference between Hamiltonian and Lagrangian mechanics?

Hamiltonian and Lagrangian mechanics are two different approaches to understanding the dynamics of physical systems. Lagrangian mechanics is based on a principle of least action, while Hamiltonian mechanics is based on the conservation of energy. Both approaches use different mathematical formalisms to describe the motion of particles and systems.

2. What are the main applications of Hamiltonian and Lagrangian mechanics?

Hamiltonian and Lagrangian mechanics are widely used in physics and engineering to study the behavior of physical systems. They are particularly useful for analyzing systems with multiple degrees of freedom, such as celestial mechanics, fluid dynamics, and quantum mechanics. They also have practical applications in fields such as robotics, control systems, and classical mechanics.

3. How do Hamiltonian and Lagrangian mechanics relate to Newtonian mechanics?

Newtonian mechanics, also known as classical mechanics, is based on Newton's laws of motion and is the most commonly used framework for understanding the dynamics of physical systems. Hamiltonian and Lagrangian mechanics are extensions of Newtonian mechanics, providing more general and elegant methods for solving complex problems.

4. What are the advantages of using Hamiltonian and Lagrangian mechanics over other approaches?

One advantage of Hamiltonian and Lagrangian mechanics is that they are based on fundamental principles that can be applied to a wide range of physical systems. They also provide a more elegant and efficient way of solving problems compared to other methods, such as using differential equations. Additionally, these approaches can help reveal deeper insights and connections between seemingly unrelated systems.

5. How can I learn more about Hamiltonian and Lagrangian mechanics?

There are many resources available for learning about Hamiltonian and Lagrangian mechanics, including textbooks, online courses, and academic lectures. It is recommended to have a solid understanding of calculus, differential equations, and classical mechanics before diving into these topics. Additionally, hands-on practice and problem-solving are crucial for gaining a deeper understanding of these concepts.

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