Lagrangian and hamiltonian formalism

In summary: I never thought about it that way. That would certainly be in the spirit of things, as far as QM goes. I never thought of it that way, but nice. So you are saying that one starts with a classical system with a parameter, quantize it, and that parameter is still there? That would be cool, but I am not sure that is what you meant.In summary, the Lagrangian and Hamiltonian formalisms are two different mathematical approaches to classical mechanics. The Lagrangian approach is useful for working with generalized coordinates and is coordinate system invariant, making it useful for dealing with tensorial quantities. The Hamiltonian approach is more natural and fundamental, and follows from the principles
  • #1
captain
164
0
what is the difference between these two formalism and when are each used? also is it true the lagrangian formalism is used more in qft, if so i am curious to know why?
 
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  • #2
Well, have you studied them yet? It's good to know at what level the answer is going to be...
 
  • #3
you need to be more respectful

genneth said:
Well, have you studied them yet? It's good to know at what level the answer is going to be...

that is rude. He only asked a question. what is it with this board.?
is this board some kind of pissing contest?
 
  • #4
melrose said:
that is rude. He only asked a question. what is it with this board.?
is this board some kind of pissing contest?


eeh what is the matter with you? He only asked at what level he study so he can give the answer in such a way that it becomes clear. You do not explain nuclear physics to a high school kid as you would do to a course mate at final year at university..
 
  • #5
melrose said:
that is rude. He only asked a question. what is it with this board.?
is this board some kind of pissing contest?

It was certainly not intended to be rude. The depth of classical mechanics gets truly dizzying at times, so it's good to know where to start. If the answer casually throws technical terms about which the questioner doesn't know, then it's hardly answered the question.
 
  • #6
captain said:
lagrangian and hamiltonian formalism

what is the difference between these two formalism and when are each used? also is it true the lagrangian formalism is used more in qft, if so i am curious to know why?

I consider H.Goldstein “Classical Mechanics” the best introduction. There is 3-rd edition, but I personally prefer 2-nd, Addison-Wesley, 1980. However, as genneth pointed out, from here to QFT is long way to go.

Regards, Dany.
 
  • #7
the langrangian formulation is particularly useful because it is true in generalized coordinates. remember, Newton's laws are formulated in cartesian coordinates. working problems in Newtonian mechanics in other coordinate systems may be very difficult at times, and may also give rise to fictionous forces that are not necessary. since the lagrangian formulation is coordinate system invariant, it lends itself well to tensorial quantities. the lagrangian formalism can also be very powerful since you don't need to introduce forces at all (unless you want to) if you rely upon the principle of least action.

the hamiltonian formulation is useful in formulating advanced principles of dynamics, such as the Liouville theorem, etc.
 
  • #8
Lagrangian mechanics works with generalized position ("configuration") and velocity variables, leading up to a system of 2nd order differential equations.
Hamiltonian mechanics works with generalized position and momentum variables ("phase space"), leading up to a system of coupled 1st order differential equations.

In some relatively simple cases, they are equivalent by the Legendre transformations.
 
  • #9
captain said:
what is the difference between these two formalism and when are each used? also is it true the lagrangian formalism is used more in qft, if so i am curious to know why?

In my opinion, the Hamiltonian formalism is more natural and fundamental. It follows immediately from the principle of relativity, the Poincare group structure, and postulates of quantum mechanics.

The Lagrangian (field) formalism was found useful in QFT. I am not sure what is the physical significance of fields, actions, Lagrangians, etc. I suspect they are just neat mathematical tricks, which allow one to construct relativistically invariant Hamiltonians in QFT.

I know that my point of view is controversial. However, you can check S. Weinberg
"The quantum theory of fields", vol. 1 and see that I am not so far off. His entire logic is based on the premise of the supremacy of Hamiltonians.

Eugene.
 
  • #10
The relationship to quantum theory is also fun. Hamiltonian approach is what we usually take, where we "quantize" the position and momenta by lifting them to be operators, and finding a decent representation of them over some Hilbert space. The Lagrangian approach turns into a sum-over-paths approach, which is nice and intuitive: the probability amplitude is the sum of the exponential of [tex]i[/tex] times the classical action.

You've probably heard of the Lagrangian approach used in QFT, where we calculate the motion and interactions of particles by summing up all the possibilities, using Feynman diagrams.

An advantage of the Lagrangian approach is that it is often transparent what symmetries are present, so it becomes easier to find theories which follow special relativity, for example. In the Hamiltonian theory (as you would find in textbooks, at least), time tends to play a dominating role, which is in violation of the spirit of special relativity.
 
  • #11
genneth said:
In the Hamiltonian theory (as you would find in textbooks, at least), time tends to play a dominating role, which is in violation of the spirit of special relativity.

This assertion is often repeated in textbooks. However, I think it is based on a misunderstanding of the true spirit of relativity. The most fundamental requirement imposed by relativity on quantum theories is that "the Hilbert space of the system must carry an unitary representation of the Poincare group". The Hamiltonian approach is perfectly consistent with this requirement. Actually, the Hamiltonian is just a generator of time translations in this representation. So, there is no inconsistency between the Hamiltonian theory and (correctly understood) relativity.

It is true that in quantum mechanics time t is a classical parameter and position x is described by a quantum operator. So, there is formal disagreement with Einstein's special relativity which declares the equivalence of t and x. However, this declaration of x-t equivalence is, actually, an assumption rather than a rigorously proven fact.

This is my personal view, which is not shared, as far as I can tell, by anybody else. So, I'll stop here. If you are interested, we can discuss these points further.

Eugene.
 
  • #12
There is no violation of special relativity perse with the Hamiltonian framework, but yes it does in a sense violate 'the spirit'. You lose manifest lorentz invariance, even though its still there deep down (embedded into the core of the theory). That shouldn't bother anyone really.

Quantization with parameter time isn't really a problem until you start playing around with gravity or certain very specific examples. There things get much more subtle (the dynamics are not in the actual hamiltonian, which is identically zero, but in the so called Hamiltonian constraint). Technical problems quickly arise. However, even in those situations, you are ok if you proceed with caution, but you will have to set up a regularization at one point.
 
  • #14
i would like to thank all contributors to this discussion
 

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

Lagrangian and Hamiltonian formalism are two different mathematical approaches used to describe the dynamics of a physical system. In Lagrangian formalism, the system's behavior is described by the Lagrangian function, which is a function of the system's position and velocity. In Hamiltonian formalism, the system is described by the Hamiltonian function, which is a function of the system's position and momentum. The main difference between the two is that Lagrangian formalism is based on the system's kinetic and potential energies, while Hamiltonian formalism is based on the system's total energy.

2. What is the principle of least action in Lagrangian formalism?

The principle of least action states that the actual path taken by a system between two points in time is the path that minimizes the action, which is the integral of the Lagrangian function over time. In other words, the system takes the path that requires the least amount of energy to move from its initial state to its final state. This principle is a fundamental concept in Lagrangian formalism and is used to derive the equations of motion for a system.

3. How is the Hamiltonian function related to the Lagrangian function?

The Hamiltonian function is related to the Lagrangian function through a mathematical transformation known as the Legendre transformation. The Hamiltonian function is equal to the Lagrangian function minus the product of the system's velocity and its conjugate momentum. This transformation allows us to describe the dynamics of a system in terms of its position and momentum, rather than its position and velocity.

4. Can Lagrangian and Hamiltonian formalism be used for all physical systems?

Yes, Lagrangian and Hamiltonian formalism can be used to describe the dynamics of all physical systems. However, the equations of motion derived from these formalisms may be more complex for certain systems, such as systems with non-conservative forces or systems with constraints. In these cases, additional techniques, such as the use of Lagrange multipliers, may be necessary to accurately describe the system's behavior.

5. What are some advantages of using Lagrangian and Hamiltonian formalism?

One major advantage of using Lagrangian and Hamiltonian formalism is that they provide a more elegant and concise mathematical description of a system's behavior compared to other methods, such as Newton's laws of motion. They also have the ability to handle complex systems with multiple degrees of freedom. Additionally, these formalisms have important applications in fields such as mechanics, electromagnetism, and quantum mechanics.

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