Principle of least action ...some confusion

[amjad-sh]

Principle of least action states that the particle choose the path which optimizes its action.Where the action is defined by S=t1∫t2(L)dt and L is the Lagrangian of the system.This leads to δS=0 and it is a condition to optimize S.

I will summarize what confuses me about this principle in a series of questions:

1-Is the principle of least action must be taken for granted and there is no proof for it?If there is a proof where I can find it and what are the mathematical prerequisites needed to grasp it?

2-Why the action is defined like this? I mean why the integrand is the Lagrangian? why it is not another function?

3-Finally why δS=0 is a condition to optimize S? and why we call it variational of S ?why we don't just write ds=0?

 

[andrewkirk]

1. The principle can be derived from the equations of motion. Any good physics text that covers Lagrangian mechanics should contain a proof.

2. The integrand is the Lagrangian because that makes the theorem work.

Note however, that the Lagrangian is not in all cases the Lagrangian to which one is first introduced via mechanics, which is T - V (kinetic minus potential energy). Where electromagnetic forces are involved, it is different, and the formula used for the Lagrangian is chosen specifically so as to make the principle of least action work. So a more robust way to think of the Lagrangian is as the function of the system variables that makes the principle of least action work. One then has to reverse the proofs referred to above, and instead prove that the Lagrangian for pure mechanical systems is T - V, and likewise derive the form of the Lagrangian for electromagnetic systems.

3. The term 'variational' comes from the calculus of variations, which is a subject in itself. Lagrangian mechanics is an application of the calculus of variations. We write δS=0 instead of dS=0 because the latter implies a change in S in response to a change in a numeric variable such as time or distance. What we are changing here is not a numeric variable but a function, which is the path of the object. Calculus of variations is a kind of extension of calculus in which functions take the roles of (some) numbers and functionals take the role of functions.

If you want to get a really thorough understanding of the topic, the best way (IMHO) is to learn a bit about calculus of variations, which places it all in a very sound theoretical framework.

 

[amjad-sh]

1. The principle can be derived from the equations of motion.

But I thought that the equations of motion are derived from the principle!

3. The term 'variational' comes from the calculus of variations, which is a subject in itself. Lagrangian mechanics is an application of the calculus of variations. We write δS=0 instead of dS=0 because the latter implies a change in S in response to a change in a numeric variable such as time or distance. What we are changing here is not a numeric variable but a function, which is the path of the object. Calculus of variations is a kind of extension of calculus in which functions take the roles of (some) numbers and functionals take the role of functions.

I think I get it here!

 

[andrewkirk]

But I thought that the equations of motion are derived from the principle!

I meant Newton's equations of motion, not Lagrange's. Sorry, I should have made that explicit.

 

[amjad-sh]

I meant Newton's equations of motion, not Lagrange's. Sorry, I should have made that explicit.

Do you mean that there is no mathematical proof of Newton 2nd law and the principle of least action is concluded from it?
because I think I heard one day that Newtons law are are stated due to experimental observations only but then Lagrange and Euler proved it using calculus of variations.
Is this true?

 

[andrewkirk]

Well Newton's laws of motion can be proven from Lagrange's and, in the particular case of mechanical-only systems (no electro-magnetism), Lagrange's equations can be proven from Newton's. So they are to some extent equivalent. But we have to start by accepting one or the other as a brute fact.

So the thing you heard probably either said or meant to say that Lagrange and Euler proved that Newton's laws could be derived from the Lagrange-Euler equation which, IIRC, is equivalent to the principle of least action.

More generally, no physical law can be proved to be true. All we can do is observe that experimental observations have so far been consistent with a law, so we accept it until something better comes along. That was Karl Popper's great insight.

 

[vanhees71]

The action principle seems very obscure from a classical point of view. As stated in the postings before, it's just working as a mathematical tool to get the known equations of motion in another form, and this is very important, because it opens a whole universe of deep mathematical concepts, most importantly symmetry principles (Noether's theorem) and the "algebraisation" of these symmetry/geometry principles in form of the Poisson brackets which form a Lie algebra on the phase-space functions and the corresponding representation of the symmetries.

But what's the physics behind the variational principle is a mystery, and it triggered a lot of "philosophical" action at the time, which sometimes reached esoterics quality (which is a strange tendency of philosophic attempts to find a deeper meaning of physical laws).

In my opinion this riddle was only solved quite recently, when Feynman discovered his path integral formulation of quantum mechanics (his PhD thesis). The classical path is precisely the one fulfilling the least-action principle, because it makes the action stationary. If the values of the action for the typical situation under consideration is very large compared to $\hbar$ (which indicates that you have a situation, where the classical laws of motion are a good approximation), you can evaluate tha path integral that describes the quantum dynamics using the stationary-phase method, and this precisely means the quantum fluctuations around tha path that defines a stationary point of the action functional are small. That explains why the classical path of the particle in phase space is precisely the one that makes the action stationary. All other paths lead to rapidly oscillating phase factors that add to 0 in the path integral.

 

[andresB]

Do you mean that there is no mathematical proof of Newton 2nd law and the principle of least action is concluded from it?
because I think I heard one day that Newtons law are are stated due to experimental observations only but then Lagrange and Euler proved it using calculus of variations.
Is this true?

Mathematical proofs have to start from somewhere, in this case all physics and the mathematics behind them are based to match experimental observations.