Principle of Least Action via Finite-Difference Method

[bolbteppa]

I have to be honest, the principle of least action seems to me more of a religious claim one takes on complete faith, though of course I'm hoping this is just because I don't understand it. I tried to explain this to a friend suffering through a mechanics class & was literally pushed to say 'one must just assume this magical religious-like principal, alright...' & I'm not happy to do that again!

I am reading Gelfand's Calculus of Variations & mathematically everything makes sense to me, it makes perfect sense to me to set up the mathematics of extremization of functionals & show that in extremizing a certain functional you can end up with Newton's laws, i.e. you could extremize some arbitrary functional L & in examining the form of Newton's laws you see that one should define L as T−V & this way of looking at things requires no magic, to me it seems as though you've just found a clever way of doing mathematics that results in Newton's laws.

However in books like Landau one must assume this magical principle of least action using the kind of thinking akin to Maupertuis, & I remember that every time I read some form of justification of this there's always a crux point where they'd say 'because it works', or worse refer to QFT or Feynman as if any of this foresight was available to Einstein when employing variational principles on [classical] manifolds etc... I'm thinking that there may be a way to explain the principle of least action if you think of extremizing functionals along the lines Euler first did & use the method of finite differences (as is done in the chapter on the Variational Derivative in Gelfand), i.e. because you're thinking of a functional as a function of n variables you can somehow incorporate the T−V naturally into what you're extremizing, but I don't really know... I'm really hoping someone in here can give me a definitive answer, to kind of go through what the thought process is in this principle once and for all!

As far as I know, one can either assume this as a principle as in Landau & use all this beautiful theory of homogeneity etc... to get some results, or else once can assume Newton's laws & then use the principle of virtual work to derive the Euler-Lagrange equations, or start from Hamilton's equations & end up with Newton's laws, or one can assume Newton's laws & set up the calculus of variations & show how extremizing one kind of functional leads to Newton's laws - but I'm not clear on exactly what's going on & would really really love some help with this.

Just to be clear I've read a ton of links online, on this site & many others & haven't found an answer to any of my questions unfortunately, thus any help is really really welcome!

 

[lightarrow]

First of all, Landau - Mechanics is very criptic and he doesn't make clear proves of what he finds; I sincerely advise you to start from another book (I started reading that book 30 years ago and I still have a lot of things to understand .)
Second, I am not sure if you are talking of what is usually called the Hamilton's principle (the action, that is the time integral of L is stationary on the real motion q(t)) or if you are talking of Maupertuis' principle which rely on the "reduced action".
Third, I believe you should formulate the question in a more direct way, for example: "why we define L as T - V in the context of...?"
Fourth, the Hamilton's principle is a real mistery, however its utility is probably greater than what you believe, even in mechanics only. Anyway the real Big powerful theorem, which comes from the Lagrangian approach, is Noether's Theorem (you can cansider it as the "first word of the Physics").

 

[bolbteppa]

A quote from Weinstock's Calculus of Variations book:

"Although it is in some places stated that Hamilton's principle may be used to replace Newton's laws of motion as the fundamental starting point for mechanical systems posessing a lagrangian, it should be realized that Newton's laws are implicitly employed in the preceding paragraphs in at least two ways: (i) The definition of mass resides in Newton's third law. (ii) In the tacit assumption that our system of coordinates is fixed relative to an inertial frame of reference, we make use of Newton's first law, by means of which an inertial frame is defined.
P75

I found a similar quote in Hertz' old book, though I can't locate it anywhere at the moment unfortunately. I'm not feeling any better

Second, I am not sure if you are talking of what is usually called the Hamilton's principle (the action, that is the time integral of L is stationary on the real motion q(t)) or if you are talking of Maupertuis' principle which rely on the "reduced action".

I apologize by 'thinking akin to Maupertuis' I meant to invoke the idea that the principle was simply god-given & unquestionable. The question is basically - how would I explain to my friend who is having problems in a Landau-level mechanics class the very beginning of Landau to him without invoking God or magic, whether you begin with Hamilton, Maupertuis, Gauss, Jacobi or Hertz doesn't matter

Third, I believe you should formulate the question in a more direct way, for example: "why we define L as T - V in the context of...?"

Well one need not define L as T - V necessarily, as you can get Newton's laws of motion from different forms of your lagrangian. However if you never knew Newton's laws how in the world would you know the crazy lagrangian in that link described a physical system? The justification seems to me that you end up with Newton's laws, but how would some gung ho mathematician who'd never studied Newtonian mechanics instead jumping into Landau know how to deal with this?

 

[DrewD]

how would I explain to my friend who is having problems in a Landau-level mechanics class the very beginning of Landau to him without invoking God or magic

The same question could be asked about Newton's Laws. One would probably answer that the reason they are used is because they make predictions that fit with experimental results. Hamilton's Principle is a similar starting point. The postulates of special relativity are as well. Of course, the definitions needed in order to start from Hamilton's Principle are greater than those for Newton's Laws, which is probably why Newton came first.

That being said, if you want to give analogies that will make him/her more comfortable, Fermat's principle is similar and is easier to visualize. I'm not sure that this is your issue though.

 

[lightarrow]

A quote from Weinstock's Calculus of Variations book:
"Although it is in some places stated that Hamilton's principle may be used to replace Newton's laws of motion as the fundamental starting point for mechanical systems posessing a lagrangian, it should be realized that Newton's laws are implicitly employed in the preceding paragraphs in at least two ways: (i) The definition of mass resides in Newton's third law. (ii) In the tacit assumption that our system of coordinates is fixed relative to an inertial frame of reference, we make use of Newton's first law, by means of which an inertial frame is defined."

As I wrote, Landau is very criptic and you can't expect mathematical rigor from that book, however, if you read it, you'll see that he tries to define the concepts of inertial frame and of mass without using Newton's laws.

The question is basically - how would I explain to my friend who is having problems in a Landau-level mechanics class the very beginning of Landau to him without invoking God or magic, whether you begin with Hamilton, Maupertuis, Gauss, Jacobi or Hertz doesn't matter

As drewD wrote, mechanics, be in the form of Newton's laws or Hamilton's principle, is based on postulates (and the same with many other branches of physics, if not all of them...) Landau says that the equations of motion can be derived from differential equations involving a function called "Lagrangian", the form of which he will find in the next pages.

Well one need not define L as T - V necessarily, as you can get Newton's laws of motion from different forms of your lagrangian. However if you never knew Newton's laws how in the world would you know the crazy lagrangian in that link described a physical system? The justification seems to me that you end up with Newton's laws, but how would some gung ho mathematician who'd never studied Newtonian mechanics instead jumping into Landau know how to deal with this?

He explains (in a criptic way but he does) all that in the first chapter, you only have to read it (more than one time, probably )

--
lightarrow

 

[WannabeNewton]

I agree with you that the premise of the principle of least action is extremely mysterious in classical mechanics. I have yet to find a satisfactory answer for exactly the question you have posed. At the moment I just accept it because "it works and fits experiments" much like many other things in physics.

 

[atyy]

A differential equation is often the solution of a variational problem, and many of those in classical mechanics are.

When does the variational formulation exist? There's discussion of the "inverse problem" of the calculus of variations in some of the papers at http://www.dic.univ.trieste.it/perspage/tonti/papers.htm , maybe try eg. http://www.dic.univ.trieste.it/perspage/tonti/DEPOSITO/Rassias.pdf . Also an interesting comment in the introduction of http://arxiv.org/abs/1008.3177 .

Although the differential equation and the variational problem have the same physical content, when one wishes to enforce symmetries, it's often easier to do so via the variational formulation. IIRC, it's also easier to enforce constraints.

 

[dimwatt]

The argument for the stationary action principle is subtle, albeit powerful. I'm sure there might be some way to derive it independent of Newton's Laws, however this particular argument does use it, but it's still illuminating, in my opinion at least.
If you begin by defining L = T-V, with V and T as functions of the vectors $\vec{r},\dot{\vec{r}}$, respectively, and t in Cartesian coordinates, it follows easily that $\frac{\partial L}{\partial x} =-\frac{\partial U}{\partial x}=F_x$, and likewise for y and z, and that $\frac{\partial L}{\partial \dot{x}}=\frac{\partial T}{\partial \dot{x}}=m\dot{x}=p_x$ etc.
Recalling that $F_x=\frac{d}{dt}$ it follows that $\frac{\partial L}{\partial x}=\frac{d}{dt}\frac{\partial L}{\partial \dot{x}}$, and likewise for y and z.
These equations take the form of an Euler-Lagrange equation, which implies (this part might require more proof) δ∫L=0.
The crux (and subtle) move here is that if you let any generalized coordinate $q_i$ (e.g. polar or spherical coordinates) be a function of the position vector $\vec{r}$, then, as long as there is a unique value of x,y,z corresponding to each point of the generalized vector $(q_1,q_2,q_3)$, then $L(x,y,z,\dot{x},\dot{y},\dot{z},t)\rightarrow L(q_1,q_2,q_3,\dot{q_1},\dot{q_2},\dot{q_3},t)$.
This doesn't change the action integral ∫L in anyway, and so, by going in reverse now, the Euler-Lagrange equations with respect to the new generalized coordinates take on the exact same form, i.e. $\frac{\partial L}{\partial q_i}=\frac{d}{dt}\frac{\partial L}{\partial \dot{q_i}}$

 

[dimwatt]

Recalling that $F_x=\frac{d}{dt}$

Sorry missed that. That should read $F_x=\frac{d p_x}{dt}$