Action Minimization for Block on Inclined Plane

[WiFO215]

If instead of writing down the Euler-Lagrange equations straight away for some system, say a block on a freely moving inclined plane, I were to first write down the action that was to be minimized,

$$S = \int_{t_{1}}^{t_{2}}L(q,q',t)dt$$

what would I take to be t₁ and t₂ supposing that I needed to solve for the accelerations of the block and plane?

Also, I would require that my q(t₁) and q(t₂) were given so that when I integrate by parts, I get the usual eqns of motion. What would I take for these values? Let us say, that in this specific problem, the block starts off at the top of the plane at t = 0.

 

[CompuChip]

For t₁ you take the starting (or "reference") time, and for t₂ the time at which you want to know these values.

The idea is that you set the variation of S to zero, and from that derive the equations of motion for q(t) and q'(t). Then you solve these with the appropriate boundary conditions, in this case:
q(t₁) = q₀, q'(t₁) = v₀
assuming that you know the position and velocity at t = t₁, and evaluate at t = t₂.

What I mean is: the construct is more of a formal one. You don't actually evaluate the integral. What you do is, you find the configuration (q, q') for which the integral is at an extremum. In a sense, your question is like: "how do I calculate f(x)?" when the physical problem is merely "for which x is f'(x) = 0?"

 

[WiFO215]

No that isn't what my question is. My question arises because of the derivation I've read in most books. Pick up Landau for example. The derivation of the Euler Lagrange equations uses the fact that you know q(t₂). Here, we don't know what it is. So how would one adjust for that fact?

 

[GRDixon]

No that isn't what my question is. My question arises because of the derivation I've read in most books. Pick up Landau for example. The derivation of the Euler Lagrange equations uses the fact that you know q(t₂). Here, we don't know what it is. So how would one adjust for that fact?

I don't know the answer to your question. But "The Feynman Lectures on Physics," V2, Chapter 19 ("The Principle of Least Action") might be worth reading.

 

[WiFO215]

I read it, but in the problems that he considers, he starts by fixing a start and end point. I think one MUST do that in all Lagrangian problems.

 

[WiFO215]

While I was reading one of the standard textbooks written by Hand & Finch, they use the Euler-Lagrange (E-L) equation to solve problems solved using Newtonian methods to show how this new method works on familiar problems. There, they straight away start by writing down the E-L equation to solve the problem, rather than start with the Action. One such example might be a block sliding down an inclined plane which is free to slide. Indeed, the E-L equation does solve this problem far faster than Newtonian methods.

However, if I were to solve the problem from scratch, i.e. start by writing down the Action, what would I have used as start and end times? I suppose I can use any time I wish. What would I have used as start and end positions? I certainly do not know where the particle might be quite some time later as the motion is quite complicated to describe without mechanics. If I do know where it is, I have already solved the problem, and needn't use the Action method at all! So how can one justify using the E-L equation to solve this problem?

The derivation of the E-L equation in textbooks always uses the fact that given a certain interval of time, I know the positions at the starting and ending and hence their variation is zero, and the boundary term that appears in the derivation after integrating the first order change in the Action by parts disappears. However, when they give examples, they give examples like the ones stated above, where I clearly do not know where the block and plane are after some time. I have been given different boundary conditions. I can tell you the initial position, but cannot tell you the final position. I can tell you the initial velocity of the block and plane. Can that information be used to solve the problem? How do I proceed?

 

[opsb]

I was struggling with the same problem, and think I have some sort of a remedy. In deriving the E-L equations, by varying a path and evaluating the stationary point of action, it is true that start and end points (define point: time and place) are set. However, it is equally relevant that these points are arbitrary. The E-L equations have the same form wherever these points are. Let's consider a general problem where we define the potential only (no starting conditions). Any curve which satisfies the E-L equations defines a possible curve for a particle in the potential defined in the problem. That's because we can pick any arbitrary start and end points and know that there is a single curve which passes through the two which minimises the action, for every start and end point combo.

So what happens when we don't necessarily know these points to begin with? In the kind of problem being discussed, we generally know the start point (x=0, t=0; that sort of thing). So we reduce the number of possible paths to all those which pass through this point. We have infinitely fewer possible paths than before, but still infinitely more that one, unfortunately. However, in the kind of problem that we're considering, we're also given the initial momentum. This momentum can only correspond to one possible path through the original point (this is a little subtle - Landau explains it lucidly - essentially the E-L equations imply that the state of a particle can be defined by the position and the momentum alone (since the Lagrangian depends only on x and dx/dt). If we know both, the particle can have only one future motion (path)). So we have only one possible curve that the particle will follow. We define an end time (the question says at time 't' where is the particle), and we can figure out the end position.

So, to deal with your problem more directly (and hopefully correctly!), in deriving the E-L equations, we pick two arbitrary points and find equations which have to be satisfied in moving between these points. Since the points can be anywhere, we define an infinite number of curves (for any given Lagrangian). In our problem, we specify one curve by defining a start point, and a start momentum. Then, given any time later, we can know where the particle is. That is to say, you wouldn't define a specific end point, you'd define an arbitrary one, then pick the path with the correct intial momentum, rather than the one which passes through some specified end point.