Using momentum conservation as holonomic constraint

[Dor_M]

< Mentor Note -- thread moved to HH from the technical physics forums, so no HH Template is shown >

My question is from an exam in analytic mechanics. The question was about an object sliding on inclined plane, the plane's angle is constant, and the plane is free to move along X axis. No friction.
The problem is originaly three degrees of freedom, (x, y) of the object and (X) of the plane. The constant angle brings you to two degrees of freedom.
Now, I used the momentum conservation along the x-axis as a constrain, and got a lagrangian with one degree of freedom. I know it's not what was expected, and that they wanted me to get the conservation of the momentum along x from the lagrangian, but is it wrong? The constrain is holonomic for my understanding. Also I received the correct results.

another thing is that the second question was "which conservations do you have in the problem". I answered that the energy is conserved, my lagrangian didn't contained X anymore, but only a coordinate that describe the distance of the object from the edge of the plane, and it wasn't cyclic. I answered that the energy alone is conserved.

So,
1. Is it a mistake to use the momentum conservation as holonomic constraint in this case, and if it is, why? (even though I got the correct results)
2. If I got rid of X in the lagrangian, should I still count it's momentum as conserved?

(I'm sorry for spelling mistakes, english is not my first language.)

 

[Orodruin]

Is it a mistake to use the momentum conservation as holonomic constraint in this case, and if it is, why? (even though I got the correct results)

Yes, you cannot just assume momentum conservation (which is what you do if you put it in as a constraint). In the cases where you do have it it should follow from translational invariance of the action and Noether's theorem.

 

[Dor_M]

I still don't understand. Which rule did I break? In the case of the constrain (Xcm(M1+m2) - X1M1-x2m2) = 0. I can't see which law it is breaking. why can't I assume momentum conservation?

 

[Orodruin]

I still don't understand. Which rule did I break? In the case of the constrain (Xcm(M1+m2) - X1M1-x2m2) = 0. I can't see which law it is breaking. why can't I assume momentum conservation?

Because momentum conservation is derived as a constant of motion from the variation of the action. Of course you could assume momentum conservation, but that either assumes something that is already present or imposes a constraint that is not part of the original formulation (in this case the former).

The question is, why do you assume momentum conservation? What rational argument do you make to assume it?

 

[Dor_M]

I know. The thing that made me do that is that I asked the professor something during the exam and he told to me to reduce the degrees of freedom as much as I can. So I did. I know it was not what was expected, but still, it does not looks like I did something wrong. I mean, that seem like a "legal" constrain, I used it as one, and most importantly, IT WORKED. I know it is not part of the formalism, but since it worked, and if it does not break any rule, it is not a mistake. I have serious doubt about if it is really "legal", but if it is not, I want to know why. Because again, it is holonomic constraint, for my understanding.
Why did I assume momentum conservation? because there is clearly no forces along the horizontal (x) axis. I assumed momentum conservation along that axis.

 

[Orodruin]

Why did I assume momentum conservation? because there is clearly no forces along the horizontal (x) axis. I assumed momentum conservation along that axis.

This is what I thought. You are a priori not allowed to think in terms of forces. They are not part of the basic formalism. The identification of forces as derivatives of momentum and the subsequent connection between translational symmetry to no external forces is something that emerges from your theory - not something you build into it by requiring it.

It "worked" only because you already knew from an equivalent formalism (Newtonian mechanics) that it should be a constant of motion and therefore imposed something that was already true in the theory.

 

[Dor_M]

I totally see your point, I understand what you are saying. specially in the case that this was an exam in analytic mechanics. What I do not understand is why it is wrong. Let's say that you faced a problem outside of an exam, and you carefully solved the problem by mixing conclusions from two formalism. for example, you used one formalism, got a conclusion, carefully and legally used that conclusion in the other formalism, found another conclusion, and carefully and legally plug it in the first formalism, and so on. I don't see a problem with doing that, as long that you don't break the rules of the formalisms.