Can't Understand the Solution of Landau-Lifschitz Mechanics Problem

[gobbles]

I'm going over Landau Lifschitz vol. 1, solving selected problems and I can't understand how to arrive at the correct solution of this one (question (part b) + solution of relevant question attached).

I did arrive at the correct expressions of the coordinates of the point m, but I can't get to the solution. What I'm doing is taking the derivative of x and y, adding their squares together to get v² and then the kinetic energy and subtracting the potential energy from that, which is just -y times mg, but I don't get the correct answer. Please help.

 

[gobbles]

Just noticed that the question is missing.
It is to find the Lagrangian of the system.

 

[mistergrinch]

The answer looks wrong to me. The first term should be phi_dot^2 -- the units aren't even correct. For the 2nd term, I get the cos and sin switched and a factor of phi_dot in place of one alpha. I guess I'm as confused as you are...

 

[lemur81]

I am stuck on this problem as well. I ended up with the same solution as mistergrinch. Surely there must be someone out there who has encountered this problem in such a famous text. I feel I am missing something small but important. Any help would be appreciated.

 

[ehild]

The total derivatives are omitted from the Lagrangian.

The second term of the square of \dot{x} which should be

-maγlsin(γt)cosφ\dot{\varphi}

can be written as

d/dt{-maγl sin(γt) (sin(φ)}+maγ²lcos(γt)sin(φ), and the derivative with respect time is omitted.


ehild

 

[lemur81]

Ah I see! Perfect, thank you very much.

 

[ehild]

It took me some time to find it out. The missing dot in the first term is a typo in your copy: The dot is present in the original Russian edition.

ehild

 

[StephenRS]

Can you explain how you go from one term to the other? I'm taking a graduate level classical mechanics course, even though my physics background is limited, but my math background goes through ODE's. I suppose I am getting as far as the others were, but if you could explain why or how we know what you said it would help extremely