Lagrange Multipliers in Classical Mechanics - exercise 1

[mcaay]

Homework Statement

The skier is skiing without friction down the mountain, being all the time in a specified plane. The skier's altitude y(x) is described as a certain defined function of parameter x, which stands for the horizontal distance of the skier from the initial position. The skier is subjected to gravity g = -g e_y.

a) Calculate forces of constraint in relation to x.
b) Prove that if the skier can detach from the mountain at a point with horizontal value x_od, then y''(x_od) < 0, i.e. function y(x) is concave in this point.

Use first kind Lagrange equations (Lagrange multipliers).

Homework Equations

The Attempt at a Solution

To be honest I am completely lost. This is so abstract with not even a function given that I don't know how to specify the initial constraint equation. Normally there would be y and x in it, so I wrote f(x,y) = y - y(x) = 0, which feels weird ...

I wrote the Lagrange equations based on that weird constraint, but even if that would be good I have no idea how to get Lambda from that :(

I started doing masters in physics after finishing bachelor in mechatronics, and compared to physics students I have huge gaps in knowledge (even though classical mechanics is a 2nd year subject here).

 

[TSny]

Hello.

It is preferable to type in your work rather than post pictures of your work. That way, it is much easier for helpers to quote specific parts of your work. (Posting pictures of diagrams is fine.)

I think you are on the right track, overall. The way you set up the constraint equation as ##f(x, y) = y - y(x) = 0## looks correct.

However, the equation circled below is not correct

 

[mcaay]

Hey, sorry for not responding immediately. Thanks for the tips, I think now I got it right, so I'll upload it for people who might lurk in this thread.

Sorry for the Polish comments, I wrote that the answer can not depend on dx/dt, but it can depend on Energy which is constant and on derivatives of g(x).
And in b) that when the skier detaches from the slope - there are no reaction forces, so my'' = -mg

 

[TSny]

OK. Part (a) looks good to me. For (b) you showed that ##\ddot y(x_{od}) < 0##. Did you show that ##y''(x_{od}) < 0## ?

 

[mcaay]

Oh damn! Thanks for lettng me realize it's not the same :p

 

[mcaay]

This should do I think

 

[TSny]

Yes. Or you could let ##\lambda = 0## in

 

[Ray Vickson]

Hey, sorry for not responding immediately. Thanks for the tips, I think now I got it right, so I'll upload it for people who might lurk in this thread.

Sorry for the Polish comments, I wrote that the answer can not depend on dx/dt, but it can depend on Energy which is constant and on derivatives of g(x).
And in b) that when the skier detaches from the slope - there are no reaction forces, so my'' = -mg

View attachment 215647

STOP posting images---it is against PF standards. Do what others have done: take the time and effort to type it out.