Alternative Kinetic Energy Formulation and Goldstein's Problem 11

[Fedor Indutny]

Hello everyone!

I have a (supposedly) calculus problem that I just can't seem to figure out. Basically, I'm trying to understand why alternative kinetic energy formulation does not yield the same equations of motion in problem 11 of Goldstein's Classic Mechanics 3 edition.

The text of problem is following:

Consider a uniform disc that rolls without slipping on a horizontal plane. A horizontal force is applied to the center of the disc and in a direction parallel to the plane of the disk.

(a) Derive Lagrange's equations and find the generalized force
(b) ...doesn't matter for this question...

I have solved the problem for kinetic energy $$T = m * (v_x^2 + v_y^2) / 2$$, and indeed the equations of motions become $$d/dt(m * v_x) = Q_x$$, where Q_x is a generalized force. Nothing unexpected here.

However, if I formulate kinetic energy as $$T = m * (v_x/cos(theta))^2 / 2$$, everything in the equation seems to change with the additional dependence on theta (the angle of disc orientation on the xy plane).

Is there anything wrong with using this alternative kinetic energy formulation (except that it blows up on $$theta = pi/2$$)?

Any help is greatly appreciated, thank you!

 

[Fedor Indutny]

Perhaps, the reason why it does not work is that partial derivative $$∂theta/∂(v_x)$$ is not 0. Is it a right guess?

 

[Fedor Indutny]

Oh, I think I figured it out. The virtual displacements $$∂q_i$$ are not independent in these coordinates, therefore if my $$T$$ depends on both $$x$$ and $$θ$$, I have to take both in account to create an equation of motion.

Please let me know if any these comments are correct :)