Of course you are correct, F_t =dU/dt is not a meaningful quantity in classical physics - it would be a time component of force - nonsensical in the 3D vector representation of a force.
However, F_x = -dU/dx and F_y=-dU/dy form the correct connection between a conservative force and its associated potential.
When I posted the musings about the problem (an illustration of blindly following the math instead of the physics), I probably should not have focused on the position (1,1) at the rim of the bowl (and the infinity that results). The nonsensical result even applies below the rim. The potential generating force is gravity (not the contact force) which only acts along the y direction. Yet - the blind use of the expression F_x = -dU/dx gives an x component of force related to the potential. The nonsense occurs from folding in the constraint y = 1 - sqrt(1-x^2), not from any problem with the expression F_x = -dU/dx per se.
Bob at PC
________________________________________
From: phys-l-bounces@carnot.physics.buffalo.edu [phys-l-bounces@carnot.physics.buffalo.edu] on behalf of Bob Sciamanda [treborsci@verizon.net]
Sent: Thursday, April 05, 2012 12:50 AM
To: Forum for Physics Educators
Subject: Re: [Phys-l] Force and potential
Interpreting this dU/dx as a force in the x direction makes no sense. mgy
is the potential energy associated with the VERTICAL gravitational force mg.
What you did would be like considering a mass in free fall, and treating the
parameter t in the same way as you are here treating the parameter x ==>
U = mgy = mg(.5gt^2)
F_t =dU/dt = mg^2t is nonsense! (Not even the dimension of force.)
From: LaMontagne, Bob
Sent: Wednesday, April 04, 2012 3:11 PM
To: Forum for Physics Educators
Subject: [Phys-l] Force and potential
Deep in the sugar shock of eating too many Easter "Peeps", I came up with
the following:
Consider a bowl formed from the lower half of a sphere. In 2D I can write
the equation for the bowl as x^2 + (y-1)^2 = 1.
This gives y = 1 - sqrt(1-x^2)
If I place a marble at the inside rim (x=1, y=1) of the bowl and release it,
the marble will oscillate back and forth, repeatedly coming back to the
release point.
The potential energy is given by U=mgh=mgy = mg[1 - sqrt(1-x^2)]
Using F_x = - dU/dx , then we have F_x = -x/sqrt(1-x^2) which implies that
F_x is infinitely strong when x = 1 (the release point).