After looking at Rod’s Solution on Reasonable Deviations, I have come to the conclusion that my solution was way to complicated. Instead of using high school physics, I decided to solve the problem using Hamiltonian Mechanics.
Question: A bead slides along a smooth wire bent in the shape of a parabola,
. The bead rotates in a circle, of radius
, when the wire is rotating about its vertical symmetry axis with angular velocity
. Find the constant c.
Answer: To solve this problem using Hamiltonian Mechanics, we first need to find the Hamiltonian (which is the Kinetic Energy + Potential) of the system.
The system can easily be shown to have the following form:

If we use cylindrical coordinates and constrain the path of the bead to the parabola, we can simplify the Hamiltonian to a single degree of freedom. Since the velocity
in cylindrical coordinates is
and
, the Hamiltonian reduces to:

and since
and
, the equation becomes:

Now that we have the Hamiltonian reduced to its generalized coordinates, we can apply it to the Euler-Lagrange equation. Since our Hamiltonian has only one degree of freedom, the Euler-Lagrange equation will have the following form:

First, lets take the derivatives needed for the differential equation. They can be shown to be:



When we put these values into the differential equation, we will get:


Since the position of the bead is confined to a circle of radius R, we can simplify the differential equation by using the following conditions:
,
, and
. The differential equation will then reduce to:

Therefore, the value of c must be:



Comment by Nikita Nikolaev
Made Friday, 25 of January , 2008 at 5:49 pm
Nice! Nicely written!
Now I think it is the time I write up my solution!
Excellent!