1. (20 points) Is the constraint
(x2 + y2 + z2)dx +2(xdx + ydy + zdz) = 0 (1)
a holonomic constraint or a non-holonomic one?
2. (30 points) A particle is subjected to the potential V (x) = −Fx, where F is a constant. The particle travels from x = 0 to x = a in a time interval t0. Assume the motion of the particle can be expressed in the form x(t) = A+Bt+Ct2. Find the values of A,B, and C such that the action is a minimum.
3. (25 points) A bead of mass m slides down a frictionless wire bent into the form of a parabola y = Ax2, determine the constraint force acting on the bead.
4. (25 points) A uniform hoop of mass m and radius r rolls under gravity from rest without slipping on top of a fixed cylinder of radius R, use the method of Lagrange multipliers to find the point where the hoop falls off the cylinder.
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