Homework #3
Assigned: Aug. 18th, 2005
Due: Aug. 25th, 2005
Points: 20

1. 10 points
   Hamilton's principle states that the actual path in configuration space followed by a holonomic dynamic system during the fixed time interval t₀ to t₁ is such that the integral
   J = int(L dt) with limits t₀ to t₁
   is stationary with respect to path variations which vanish at the end-points.
   Apply this principle to a spherical pendulum whose configuration space has two angles q1 and q2 and L is given by
   0.5*m*L^2( q1dot^2 + q2dot^2*sin(q1)^2 ) - m*g*L*cos(q1)
   and derive the equations of motion. State the boundary conditions on q1 and q2 and interpret them physically. Note that a spherical pendulum swings in 3-D space and hence needs two angles q1 and q2 to describe its position.
2. 5 points
   A chain of length L and mass per unit length m is hanging between two points (x₁, y₁) and (x₂, y₂). Solve for its shape by using the principle of minimum potential energy.
   Do you see any similarity between this problem and another one you have already seen?
3. 5 points
   Derive the expression for the Euler's buckling load for a fixed-free column by posing it as a problem of calculus of variations.