Classical Mechanics
For this problem, do not use the method of Lagrange multipliers but include instead the constraints directly into the expressions of the kinetic and potential energies.
Consider a planar double pendulum with lengths li,l2 and masses m1,m2 (see figure)
mm
2
m2
y
1. Write the constraints of the double pendulum system.
2. Use the variables i and 2 (and their time derivatives) to express the kinetic an potential energies and construct the Lagrangian of the system.
3. Derive the Lagrange equations for the system.
In the following, consider that m1 = m2 = m and l1 = l2 = l.
4. Write the equations of motion under these assumptions and express the equations in the limit of small oscillations.
Assume that the solutions of these equations have the form ;(t) = A;exp(iwt) with the same value of for j = 1, 2.
5. Determine the expressions of such that the resulting system of equations has a solution different than the trivial one, A = A2 = 0.
2
6. Express the relation between A and A2 for the possible solutions of and describe briefly what are the resulting motions of the two masses.
Bonus points] Determine the expressions of without doing the simplifying assumptions, m1 = m2 and l1 = l2. The use of a computer-algebra system (CAS) is allowed. If you use a CAS, you dont need to provide intermediate steps but show how the problem was formulated with the CAS
