Lesson 15
Lagrange's Equations with Undetermined Multipliers

1) Consider the system shown to the right. The orange disk rolls without slipping down the blue ramp. It would be possible to write the Lagrangian for this system either (a) using both y and q as generalized coordinates, or (b) using only y or q as the proper coordinate. Explain first of all why this is true (it really is). Then explain what would be different about the way you'd have to set up your equations in the two cases (a) and (b).

2) Consider the double Atwood machine shown in the figure and discussed in Example 7.8. The ropes on pulley 1 and pulley 2 have lengths l₁ and l₂ respectively. Suppose you want to determine the equations of motion for the masses in this system. It is possible to use Lagrange undetermined multipliers rather than setting it up as in Example 7.8. For the system as shown in this figure,

• what would be an expression for the kinetic energy T?
• what would be an expression for the potential energy U?
• what would be the constraint equations?

  and, most importantly,

• Consider the l's that you introduce when you write out the Lagrange equations. What do you think those l quantities actually physically represent?

3) Why are Lagrange undetermined multipliers useful?

1. they represent the forces of constraint
2. they permit use of a greater number of coordinates by providing the ability to include constraints
3. both a and b above
4. none of the above

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