1 Test #2 of 4 Thurs. 10/17/02 – in class Bring an index card 3”x5”. Use both sides. Write anything you want that will help. You may bring your last index card as well. Lagrangian method (most of exam) Line integrals / curls Generalized forces / Lagrange multipliers / constraint forces Tuesday 10/15 – Real-time review / problem session :08

2 Class #14 of 30 Extensions of the Lagrangian Method Generalized force and momenta  Plausibility of the Lagrange equations Lagrange Multipliers  Calculating forces of constraint  Worked example  You solve it (Taylor 7-50). :72

3 m1m1 m2m2 Atwood’s Machine Lagrangian recipe :40

4 T7-16 Rolling mass on ramp :65 m y q x

5 :72 Generalized Force and Momentum Traditional Generalized Force Momentum Newton’s Law

6 Lagrange Multipliers Lagrangian method as practiced so far eliminates the need to write down forces of constraint BECAUSE it assumes that are consistent with forces of constraint!! Forces of constraint of form may be found by method of Lagrange multipliers :12

7 Lagrange Multipliers Lagrangian method as practiced so far eliminates the need to write down forces of constraint BECAUSE it assumes that are consistent with forces of constraint!! Forces of constraint of form may be found by method of Lagrange multipliers :12

8 1) Write down T and U in any convenient coordinate system. 2) Write down constraints of form 3) Define the generalized coordinates 4) Rewrite in terms of 5) Calculate 6) Plug into 7) Solve ODE’s and solve for lambda 8) The terms give the constraint forces Lagrange’s Dining Room :17 Mechanics “Cookbook” for Lagrange Multipliers

9 m1m1 m2m2 Atwood’s Machine Lagrangian recipe :40

10 m1m1 m2m2 Atwood’s Machine Lagrange Multiplier Recipe :40

11 m1m1 m2m2 Atwood’s Machine Lagrange Multiplier Recipe :40

12 m1m1 m2m2 Atwood’s Machine Lagrange Multiplier Recipe :40

13 Taylor 7-50 A mass m1 rests on a frictionless horizontal table. Attached to it is a string which runs horizontally to the edge of the table, where it passes over a frictionless small pulley and down to where it supports a mass m2. Use as coordinates x and y, the distances of m1 and m2 from the pulley. These satisfy the constraint equations f(x,y)=x+y=const. Write down the two modified Lagrange equations and solve them for x’’, y’’ and the Lagrange multiplier Lambda. Find the tension forces on the two masses. :12

14 Class #14 Windup Exam next Thursday :72