1. Constraints

2. Pendulum Tension  The Newtonian view is from balanced forces. Tension balances radial component of gravity  The tension force is conservative. Unknown potential Narrow and deep x y R  m T F = mg mgcos  mgsin  r = R

3. Constraint Force  Allow the radial coordinate to vary in the Lagrangian. Tension potential V C (r)Tension potential V C (r)  Find the radial EL equation.  The extra term is the force of tension. Constant radius RConstant radius R Tension force includes gravity and centripetal forceTension force includes gravity and centripetal force

4. Generalized Constraints  Holonomic constraints can be expressed in terms of coordinates and time. Assume C constraints. New coordinates are 0  The constraints form a potential. Modified Lagrangian New equations of motion for constraint coordinates k = 1 … 3N j = 1 … C

5. Undetermined Multipliers  A holonomic constraint can be expressed as a time differential. Up to k constraintsUp to k constraints  If time is not explicit then these can be included in the EL equations. Functions (t) are undermined multipliersFunctions (t) are undermined multipliers Equivalent to forces of constraintEquivalent to forces of constraint

6. Rolling Disk  A rolling disk has a constraint. Linear distance vs. angle turned  Two variables could be reduced to one.  m  R y l

7. Rolling Constraints  Set up the Lagrangian.  Find both EL equations with an undetermined multiplier.  The constraint equation is used to reduce the differential equations. Solve forSolve for

8. Generalized Rolling Forces  The generalized rolling forces derive from the multiplier. The y -component gives a force The  -component gives a torque