1. Work

2. Generalized Coordinates  Transformation rules define alternate sets of coordinates. 3N Cartesian coordinates x i f generalized coordinates q m Select f degrees of freedom  Small changes in a coordinate can be expressed by the chain rule. polar coordinate example

3. Generalized Velocity  The differentials for the coordinates don’t depend on the coordinates themselves.  The time derivative gives generalized velocities. One per generalized coordinateOne per generalized coordinate  A general identity relates the generalized coordinates and velocities.

4. Generalized Force  Force acting over a small displacement is the work. Express in generalized coordinates  Rewrite the work in terms of the generalized force components, Q m, Q t Last term for time dependence

5. Constraint Forces  All the Q m are applied forces. No dependence on constraint coordinatesNo dependence on constraint coordinates Not forces of constraintNot forces of constraint Constraint forces do no workConstraint forces do no work  Forces of constraint are often unknown. Newtonian problem complicated by themNewtonian problem complicated by them

6. Potential Energy  A conservative force derives from a potential V.  The generalized force can be derived from the same potential. Work expressed in terms of potential energy if conservative forceWork expressed in terms of potential energy if conservative force

7. Acceleration and Velocity  The work can be expressed by mass and acceleration. Mass m (i) related to xi  The Cartesian coordinate is transformed to the generalized coordinate. Use the boxed identity Work expanded in terms

8. Kinetic Energy  The velocity can be used to find the kinetic energy T. Rearranging summationRearranging summation

9. Work Compared  Work in terms of the kinetic energy must equal the work in terms of the force. Each generalized component considered separatelyEach generalized component considered separately Time-dependent part just Newton’s 2 nd lawTime-dependent part just Newton’s 2 nd law trivial identity; ma=F

10. Lagrangian Function  Conservative forces depend only on position. Leave non-conservative forces on the right side of the equationLeave non-conservative forces on the right side of the equation  The quantity L = T  V is the Lagrangian.  This gives Lagrange’s equations of motion. For f equations, 2 f constantsFor f equations, 2 f constants next