1. Physics 319 Classical Mechanics
G. A. Krafft
Old Dominion University
Jefferson Lab
Lecture 14
G. A. Krafft
Jefferson Lab

2. Lagrangian Method
Choose coordinates to incorporate constraint automatically.
Write down the Euler-Lagrange Equation.
Solve the Euler-Lagrange Equation.
Double Pendulum example
Analyzed in detail in Chapter 11
(0,0) l1 l2

3. Ignorable or Cyclic Coordinates
Lagrangian for particle acted on by gravity
Does not depend on x or y. Euler-Lagrange imply
When the Lagrangian is independent of a generalized coordinate that coordinate is said to by ignorable or cyclic. Automatically, generalized momentum is conserved. Usually this degree of freedom can be integrated completely using the conservation of generalized momentum

4. Conservation Laws/Nöther’s Theorem
Suppose Langrangian is translationally invariant
This invariance in Lagrangian implies momentum conserved
Example: N interacting bodies

5. Momentum Conservation
Total momentum conserved
When Lagrangian invariant to (infinitesimal) rotation

6. Angular Momentum Conservation
Total angular momentum conserved
When Lagrangian independent of time

7. Hamiltonian Function
The negative of the preceding combination is called the Hamiltonian function
The Hamiltonian is conserved
Taylor shows for time-independent generalized coordinate transformations, leading to quadratic dependence of the Lagrangian on the
Energy conserved!

8. Lagrangian for Electromagnetic Force
Electromagnetic Lorentz force follows from the Lagrangian
Canonical momentum

10. Lagrange Multipliers
Sometimes constraints are not easily accounted for only by coordinate choice. If the constraint can be put in the form of a time-integral along the path, the problem can be solved using Lagrange multipliers
Find stationary condition using Euler-Lagrange for
Works because the constrained variation of the added λf is zero (it integrates to a constant) and does not change the fact that the constrained solution is still has stationary Lagrangian

11. Snell’s Law 3!
In terms of two angles