Physics 319 Classical Mechanics G A Krafft Old

Physics 319 Classical Mechanics G. A. Krafft Old Dominion University Jefferson Lab Lecture 14 Undergraduate Classical Mechanics Spring 2017

Lagrangian Method 1. Choose coordinates to incorporate constraint automatically. 2. Write down the Euler-Lagrange Equation. (0, 0) 3. Solve the Euler-Lagrange Equation. l 1 • Double Pendulum example l 2 • Analyzed in detail in Chapter 11 Undergraduate Classical Mechanics Spring 2017

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 Undergraduate Classical Mechanics Spring 2017

Conservation Laws/Nöther’s Theorem • Suppose Langrangian is translationally invariant This invariance in Lagrangian implies momentum conserved • Example: N interacting bodies Undergraduate Classical Mechanics Spring 2017

Momentum Conservation • Total momentum conserved • When Lagrangian invariant to (infinitesimal) rotation Undergraduate Classical Mechanics Spring 2017

Angular Momentum Conservation • Total angular momentum conserved • When Lagrangian independent of time Undergraduate Classical Mechanics Spring 2017

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! Undergraduate Classical Mechanics Spring 2017

Lagrangian for Electromagnetic Force • Electromagnetic Lorentz force follows from the Lagrangian • Canonical momentum Undergraduate Classical Mechanics Spring 2017

Undergraduate Classical Mechanics Spring 2017

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 Undergraduate Classical Mechanics Spring 2017

Snell’s Law 3! • In terms of two angles Undergraduate Classical Mechanics Spring 2017