Physics 319 Classical Mechanics G. A. Krafft Old Dominion University Jefferson Lab Lecture 21 G. A. Krafft Jefferson Lab

Lagrangian Small Oscillation Theory Method for solving problems where several coupled oscillations present Steps are Write Lagrangian for several oscillations including coupling. If needed go into small oscillation limit Solve for system oscillation “normal mode” frequencies Solve for oscillation amplitude vector for each normal mode Go into coordinates, the so-called normal mode coordinates, where the oscillations de-couple, to solve initial conditions and time dependences

Two Masses and Three Springs Forces and equation of motion are Introduce 2 component “vector” describing state of system

Equations of Motion in Vector Form Equations of motion are

Sinusoidal Ansatz As we have done many times before assume sinusoidal solutions of general form Simultaneous Linear Equations! Solution method from Linear Algebra Gives possible “normal mode” oscillation frequencies. Then solve for associated (eigen)vector.

Case of Identical Masses and Springs Normal mode frequency problem an eigenvalue problem. Solve normal mode (also called secular) equation

First Normal Mode Take minus sign solution Back in original matrix equation Such an oscillation in the system is the symmetric mode Masses move in the same direction with the middle spring unextended. Oscillation frequency “obviously” satisfies

Second Normal Mode Take plus sign solution Now normal mode eigenvector is Such an oscillation in the system is the antisymmetric mode Masses move in the opposite directions with the middle spring extended twice as much as the other two.

In Pictures

General Solution General solution for motion determined by 4 initial conditions, giving the real and imaginary parts of A and B Picture of general motion

Normal Mode Coordinates General motion is simplified if go into coordinates tied to the normal mode eigenvector pattern. Define These combinations will only oscillate at the normal mode frequencies ω± separately, ξ1 at ω− and ξ2 at ω+ By going into the normal mode coordinates, the coupled oscillations problem becomes decoupled!

Case of Weak Coupling Expect slight frequency shifts in oscillators Normal mode eigenvectors are the same symmetric and antisymmetric combinations that we saw before.

General Solution Place following boundary conditions on solution Then get Phase delayed oscillations with amplitude that goes from one degree of freedom to the other and back again