27-dynsys2 - CONTINUOUS DYNAMICAL SYSTEMS II Math 21b, O....

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: CONTINUOUS DYNAMICAL SYSTEMS II Math 21b, O. Knill Homework: section 9.1: 54 and section 9.2: 12,18,22-26,34 COMPLEX LINEAR 1D CASE. x = x for = a + ib has solution x ( t ) = e at e ibt x (0) and length | x ( t ) | = e at | x (0) | . THE HARMONIC OSCILLATOR: The system x =- cx has the solution x ( t ) = cos( ct ) x (0) + sin( ct ) x (0) / c . DERIVATION. x = y, y =- x and in matrix form as x y =- 1 x y = A x y and because A has eigenvalues i , the new coordinates move as a ( t ) = e i ct a (0) and b ( t ) = e- i ct b (0). Writing this in the original coordinates x ( t ) y ( t ) = S a ( t ) b ( t ) and fixing the constants gives x ( t ) , y ( t ). EXAMPLE. THE SPINNER. The spinner is a rigid body attached to a spring aligned around the z-axes. The body can rotate around the z-axes and bounce up and down. The two motions are coupled in the following way: when the spinner winds up in the same direction as the spring, the spring gets tightened and the body gets a lift. If the spinner winds up to the other direction, the spring becomes more relaxed and the body is lowered. Instead of reducing thespring becomes more relaxed and the body is lowered....
View Full Document

This note was uploaded on 04/06/2008 for the course MATH 21B taught by Professor Judson during the Spring '03 term at Harvard.

Ask a homework question - tutors are online