1550078481-Ordinary_Differential_Equations__Roberts_

364 Ordinary Differential Equations

Letting uI = YI (the position of pendulum 1), u2 = y~ (the velocity of

pendulum 1), u3 = Y2 (the position of pendulum 2), and U4 = y~ (the ve-

locity of pendulum 2) and differentiating each of these equations, we find

u~ = y~

u / = y /1 =--Yg I - -k ( YI -y2 ) = - (g - + -k) YI+ -yk 2

2 I f, m f, m m

u~ = y~

U4 / = Y2 II = - -g y2 - -k ( Y2 -yI ) = -k YI - (g - + -k) Y2·

f, m m f, m

Next, replacing YI by uI, y~ by u2, Y2 by u3, and y~ by u 4 on the right-
hand side of the equation above, we obtain the following first-order system of
differential equations

U~ = - (-g + -k) UI + -Uk 3

f, m m

u~ = -k uI - (g - + -k) U3.

m f, m

Written in matrix notation this system of equations is

0 1 0 0

u' I UI

-(~ + ~)^0

k

u~

0

m U2

u~^0 0 0 1 U3

u~ k 0

• (~ + ~)
  0 U4
  m

Observe that the 4 x 4 matrix appearing in this system is a constant matrix.