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Unformatted text preview: Physics 235 Chapter 11 !" 1 !
= $ cos # sin %
!#
!" 2 !
= $ cos # cos %
!#
!" 3
!
= &$ sin #
!# !" 1
= cos %
!
!#
!" 2
= & sin %
!
!#
!" 3
=0
!
!# and Lagrange's equation becomes
d
!
! ({ I1" 1 sin # + I 2" 2 cos # } cos $ % I 3" 3 sin $ ) % { I1" 1 cos # % I 2" 2 sin # } = 0
dt • The Euler angle ψ: Lagrange's equation for the coordinate ψ is 0= !$ i d
!$ i
!T d !T
!T !$ i d
!T !$ i
#
=%
#%
= % I i$ i
# % I i$ i
=0
!
!
!
!" dt !"
dt i !$ i !"
!" dt i
!"
i !$ i !"
i Differentiating the angular velocity with respect to the coordinate ψ we find
!" 1 !
!
= $ sin % cos # & % sin # = " 2
!#
!" 2
!
!
= &$ sin % sin # & % cos # = &" 1
!#
!" 3
=0
!# !" 1
=0
!
!%
!" 2
=0
!
!%
!" 3
=1
!
!% and Lagrange's equation becomes
I1! 1! 2 " I 2! 2! 1 " d
d
!
{ I 3! 3 } = ( I1 " I 2 )!1! 2 " dt { I 3! 3 } = ( I1 " I 2 )!1! 2 " I 3! 3 = 0
dt Of all three equations of motion, the last one is the only one to contain just the components of the
angular velocity. Since our choice of the x3 axis was arbitrary, we expect that similar relations
should exist for the other two axes. The set of three equation we obtain in this way are called the
Euler equations:
!
( I1 ! I 2 )"1" 2 ! I 3" 3 = 0
!
( I 2 ! I 3 )" 2" 3 ! I1"1 = 0  16  ...
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This note was uploaded on 01/08/2012 for the course PHY 235 taught by Professor Morgan during the Spring '09 term at SUNY Stony Brook.
 Spring '09
 MORGAN
 Physics

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