346 



GENERALIZED COORDINATES 



EULER'S EQUATIONS FOR A RIGID BODY 



278. Euler's equations ( 252) can be derived from those of 

 Lagrange. 



Let the moments of a rigid body about its principal axes of 

 inertia at a point 0, which is fixed in the body and is also 

 either fixed in space or is the center of gravity of the body, be 

 A, B, C. Then if o^, &) 2 , o> 3 are the components of rotation about 

 these axes, we have, as in 248, 



T = %(Aa)\ +B(D\ + Ca)l). (184) 



As Lagrangian coordinates, let us take 6, <f> the spherical polar 

 coordinates of the third axis OC of the body, and ty a third coor- 

 dinate which measures the angle between the first axis OA of the 



rigid body and the plane 

 through OC and the axis 

 6 = 0, say the plane COz. 

 We have first to find 

 <o l9 G> 2 and < 3 in terms of 

 6, < and ty, so as to ex- 

 press 2T as a function of 

 these coordinates. The 

 motion of the body is com- 

 pounded of the motion 

 relative to the plane COz, 

 together with the motion 

 of the plane COz relative 

 to fixed axes. The former 

 motion consists of a rotation ijr about OC, and this, resolved along 

 the axes OA, OB, OC, has components 



0, 0, i^. 



The motion of the plane COz is compounded of 



(a) a rotation 6 about an axis at right angles to its plane ; 



(b) a rotation (j> about the axis = 0. 



FIG. 156 



