please explain the correct answer(option); in the absence of an applied torque a

Question

please explain the correct answer(option);

in the absence of an applied torque a rigid body with three distinct principal moments of inertia given by I1, I2 and I3 is rotating freely about a fixed point inside the body.the Euler equations for its components of its angular velocity (1,2,3)are
d1/dt=[(I2-I3)/I1]23, d2/dt =[(I3-I1)/I2]13 , d3/dt =[(I1-I2)/I3]12
the equilibrium points in(1,3,3) space are:
1. (1,-1,0) , (-1,0,1) , (0,-1,1)
2. (1,1,0) , (1,0,1) , (0,1,1)
3. (1,0,0) , (0,1,0) , (0,0,1)
4. (1,1,1) , (-1,-1,-1) , (0,0,0)

Explanation / Answer

so for the derivatives to be zero and Is be nonzero and nonequal w1w2=0 this means either w1 or w2=0 w2w3=0 this means either w2 or w3=0 w1w3=0 this means either w1 or w3=0 so how can we satisfy all three. well lets first assume w1 is nonzero, that means from the first equation w2 =0 and from the third w3=0 so that means(1,0,0) is one case then lets try with w2 being nonzero, well from the first one w1=0 and from the second w3=0 so that means (0,1,0) is this case then finally lets try w3 nonzero that means from third that w1=0 and from second w2=0 so this is (0,0,1) this is obviously answer 3.

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