# Motion under eccentric torque

Hello everyone. I have the following problem: Find the equations of motion of a rigid body under the effect of a field that applies a torque around a point which is different from its center of mass.

Let's say for simplicity that the particle is uniaxial and the torque is applied somewhere on its axis (but not in the center). The coordinates of the particle are the position of the center of mass (R) and an orientation unit vector (u). We know that the rotational kinetic energy is given in terms of the angular velocity of the body (w) as wIw/2, where I is the moment of inertia, not in terms of the conjugate momenum of the coordinates u, while the potential energy is a function of u. The relation between the orientation and angular velocity is:

(du/dt)=w x u.

I would like to know which set of coordinates and conjugate momenta would you use to solve such a problem?

Thank you in advance for any help :).

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A pure torque (with 0 net force) does not have any specific point where it "acts on". This should simplify the problem.

Your orientation unit vector has 2 degrees of freedom, the orientation of the body has 3. How do you describe any rotation with your vector?

Thank you for your answer :) You are right, I forgot to mention a critical point, the field can produce a torque AND a force (think about dipole-dipole interactions for example).

Being a uniaxial particle the orientation involves just two degrees of freedom, to symplify the problem. The third one is just degenerate because of the symmetry, but you can use any standard method for rotating the frame of the particle.