General external torque on a gyro

I will get to work on this tomorrow. From dielectric relaxation theory the torque (Tq) exerted by an external static electric field E on a molecular dipole moment (mu) is

Tq = mu x E

and this corresponds to a potential energy

E = – mu dot E

In dynamics the external torque on a gyro is defined by Eq. (8.30) of Marion and Thornton as

Tq. (external) = sigma over i (r x F) sub i

so the potential energy is

U (external) = – sigma over i r dot F

One simple example of an external potential energy is given by U = – mgh cos theta in the symmetric top with one point fixed (Section 10.10). So it is easier to set up the lagrangian and the Euler Lagrange equations will do all the rest. We can find everything from theta(t), phi(t) and chi(t). This avoids the complexities of frame transformation because we are dealing with energies (rotational kinetic and potential).


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