A coordinate representing rotation about a variable axis and kinetic energy

Question

If a system is represented by a set of generalized coordinates ##q_i## in which one coordinate say ##\theta## is such that ##d \theta## represents a rotation of the system about a fixed axis( an axis whose orientation remains fixed in space) then the kinetic energy ##T## shouldn't depend on it. I understand why this is true.



What if the rotation axis isn't fixed and it's orientation changes in space?  What can we say about ##T##? Will it depend on ##\theta## or not?



Thank you in advance :)