Lets assume a body rotating and m*du/dt=F, where u=(ux,uy,uz) and F=(Fx,Fy,Fz). If I would like to express the velocity in cylindrical coordinates, should I add an additional inertial force, such as m*du/dt = F + Fin, where F=(Fr,Ff,Fz) and u=(ur,uf,uz)? Note that the new cylindrical system is not the local coordinate system of the body itself but a global one in order to (x=0,y=0) transforms to r=0 (phi doesn't play any role here since it's a singular point).
It depends whether you are using stationary systems or one that is rotating with your body. If you use inertial systems (stationary or moving with linear velocity) then there is no need for any other additional forces.
I am trying to describe the motion of a charged particle under the influence of Lorentz force. So far I have a global cartesian coordinate system and I would like to go to a global cylindrical system. I assume that I will need additional inertial force to do that.
In order to transform between self-consistent and conventional coordinate systems such as cartesian or cylindrical (polar?), no addition of any force term should ever be needed.
Here it is important to understand not only how the origin of your coordinate frame is defined (fixed to what, translating with what, assigned to what...?), but more importantly, is one frame rotating or accelerating vs. another frame.
Its all about the acceleration. A Newtonian frame is non-accelerating, and is therefore also called an "Inertial" inertial frame. Rotation of a reference frame would cause acceleration of a particle fixed in that frame (fixed to any point but the instantaneous center of rotation or ICOR), and so a rotating reference frame is accelerating or "non-inertial".
The coordinate frame transforms that generate additional terms are transforms going from a non-inertial (i.e. accelerating) to an inertial (non-accelerating) frame. A rotating frame is a classic example of an accelerating or non-inertial frame. In any Dynamics problem, or more broadly in any Physical Mechanics problem, the transform from one frame to another must account for acceleration of the non-inertial frame as expressed in terms of the inertial frame, thus the added terms typically required within the inertial frame for motion initially described in a rotating frame.
Take one generic point defined in the rotating or accelerating ("non-inertial") reference frame. Then, develop the definition of that point in terms of the reference frame that you would like to transform to. The key is that differentiation must hold for the new frame and that is how you will find the "added" terms required in whatever inertial or non-accelerating frame you are trying to transform to.
The best way to do this is graphically, with vectors representing particle position and velocity, and components of those vectors along whatever axes define the initial coordinate system. Different color chalk on the black board, or different color lines in the computer plot, are both good ways to track position vs. velocity.
In the end you should have a set of equations that transforms cleanly back and forth between one frame and the other. Most importantly, the way to know you have a good answer is by continuity and F=MA, which has to hold in both frames in order for your transform to be valid. There are many Dynamics text books that treat this coordinate frame transformation problem. Analytically the added terms come out of vector differentiation and may be derived likewise. Setting up the problem graphically will always give you a better understanding of the physical reality resulting in additional terms in one vs. another coordinate frame.
Always label your coordinate frames individually and track what frame definition applies to what case(s). This way you will never fall victim to the dreaded "rotating frame transformation error" that invalidates all subsequent solutions after it is applied. One of the biggest of such errors is the error of omission, which you have already avoided by your realization that a transform is required - good job. Now the fun begins - you get to derive the transform. Find a Dynamics textbook and soak your brain in it, then things will make more sense. Remember to carefully mind you notation and label each frame clearly, the secret to success as a Dynamicist.
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