How is the Coriolis generalized potential obtained

[AlephClo]

The Coriolis potential last term of (42) is obtained by integration through r and R from last term of (40).
I do not understand why we do not need to integrate through v as well, since the Coriolis force depends on v?

Homework Equations

Equation (41) is wrong I think, L must be replaced by U.
The forces for the 2 springs are F(r)= -kr, and F(R)= -kR (bold are vectors)
The generalized force Qj = Fi ⋅ δri/δqj (δ is del the partial derivative; j and i are indices)

The Attempt at a Solution

the 4 terms of (42) are obtainable from the intergration relatively to r and R.
Since the Coriolis force is dependent of the velocity v, why we do not need to Integration relative to v = (dr/dt, dR/dt) as well?

Or more generally when is it required that we integrate through position and velocity the force that depends on position and velocity to obtain a generalized potential.

Thank you.

 

[Nate810]

The Coriolis force is a type of inertial force that depends on the velocity of the object. It is an effect of the Coriolis acceleration, which is caused by the rotation of the Earth. Therefore, there is no need to integrate through velocity for the Coriolis potential because it is already taken into account by the rotation of the Earth. However, if you were considering a different type of inertial force, such as a centrifugal force, then you would need to integrate both position and velocity in order to obtain the generalized potential.