In non-inertial frames fictitious forces appear. Consider a particle moving with velocity v in a reference frame K which moves with velocity V(t) relative to the inertial frame K0 and rotates with angular velocity W(t).
The Lagrangian of the particle is
L = (1/2)mv2 + mv
×
(W
´ r) + (1/2)m(W
´ r)2 - mW×r
- U, with W = dV(t)/dt.
¶L/¶v
= mv +
m(W
´ r), ¶L/¶r
= m(v
´ W) +
m(W
´ r) ´ W
- mW - ¶U/¶r.
The equations of motion are
mdv/dt = -¶U/¶r - mdV/dt + mr ´ dW/dt - 2mW ´ v - mW ´ (W ´ r).
Here
| -mdV/dt = fictitious force due to acceleration of frame |
| mr ´ dW/dt = fictitious force due to non-uniform rotation of frame |
| -2mW ´ v = Coriolis force |
| -mW ´ (W ´ r) = Centifugal force |
For a uniformly rotating frame dW/dt = 0, dV/dt, and the equations of motion are
mdv/dt = -¶U/dr - 2mW ´ v - mW ´ (W ´ r).