Rotating reference frame

To derive these fictitious forces, it's helpful to be able to convert between the coordinates ${\displaystyle \left(x',y',z'\right)}$ of the rotating reference frame and the coordinates ${\displaystyle \left(x,y,z\right)}$ of an inertial reference frame with the same origin. If the rotation is about the ${\displaystyle z}$ axis with a constant angular velocity ${\displaystyle \Omega }$, or ${\displaystyle \theta (t){=}\Omega t}$, and the two reference frames coincide at time ${\displaystyle t=0}$, the transformation from rotating coordinates to inertial coordinates can be written

${\displaystyle x=x'\cos \left(\theta (t)\right)-y'\sin \left(\theta (t)\right)}$

${\displaystyle y=x'\sin \left(\theta (t)\right)+y'\cos \left(\theta (t)\right)}$

whereas the reverse transformation is

${\displaystyle x'=x\cos \left(-\theta (t)\right)-y\sin \left(-\theta (t)\right)}$

${\displaystyle y'=x\sin \left(-\theta (t)\right)+y\cos \left(-\theta (t)\right)}$

This result can be obtained from a rotation matrix.

Introduce the unit vectors ${\displaystyle {\hat {\boldsymbol {\imath }}},\ {\hat {\boldsymbol {\jmath }}},\ {\hat {\boldsymbol {k}}}}$ representing standard unit basis vectors in the rotating frame. The time-derivatives of these unit vectors are found next. Suppose the frames are aligned at t = 0 and the z-axis is the axis of rotation. Then for a counterclockwise rotation through angle Ωt:

${\displaystyle {\hat {\boldsymbol {\imath }}}(t)=(\cos \theta (t),\ \sin \theta (t))}$

where the (x, y) components are expressed in the stationary frame. Likewise,

${\displaystyle {\hat {\boldsymbol {\jmath }}}(t)=(-\sin \theta (t),\ \cos \theta (t))\ .}$

Thus the time derivative of these vectors, which rotate without changing magnitude, is

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\hat {\boldsymbol {\imath }}}(t)=\Omega (-\sin \theta (t),\ \cos \theta (t))=\Omega {\hat {\boldsymbol {\jmath }}}\$ ;}

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\hat {\boldsymbol {\jmath }}}(t)=\Omega (-\cos \theta (t),\ -\sin \theta (t))=-\Omega {\hat {\boldsymbol {\imath }}}\ ,}$

where ${\displaystyle \Omega \equiv {\frac {\mathrm {d} }{\mathrm {d} t}}\theta (t)}$. This result is the same as found using a vector cross product with the rotation vector ${\displaystyle {\boldsymbol {\Omega }}}$ pointed along the z-axis of rotation ${\displaystyle {\boldsymbol {\Omega }}=(0,\ 0,\ \Omega )}$, namely,

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\hat {\boldsymbol {u}}}={\boldsymbol {\Omega \times }}{\hat {\boldsymbol {u}}}\ ,}$

where ${\displaystyle {\hat {\boldsymbol {u}}}}$ is either ${\displaystyle {\hat {\boldsymbol {\imath }}}}$ or ${\displaystyle {\hat {\boldsymbol {\jmath }}}}$.

Introduce the unit vectors ${\displaystyle {\hat {\boldsymbol {\imath }}},\ {\hat {\boldsymbol {\jmath }}},\ {\hat {\boldsymbol {k}}}}$ representing standard unit basis vectors in the rotating frame. As they rotate they will remain normalized. If we let them rotate at the speed of ${\displaystyle \Omega (t)}$ about an axis ${\displaystyle {\boldsymbol {\Omega }}(t)}$ then each unit vector ${\displaystyle {\hat {\boldsymbol {u}}}}$ of the rotating coordinate system abides by the following equation:

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\hat {\boldsymbol {u}}}={\boldsymbol {\Omega }}(t)\times {\boldsymbol {\hat {u}}}\ .}$

Then if we have a vector function ${\displaystyle {\boldsymbol {f}}}$,

${\displaystyle {\boldsymbol {f}}(t)=f_{x}(t){\hat {\boldsymbol {\imath }}}+f_{y}(t){\hat {\boldsymbol {\jmath }}}+f_{z}(t){\hat {\boldsymbol {k}}}\ ,}$

and we want to examine its first derivative we have (using the product rule of differentiation):[2][3]

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}{\boldsymbol {f}}&={\frac {\mathrm {d} f_{x}}{\mathrm {d} t}}{\hat {\boldsymbol {\imath }}}+{\frac {\mathrm {d} {\hat {\boldsymbol {\imath }}}}{\mathrm {d} t}}f_{x}+{\frac {\mathrm {d} f_{y}}{\mathrm {d} t}}{\hat {\boldsymbol {\jmath }}}+{\frac {\mathrm {d} {\hat {\boldsymbol {\jmath }}}}{\mathrm {d} t}}f_{y}+{\frac {\mathrm {d} f_{z}}{\mathrm {d} t}}{\hat {\boldsymbol {k}}}+{\frac {\mathrm {d} {\hat {\boldsymbol {k}}}}{\mathrm {d} t}}f_{z}\\&={\frac {\mathrm {d} f_{x}}{\mathrm {d} t}}{\hat {\boldsymbol {\imath }}}+{\frac {\mathrm {d} f_{y}}{\mathrm {d} t}}{\hat {\boldsymbol {\jmath }}}+{\frac {\mathrm {d} f_{z}}{\mathrm {d} t}}{\hat {\boldsymbol {k}}}+\left[{\boldsymbol {\Omega }}(t)\times \left(f_{x}{\hat {\boldsymbol {\imath }}}+f_{y}{\hat {\boldsymbol {\jmath }}}+f_{z}{\hat {\boldsymbol {k}}}\right)\right]\\&=\left({\frac {\mathrm {d} {\boldsymbol {f}}}{\mathrm {d} t}}\right)_{r}+{\boldsymbol {\Omega }}(t)\times {\boldsymbol {f}}(t)\end{aligned}}}$

where ${\displaystyle \left({\frac {\mathrm {d} {\boldsymbol {f}}}{\mathrm {d} t}}\right)_{r}}$ is the rate of change of ${\displaystyle {\boldsymbol {f}}}$ as observed in the rotating coordinate system. As a shorthand the differentiation is expressed as:

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\boldsymbol {f}}=\left[\left({\frac {\mathrm {d} }{\mathrm {d} t}}\right)_{r}+{\boldsymbol {\Omega }}(t)\times \right]{\boldsymbol {f}}\ .}$

This result is also known as the Transport Theorem in analytical dynamics and is also sometimes referred to as the Basic Kinematic Equation.[4]

A velocity of an object is the time-derivative of the object's position, or

${\displaystyle \mathbf {v} \ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}}$

The time derivative of a position ${\displaystyle {\boldsymbol {r}}(t)}$ in a rotating reference frame has two components, one from the explicit time dependence due to motion of the particle itself, and another from the frame's own rotation. Applying the result of the previous subsection to the displacement ${\displaystyle {\boldsymbol {r}}(t)}$, the velocities in the two reference frames are related by the equation

${\displaystyle \mathbf {v_{i}} \ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}=\left({\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\right)_{\mathrm {r} }+{\boldsymbol {\Omega }}\times \mathbf {r} =\mathbf {v} _{\mathrm {r} }+{\boldsymbol {\Omega }}\times \mathbf {r} \ ,}$

where subscript i means the inertial frame of reference, and r means the rotating frame of reference.

Acceleration is the second time derivative of position, or the first time derivative of velocity

${\displaystyle \mathbf {a} _{\mathrm {i} }\ {\stackrel {\mathrm {def} }{=}}\ \left({\frac {\mathrm {d} ^{2}\mathbf {r} }{\mathrm {d} t^{2}}}\right)_{\mathrm {i} }=\left({\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}\right)_{\mathrm {i} }=\left[\left({\frac {\mathrm {d} }{\mathrm {d} t}}\right)_{\mathrm {r} }+{\boldsymbol {\Omega }}\times \right]\left[\left({\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\right)_{\mathrm {r} }+{\boldsymbol {\Omega }}\times \mathbf {r} \right]\ ,}$

where subscript i means the inertial frame of reference. Carrying out the differentiations and re-arranging some terms yields the acceleration relative to the rotating reference frame, ${\displaystyle \mathbf {a} _{\mathrm {r} }}$

${\displaystyle \mathbf {a} _{\mathrm {r} }=\mathbf {a} _{\mathrm {i} }-2{\boldsymbol {\Omega }}\times \mathbf {v} _{\mathrm {r} }-{\boldsymbol {\Omega }}\times ({\boldsymbol {\Omega }}\times \mathbf {r} )-{\frac {\mathrm {d} {\boldsymbol {\Omega }}}{\mathrm {d} t}}\times \mathbf {r} }$

where ${\displaystyle \mathbf {a} _{\mathrm {r} }\ {\stackrel {\mathrm {def} }{=}}\ \left({\frac {\mathrm {d} ^{2}\mathbf {r} }{\mathrm {d} t^{2}}}\right)_{\mathrm {r} }}$ is the apparent acceleration in the rotating reference frame, the term ${\displaystyle -{\boldsymbol {\Omega }}\times ({\boldsymbol {\Omega }}\times \mathbf {r} )}$ represents centrifugal acceleration, and the term ${\displaystyle -2{\boldsymbol {\Omega }}\times \mathbf {v} _{\mathrm {r} }}$ is the Coriolis acceleration. The last term (${\displaystyle -{\frac {\mathrm {d} {\boldsymbol {\Omega }}}{\mathrm {d} t}}\times \mathbf {r} }$) is the Euler acceleration and is zero in uniformly rotating frames.

When the expression for acceleration is multiplied by the mass of the particle, the three extra terms on the right-hand side result in fictitious forces in the rotating reference frame, that is, apparent forces that result from being in a non-inertial reference frame, rather than from any physical interaction between bodies.

Using Newton's second law of motion ${\displaystyle \mathbf {F} =m\mathbf {a} }$, we obtain:[1][2][3][5][6]

• the Coriolis force

${\displaystyle \mathbf {F} _{\mathrm {Coriolis} }=-2m{\boldsymbol {\Omega }}\times \mathbf {v} _{\mathrm {r} }}$

• the centrifugal force

${\displaystyle \mathbf {F} _{\mathrm {centrifugal} }=-m{\boldsymbol {\Omega }}\times ({\boldsymbol {\Omega }}\times \mathbf {r} )}$

• and the Euler force

${\displaystyle \mathbf {F} _{\mathrm {Euler} }=-m{\frac {\mathrm {d} {\boldsymbol {\Omega }}}{\mathrm {d} t}}\times \mathbf {r} }$

where ${\displaystyle m}$ is the mass of the object being acted upon by these fictitious forces. Notice that all three forces vanish when the frame is not rotating, that is, when ${\displaystyle {\boldsymbol {\Omega }}=0\ .}$

For completeness, the inertial acceleration ${\displaystyle \mathbf {a} _{\mathrm {i} }}$ due to impressed external forces ${\displaystyle \mathbf {F} _{\mathrm {imp} }}$ can be determined from the total physical force in the inertial (non-rotating) frame (for example, force from physical interactions such as electromagnetic forces) using Newton's second law in the inertial frame:

${\displaystyle \mathbf {F} _{\mathrm {imp} }=m\mathbf {a} _{\mathrm {i} }}$

Newton's law in the rotating frame then becomes

${\displaystyle \mathbf {F_{r}} =\mathbf {F} _{\mathrm {imp} }+\mathbf {F} _{\mathrm {centrifugal} }+\mathbf {F} _{\mathrm {Coriolis} }+\mathbf {F} _{\mathrm {Euler} }=m\mathbf {a_{r}} \ .}$

In other words, to handle the laws of motion in a rotating reference frame:[6][7][8]

Treat the fictitious forces like real forces, and pretend you are in an inertial frame.

— Louis N. Hand, Janet D. Finch Analytical Mechanics, p. 267

Obviously, a rotating frame of reference is a case of a non-inertial frame. Thus the particle in addition to the real force is acted upon by a fictitious force...The particle will move according to Newton's second law of motion if the total force acting on it is taken as the sum of the real and fictitious forces.

— HS Hans & SP Pui: Mechanics; p. 341

This equation has exactly the form of Newton's second law, except that in addition to F, the sum of all forces identified in the inertial frame, there is an extra term on the right...This means we can continue to use Newton's second law in the noninertial frame provided we agree that in the noninertial frame we must add an extra force-like term, often called the inertial force.

— John R. Taylor: Classical Mechanics; p. 328