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An inertial frame is a nonrotating frame in which all force-free test bodies follow straight trajectories with constant velocity. The qualification “all” matters: a specially chosen coordinate system can make one accelerated trajectory look straight without making the frame inertial. In practice, approximately isolated test bodies and gyroscopes help identify inertial motion; identifying the relevant forces and calibrating instruments are part of testing the theory. The definition does not select a privileged state of uniform rest.

In its usual coordinate formulation, Newtonian mechanics states $m\ddot{\mathbf{x}}=\mathbf{F}$ in an inertial frame. A Galilean transformation, $t^{\prime }=t+b$, ${\mathbf{x}}^{\prime }=R\mathbf{x}+\mathbf{V}t+\mathbf{a}$ with constant orthogonal $R$ and constant $\mathbf{V}$, maps the class of inertial frames to itself. It preserves zero acceleration; a time-dependent translation generally does not. In the translating frame ${\mathbf{x}}^{\prime }=\mathbf{x}-\mathbf{a}(t)$, the same physical law reads $m{\ddot{\mathbf{x}}}^{\prime }=\mathbf{F}-m\ddot{\mathbf{a}}$. The extra fictitious force records the frame's acceleration, rather than a new interaction.

It is possible to state Newtonian mechanics without mentioning an inertial frame explicitly. On Galilean spacetime, let $u$ be a trajectory tangent normalized by the absolute time covector, $t_a{u}^a=1$. Thenuses an affine connection to define acceleration. The free-particle equation is ${\nabla }_{u}u=0$. This formulation removes a chosen coordinate frame but retains the geometric distinction between inertial motion and accelerated motion. Omitting both the inertial frame language and all equivalent inertial structure would leave the law unable to specify which trajectories count as unforced.

The same distinction applies to special relativity. Einstein's original presentation uses a class of inertial frames and calibrated rods, clocks and light signals.^[ref] A geometric presentation instead uses Minkowski spacetime, its Lorentzian metric and its Levi-Civita connection. A free massive particle obeys ${\nabla }_{u}u=0$, now with proper time parametrization and $g(u,u)=-c^2$. There is no preferred inertial rest frame, but there is a distinguished family of inertial trajectories and a physical Minkowski interval.

Accelerating coordinates are allowed in special relativity. They produce nonzero connection coefficients even though the Riemann curvature tensor remains zero. General relativity is needed when spacetime curvature and dynamical gravitation matter, not merely because the chosen observer accelerates. Thus neither theory requires a particular inertial frame in its coordinate-free statement; each still needs the structure that determines inertial motion.

A fictitious force is a term required when writing Newton's second law in an accelerating frame; it comes from the coordinate choice rather than a new interaction. For axes rotating at constant $\mathbf{\Omega }$ about a fixed inertial origin,Apply this identity twice to position to get ${\mathbf{a}}_I={\mathbf{a}}_R+2\mathbf{\Omega }\times {\mathbf{v}}_R+\mathbf{\Omega }\times (\mathbf{\Omega }\times \mathbf{r})$. Moving the last two terms to the right of $m{\mathbf{a}}_I=\mathbf{F}$ gives the stated Coriolis and centrifugal forces. An accelerating origin or changing rotation rate would add further terms.

A slowly rotating fluid planet follows an equipotential of gravitational plus centrifugal potential. The centrifugal potential is $-{\Omega }^2{r}^2{\cos }^2\lambda /2$, most negative at the equator. An outward equatorial shift raises gravitational potential and restores the balance, producing an oblate figure. Self-gravity also changes with the shape; no flattening coefficient follows from rotation alone without a density model.

Using local north and radial-up unit vectors, the centrifugal acceleration on a stationary mass isIt points away from the rotation axis: upward and equatorward, at angle $\lambda$ to radial up in the northern hemisphere. Its vertical fraction of gravitational force isusing a $24\mathrm{h}$ day for this approximate number. The resultant effective weight tilts equatorward relative to radial down. A plumb-defined local vertical already follows effective gravity; the component calculation uses the spherical radial vertical. A supported stationary mass of course has zero net acceleration.

For the Coriolis deflection of vertical launch and drop, retain only first order in $\Omega$, use effective constant $g$, and neglect horizontal velocities in the Coriolis term as instructed. With east positive, vertical velocity gives $\ddot{x}=-2\Omega \cos \lambda \dot{z}$ and no northward acceleration. For launch, $z=ut-g{t}^2/2$ and the flight time is $2u/g$. Integrating with zero initial horizontal velocity givesThe landing is westward. The westward velocity accumulated on ascent is only recovered on descent, so the two displacement contributions do not cancel.

For release from rest at $h_0=u^2/(2g)$, $z=h_0-g{t}^2/2$, $t_{\mathrm{fall}}=u/g$, and $\ddot{x}=2\Omega gt\cos \lambda$. ThusThe drop lands eastward with one quarter the launch-return displacement magnitude. North–south corrections from horizontal Coriolis coupling are beyond the retained approximation.

A fictitious force accounts for acceleration of the chosen reference frame rather than an additional interaction. For axes rotating at constant $\mathbf{\Omega }$, differentiating a vector uses $(d/dt{)}_{\mathrm{lab}}=(d/dt{)}_{\mathrm{rot}}+\mathbf{\Omega }\times$. Twice applying this givesso moving the last two terms to the force side produces the Coriolis and centrifugal forces. Here $\mathbf{v}$ is measured in rotating axes.

The rotating saddle trap has potential force $(-m{\omega }^{2}x,+m{\omega }^{2}y)$. ThereforeThe trial solution givesBoth modes are bounded for arbitrary sufficiently small initial displacements and velocities precisely when $|\Omega |>\omega$. At equality there is a zero-frequency secular mode; the exactly stationary solution does not establish stability.

For rapidly rotating saddle confinement with $\Omega \gg \omega >0$, real mode ellipses can be writtenThey are nearly circular clockwise motions at rates ${ϵ}_{\pm }\simeq \Omega \pm {\omega }^2/(2\Omega )$. Rotating these axes back to the laboratory cancels the large angular rate: the leading laboratory motion is a slow approximately harmonic ellipse with frequency ${\omega }^2/(2\Omega )$, plus small rapid micromotion. It results from the changing saddle force, not a static potential minimum. At next order the ellipse precesses prograde at ${\omega }^4/(8{\Omega }^3)$. The rotating-potential model and this guiding-centre distinction are analysed in the primary work cited here.^[ref]

Let a coordinate frame rotate with angular velocity $\mathbf{\Omega }$ relative to an inertial frame. For any vector $\mathbf{u}$,Applying this twice to position, with coincident nonaccelerating origins, givesNewton's second law written using rotating coordinates therefore contains the Coriolis forceThe other inertial corrections are the centrifugal force and, for changing rotation rate, the Euler force. An accelerating origin introduces an additional translational fictitious force.

The Coriolis force depends on velocity and is perpendicular to it, so ${\mathbf{F}}_C\cdot {\mathbf{v}}_{\mathrm{rotating}}=0$: the Coriolis force does no work in the rotating frame. It bends trajectories without directly changing their rotating-frame speed. It is absent for motion parallel to the rotation axis.

For locally horizontal motion on Earth at latitude $\lambda$, take $x$ east, $y$ north and $z$ up. Keeping the vertical component $\Omega \sin \lambda$ givesHorizontal motion is deflected right in the northern hemisphere and left in the southern hemisphere. In the ideal absence of other horizontal forces these equations give inertial circles with angular frequency $|f|$. The same velocity-dependent correction explains the precession of a Foucault pendulum, whose precession-rate magnitude is $\Omega |\sin \lambda |$. Terrestrial vertical motions also involve Earth's horizontal rotation component, so the reduced horizontal equations are not the full force law.

Let the rotating frame have origin acceleration ${\mathbf{A}}_0$ and angular velocity $\mathbf{\Omega }(t)$ relative to an inertial frame. For a vector referred to the moving axes, the transport rule isApply it twice to position. With rotating-frame velocity $\mathbf{v}$ and acceleration ${\mathbf{a}}_R$,Thus Newton's equation expressed in the rotating frame isThe added terms are translational, Coriolis, Euler and centrifugal fictitious forces. They reflect the frame's acceleration, rather than additional interactions with nearby objects.

The centrifugal force points outward from the rotation axis and, for constant $\mathbf{\Omega }$, has potential $-m|\mathbf{\Omega }\times \mathbf{r}{|}^2/2$. The Coriolis force is perpendicular to velocity, so does no work on the particle measured in that rotating frame; it changes direction rather than speed directly. The Euler force is present when the frame's rotation rate or direction changes.

On Earth, gravity and centrifugal acceleration combine into effective gravity. For slow horizontal motion at latitude $\lambda$, the vertical component of Earth's rotation gives Coriolis parameter $f=2\Omega \sin \lambda$: an initially northward motion deflects eastward in the northern hemisphere. This supplies a useful local explanation of rotating flows and Foucault precession. A bucket of rotating liquid illustrates the centrifugal potential: its steady free surface is an equipotential of $gz-{\Omega }^2{r}^2/2$, giving $z={\Omega }^2{r}^2/(2g)+\mathrm{constant}$. These equations require a declared frame; adding a centrifugal force while retaining inertial-frame acceleration would count the same kinematic effect twice.