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Reference Frame

A reference frame is a coordinate system used to describe the position and motion of objects, providing a consistent point of view.
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The statement of the theorem

Let SIS_I be the inertial reference frame defined by the basis vectors eI=(eIx,eIy,eIz)\mathbf{e}_I = (\mathbf{e}_{Ix}, \mathbf{e}_{Iy}, \mathbf{e}_{Iz}) and the origin RI(t)\mathbf{R}_I(t). Let SS be the moving reference frame, whose origin R(t)\mathbf{R}(t) and basis vectors e=(ex,ey,ez)\mathbf{e} = (\mathbf{e}_x, \mathbf{e}_y, \mathbf{e}_z) are time-dependent. The position vector r\mathbf{r} of a point PP in SS is related to the position vector R\mathbf{R} of the origin of SS in SIS_I by the transformation: r(t)=R(t)+ρ(t)\mathbf{r}(t) = \mathbf{R}(t) + \boldsymbol{\rho}(t), where ρ(t)\boldsymbol{\rho}(t) is the vector from the origin of SS to PP, expressed in SS's coordinates. The transformation between the coordinate systems is given by the rotation matrix RSI(t)∈SO(3)\mathbf{R}_{SI}(t) \in SO(3), such that e=RSI(t)eI\mathbf{e} = \mathbf{R}_{SI}(t) \mathbf{e}_I. The velocity v\mathbf{v} of PP in SIS_I is then derived using the transport theorem (or relative velocity formula): v=drdt=dRdt+vrel+∇R⋅vrot\mathbf{v} = \frac{d\mathbf{r}}{dt} = \frac{d\mathbf{R}}{dt} + \mathbf{v}_{rel} + \boldsymbol{\boldsymbol{\nabla}}\mathbf{R} \cdot \mathbf{v}_{rot} where vrel\mathbf{v}_{rel} is the velocity of PP relative to SS, and ∇R⋅vrot\boldsymbol{\boldsymbol{\nabla}}\mathbf{R} \cdot \mathbf{v}_{rot} represents the contribution from the rotation of the frame SS itself, defined by the angular velocity ω=12(ex×deydt+ey×dezdt+ez×dexdt)\boldsymbol{\omega} = \frac{1}{2} \left(\mathbf{e}_x \times \frac{d\mathbf{e}_y}{dt} + \mathbf{e}_y \times \frac{d\mathbf{e}_z}{dt} + \mathbf{e}_z \times \frac{d\mathbf{e}_x}{dt}\right).