Relational Magnitudes
Relational Magnitudes are invariant vectorial quantities that conserve their value and form under transformations of translation and rotation.
I. Definitions I (Relational Magnitudes)
The relational position ($\mathbf{r}_i$), relational velocity ($\mathbf{v}_i$), and relational acceleration ($\mathbf{a}_i$) of a particle $i$ with respect to an Auxiliary Reference Frame, are given by:
$\mathbf{r}_i \doteq \vec{r}_i$
$\mathbf{v}_i \doteq \dfrac{d(\vec{r}_i)}{dt} = \vec{v}_i$
$\mathbf{a}_i \doteq \dfrac{d^2(\vec{r}_i)}{dt^2} = \vec{a}_i$
Where $\vec{r}_i$, $\vec{v}_i$ and $\vec{a}_i$ are the ordinary vectorial position, velocity, and acceleration of the particle $i$ with respect to the Auxiliary Reference Frame.
Note
The Relational (Vectorial) Magnitudes are always the same as the Ordinary (Vectorial) Magnitudes in the Auxiliary Reference Frame.
II. Definitions II (Relational Magnitudes)
The relational position ($\mathbf{r}_i$), relational velocity ($\mathbf{v}_i$), and relational acceleration ($\mathbf{a}_i$) of a particle $i$ with respect to any Reference Frame $S$, are given by:
$\mathbf{r}_i \doteq \vec{r}_i - \vec{R}$
$\mathbf{v}_i \doteq (\vec{v}_i - \vec{V}) - \vec{\omega} \times (\vec{r}_i - \vec{R})$
$\mathbf{a}_i \doteq (\vec{a}_i - \vec{A}) - 2\vec{\omega} \times (\vec{v}_i - \vec{V}) + \vec{\omega} \times [\ \vec{\omega} \times (\vec{r}_i - \vec{R})\ ] - \vec{\alpha} \times (\vec{r}_i - \vec{R})$
Where:
- $\vec{r}_i, \vec{v}_i, \vec{a}_i$ are the ordinary vectorial position, velocity, and acceleration of particle $i$ with respect to the Frame $S$.
- $\vec{R}, \vec{V}, \vec{A}$ are the position, velocity, and acceleration of the Auxiliary Frame's origin with respect to $S$.
- $\vec{\omega}$ and $\vec{\alpha}$ are the angular velocity and angular acceleration of the Auxiliary Frame with respect to $S$.
III. Transformations (Invarianza$\cdot$Relations)
The transformations of relational position, relational velocity and relational acceleration of a particle $i$ between a Reference Frame $S$ and another Reference Frame $S'$, are given by:
$\mathbf{r}_i \doteq (\vec{r}_i - \vec{R}) = \mathbf{r}'_i$
$\mathbf{r}'_i \doteq (\vec{r}'_i - \vec{R}') = \mathbf{r}_i$
$\mathbf{v}_i \doteq (\vec{v}_i - \vec{V}) - \vec{\omega} \times (\vec{r}_i - \vec{R}) = \mathbf{v}'_i$
$\mathbf{v}'_i \doteq (\vec{v}'_i - \vec{V}') - \vec{\omega}' \times (\vec{r}'_i - \vec{R}') = \mathbf{v}_i$
$\mathbf{a}_i \doteq (\vec{a}_i - \vec{A}) - 2\vec{\omega} \times (\vec{v}_i - \vec{V}) + \vec{\omega} \times [\ \vec{\omega} \times (\vec{r}_i - \vec{R})\ ] - \vec{\alpha} \times (\vec{r}_i - \vec{R}) = \mathbf{a}'_i$
$\mathbf{a}'_i \doteq (\vec{a}'_i - \vec{A}') - 2\vec{\omega}' \times (\vec{v}'_i - \vec{V}') + \vec{\omega}' \times [\ \vec{\omega}' \times (\vec{r}'_i - \vec{R}')\ ] - \vec{\alpha}' \times (\vec{r}'_i - \vec{R}') = \mathbf{a}_i$
IV. Bibliography
- A. Blatter, A Reformulation of Classical Mechanics (2015). [PDF]
- A. Tobla, A Reformulation of Classical Mechanics (2024). [PDF]