Scalar Magnitudes

Scalar Magnitudes are invariant scalar quantities that conserve their value and form under transformations of translation and rotation, or changes between coordinate systems (Cartesian, polar, spherical, etc.).

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I. Definitions

0. Vectorial Magnitudes

The vectorial position ($\vec{r}_{ij}$), vectorial velocity ($\vec{v}_{ij}$), and vectorial acceleration ($\vec{a}_{ij}$) of two particles $i$ and $j$ are given by:

Vectorial Magnitude	Definition	Derivation
Position ($\vec{r}_{ij}$)	$\vec{r}_{ij} \doteq (\vec{r}_i - \vec{r}_j)$	(Fundamental definition)
Velocity ($\vec{v}_{ij}$)	$\vec{v}_{ij} \doteq (\vec{v}_i - \vec{v}_j)$	$\vec{v}_{ij} \doteq \frac{d(\vec{r}_{ij})}{dt}$
Acceleration ($\vec{a}_{ij}$)	$\vec{a}_{ij} \doteq (\vec{a}_i - \vec{a}_j)$	$\vec{a}_{ij} \doteq \frac{d^2(\vec{r}_{ij})}{dt^2}$

1. Scalar Magnitudes

The scalar position ($\tau_{ij}$), scalar velocity ($\dot{\tau}_{ij}$), and scalar acceleration ($\ddot{\tau}_{ij}$) of two particles $i$ and $j$ are given by:

Scalar Magnitude	Definition	Derivation
Position ($\tau_{ij}$)	$\tau_{ij} \doteq \frac{1}{2} \vec{r}_{ij} \cdot \vec{r}_{ij}$	(Fundamental definition)
Velocity ($\dot{\tau}_{ij}$)	$\dot{\tau}_{ij} \doteq \vec{v}_{ij} \cdot \vec{r}_{ij}$	$\dot{\tau}_{ij} \doteq \frac{d(\tau_{ij})}{dt}$
Acceleration ($\ddot{\tau}_{ij}$)	$\ddot{\tau}_{ij} \doteq \vec{a}_{ij} \cdot \vec{r}_{ij} + \vec{v}_{ij} \cdot \vec{v}_{ij}$	$\ddot{\tau}_{ij} \doteq \frac{d^2(\tau_{ij})}{dt^2}$

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II. Scalar Invariance Demonstrations

0. Vectorial Transformations (Absolute)

The vectorial position ($\vec{r}'_i$), vectorial velocity ($\vec{v}'_i$), and vectorial acceleration ($\vec{a}'_i$) of a particle $i$ with respect to a Reference Frame $S'$, whose origin $O'$ is at the vectorial position $\vec{r}_{O'}$ with respect to another Reference Frame $S$, are given by:

Where $\vec{r}_i$, $\vec{v}_i$, and $\vec{a}_i$ are the vectorial position, velocity, and acceleration of particle $i$ with respect to Frame $S$; and $\vec{\omega}$ and $\vec{\alpha}$ are the angular velocity and angular acceleration of Frame $S'$ with respect to Frame $S$.

Note

If $\vec{m}'_i = \vec{n}_i$ then:

$\dfrac{d(\vec{m}'_i)}{dt} = \dfrac{d(\vec{n}_i)}{dt} - \vec{\omega} \times \vec{n}_i$

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1. Scalar Position Invariance ($\tau_{ij}$)

The Scalar Position $\tau_{ij}$ is invariant under rotation and translation because the magnitude of the relative vector is preserved.

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2. Scalar Velocity Invariance ($\dot{\tau}_{ij}$)

The Scalar Velocity $\dot{\tau}_{ij}$ is invariant because the cross-product generated by the angular velocity ($\vec{\omega}$) is perpendicular to the relative position vector, resulting in a zero scalar product.

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3. Scalar Acceleration Invariance ($\ddot{\tau}_{ij}$)

The Scalar Acceleration $\ddot{\tau}_{ij}$ is invariant because all inertial terms (Angular Acceleration, Coriolis, and Centrifugal) mutually cancel due to the properties of the vector and scalar products.

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III. Fundamental Relations

1. Radial Magnitudes Relations

The scalar magnitudes ($\tau_{ij}, \dot{\tau}_{ij}, \ddot{\tau}_{ij}$) expressed using radial magnitudes ($r_{ij}$) are given by:

Scalar Magnitude	Relation with Radial Magnitudes
$\tau_{ij}$	$\tau_{ij} = \frac{1}{2} r_{ij}^2$
$\dot{\tau}_{ij}$	$\dot{\tau}_{ij} = r_{ij} \dot{r}_{ij}$
$\ddot{\tau}_{ij}$	$\ddot{\tau}_{ij} = r_{ij} \ddot{r}_{ij} + \dot{r}_{ij}^2$

2. Polar Magnitudes Relations

The scalar magnitudes ($\tau_{ij}, \dot{\tau}_{ij}, \ddot{\tau}_{ij}$) expressed using polar magnitudes ($r_{ij}$) are given by:

Scalar Magnitude	Relation with Polar Magnitudes
$\tau_{ij}$	$\tau_{ij} = \frac{1}{2} r_{ij}^2$
$\dot{\tau}_{ij}$	$\dot{\tau}_{ij} = r_{ij} \dot{r}_{ij}$
$\ddot{\tau}_{ij}$	$\ddot{\tau}_{ij} = r_{ij} \ddot{r}_{ij} + \dot{r}_{ij}^2$

3. Cylindrical Magnitudes Relations

The scalar magnitudes ($\tau_{ij}, \dot{\tau}_{ij}, \ddot{\tau}_{ij}$) expressed using cylindrical magnitudes ($r_{ij}$) are given by:

Scalar Magnitude	Relation with Cylindrical Magnitudes
$\tau_{ij}$	$\tau_{ij} = \frac{1}{2} r_{ij}^2$
$\dot{\tau}_{ij}$	$\dot{\tau}_{ij} = r_{ij} \dot{r}_{ij}$
$\ddot{\tau}_{ij}$	$\ddot{\tau}_{ij} = r_{ij} \ddot{r}_{ij} + \dot{r}_{ij}^2$

4. Circular Magnitudes Relations

The scalar magnitudes ($\tau_{ij}, \dot{\tau}_{ij}, \ddot{\tau}_{ij}$) expressed using circular magnitudes ($r_{ij}$) are given by:

Scalar Magnitude	Relation with Circular Magnitudes
$\tau_{ij}$	$\tau_{ij} = \frac{1}{2} r_{ij}^2$
$\dot{\tau}_{ij}$	$\dot{\tau}_{ij} = r_{ij} \dot{r}_{ij}$
$\ddot{\tau}_{ij}$	$\ddot{\tau}_{ij} = r_{ij} \ddot{r}_{ij} + \dot{r}_{ij}^2$

5. Spherical Magnitudes Relations

The scalar magnitudes ($\tau_{ij}, \dot{\tau}_{ij}, \ddot{\tau}_{ij}$) expressed using spherical magnitudes ($r_{ij}$) are given by:

Scalar Magnitude	Relation with Spherical Magnitudes
$\tau_{ij}$	$\tau_{ij} = \frac{1}{2} r_{ij}^2$
$\dot{\tau}_{ij}$	$\dot{\tau}_{ij} = r_{ij} \dot{r}_{ij}$
$\ddot{\tau}_{ij}$	$\ddot{\tau}_{ij} = r_{ij} \ddot{r}_{ij} + \dot{r}_{ij}^2$

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IV. Bibliography

1. A. Torassa, A Group of Invariant Equations (2014). [PDF]
2. A. Torassa, A Reformulation of Classical Mechanics (2014). [PDF]
3. A. Tobla, Linear, Radial & Scalar Magnitudes (2015). [PDF]
4. A. Tobla, A Reformulation of Classical Mechanics (2024). [PDF]