Scalar Magnitudes, Definitions, and Relations

Scalar Magnitudes are invariant 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

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 ($\vec{r}_{ij}$)	$\vec{r}_{ij} = (\vec{r}_i - \vec{r}_j)$	(Fundamental definition)
Velocity ($\vec{v}_{ij}$)	$\vec{v}_{ij} = (\vec{v}_i - \vec{v}_j)$	$\vec{v}_{ij} = \frac{d(\vec{r}_{ij})}{dt}$
Acceleration ($\vec{a}_{ij}$)	$\vec{a}_{ij} = (\vec{a}_i - \vec{a}_j)$	$\vec{a}_{ij} = \frac{d^2(\vec{r}_{ij})}{dt^2}$

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

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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 system $S'$, whose origin $O'$ is at the vectorial position $\vec{r}_{O'}$ with respect to another System $S$, are given by:

$\vec{r}'_i = \vec{r}_i - \vec{r}_{O'}$

$\vec{v}'_i = \vec{v}_i - \vec{v}_{O'} - \vec{\omega} \times (\vec{r}_i - \vec{r}_{O'})$

$\vec{a}'_i = \vec{a}_i - \vec{a}_{O'} - 2 \vec{\omega} \times (\vec{v}_i - \vec{v}_{O'}) + \vec{\omega} \times (\vec{\omega} \times (\vec{r}_i - \vec{r}_{O'})) - \dot{\vec{\omega}} \times (\vec{r}_i - \vec{r}_{O'})$

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

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.

$\tau_{ij} = \frac{1}{2} (\vec{r}_i - \vec{r}_j) \cdot (\vec{r}_i - \vec{r}_j)$

$\tau'_{ij} = \frac{1}{2} (\vec{r}'_i - \vec{r}'_j) \cdot (\vec{r}'_i - \vec{r}'_j)$

$\text{Since } (\vec{r}_i - \vec{r}_j) = (\vec{r}'_i - \vec{r}'_j)$ $\text{Because } \vec{r}'_i = \vec{r}_i - \vec{r}_{O'} \text{ and } \vec{r}'_j = \vec{r}_j - \vec{r}_{O'} \text{ (The relative position vector is independent of the frame origin.)}$

$\tau'_{ij} = \frac{1}{2} (\vec{r}_i - \vec{r}_j) \cdot (\vec{r}_i - \vec{r}_j)$

$\therefore \tau'_{ij} = \tau_{ij}$

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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.

$\dot{\tau}_{ij} = (\vec{v}_i - \vec{v}_j) \cdot (\vec{r}_i - \vec{r}_j)$

$\dot{\tau}'_{ij} = (\vec{v}'_i - \vec{v}'_j) \cdot (\vec{r}'_i - \vec{r}'_j)$

$\dot{\tau}'_{ij} = \big( (\vec{v}_i - \vec{v}_j) - \vec{\omega} \times (\vec{r}_i - \vec{r}_j) \big) \cdot (\vec{r}_i - \vec{r}_j)$

$\text{Since } \left( - \vec{\omega} \times (\vec{r}_i - \vec{r}_j) \right) \cdot (\vec{r}_i - \vec{r}_j) = 0 \text{ (The rotational term is orthogonal to the relative position vector.)}$ $\text{Because } (\vec{A} \times \vec{B}) \cdot \vec{B} = 0 \text{ (Property of the Scalar Triple Product)}$

$\dot{\tau}'_{ij} = (\vec{v}_i - \vec{v}_j) \cdot (\vec{r}_i - \vec{r}_j)$

$\therefore \dot{\tau}'_{ij} = \dot{\tau}_{ij}$

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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.

$\ddot{\tau}_{ij} = (\vec{a}_i - \vec{a}_j) \cdot (\vec{r}_i - \vec{r}_j) + (\vec{v}_i - \vec{v}_j) \cdot (\vec{v}_i - \vec{v}_j)$

$\ddot{\tau}'_{ij} = (\vec{a}'_i - \vec{a}'_j) \cdot (\vec{r}'_i - \vec{r}'_j) + (\vec{v}'_i - \vec{v}'_j) \cdot (\vec{v}'_i - \vec{v}'_j)$

$\ddot{\tau}'_{ij} = \left[ (\vec{a}_i - \vec{a}_j) - 2 \vec{\omega} \times (\vec{v}_i - \vec{v}_j) + \vec{\omega} \times \left( \vec{\omega} \times (\vec{r}_i - \vec{r}_j) \right) - \dot{\vec{\omega}} \times (\vec{r}_i - \vec{r}_j) \right] \cdot (\vec{r}_i - \vec{r}_j) + \left[ (\vec{v}_i - \vec{v}_j) - \vec{\omega} \times (\vec{r}_i - \vec{r}_j) \right] \cdot \left[ (\vec{v}_i - \vec{v}_j) - \vec{\omega} \times (\vec{r}_i - \vec{r}_j) \right]$

$\text{Since } - (\dot{\vec{\omega}} \times (\vec{r}_i - \vec{r}_j)) \cdot (\vec{r}_i - \vec{r}_j) = 0 \text{ (Angular acceleration term cancels)}$ $\text{Because } (\vec{A} \times \vec{B}) \cdot \vec{B} = 0 \text{ (Property of the Scalar Triple Product)}$

$\text{Since } - 2 (\vec{\omega} \times (\vec{v}_i - \vec{v}_j)) \cdot (\vec{r}_i - \vec{r}_j) - 2 (\vec{v}_i - \vec{v}_j) \cdot (\vec{\omega} \times (\vec{r}_i - \vec{r}_j)) = 0 \text{ (Coriolis terms cancel)}$ $\text{Because } (\vec{A} \times \vec{B}) \cdot \vec{C} = \vec{A} \cdot (\vec{B} \times \vec{C}) \text{ (Cyclic Permutation Property of the Scalar Triple Product)}$

$\text{Since } + (\vec{\omega} \times (\vec{\omega} \times (\vec{r}_i - \vec{r}_j))) \cdot (\vec{r}_i - \vec{r}_j) + (\vec{\omega} \times (\vec{r}_i - \vec{r}_j)) \cdot (\vec{\omega} \times (\vec{r}_i - \vec{r}_j)) = 0 \text{ (Centrifugal terms cancel) } [\vec{P} \cdot (\vec{r}_i - \vec{r}_j) + E = 0]$ $\text{Because } \vec{P} = \vec{A} \times (\vec{B} \times \vec{C}) = (\vec{A} \cdot \vec{C}) \ \vec{B} - (\vec{A} \cdot \vec{B}) \ \vec{C} \text{ (Vector Triple Product)}$ $\text{Because } E = (\vec{A} \times \vec{B}) \cdot (\vec{A} \times \vec{B}) = (\vec{A} \cdot \vec{A}) \ (\vec{B} \cdot \vec{B}) - (\vec{A} \cdot \vec{B})^2 \text{ (Lagrange's Identity)}$

$\ddot{\tau}'_{ij} = (\vec{a}_i - \vec{a}_j) \cdot (\vec{r}_i - \vec{r}_j) + (\vec{v}_i - \vec{v}_j) \cdot (\vec{v}_i - \vec{v}_j)$

$\therefore \ddot{\tau}'_{ij} = \ddot{\tau}_{ij}$

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III. Fundamental Relations

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
$\tau_{ij}$	$\tau_{ij} = \frac{1}{2} r_{ij}^2$
$\dot{\tau}_{ij}$	$\dot{\tau}_{ij} = r_{ij} \dot{r}_{ij}$
$\ddot{\tau}_{ij}$	$\ddot{\tau}_{ij} = r_{ij} \ddot{r}_{ij} + \dot{r}_{ij}^2$

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
$\tau_{ij}$	$\tau_{ij} = \frac{1}{2} r_{ij}^2$
$\dot{\tau}_{ij}$	$\dot{\tau}_{ij} = r_{ij} \dot{r}_{ij}$
$\ddot{\tau}_{ij}$	$\ddot{\tau}_{ij} = r_{ij} \ddot{r}_{ij} + \dot{r}_{ij}^2$

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
$\tau_{ij}$	$\tau_{ij} = \frac{1}{2} r_{ij}^2$
$\dot{\tau}_{ij}$	$\dot{\tau}_{ij} = r_{ij} \dot{r}_{ij}$
$\ddot{\tau}_{ij}$	$\ddot{\tau}_{ij} = r_{ij} \ddot{r}_{ij} + \dot{r}_{ij}^2$

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
$\tau_{ij}$	$\tau_{ij} = \frac{1}{2} r_{ij}^2$
$\dot{\tau}_{ij}$	$\dot{\tau}_{ij} = r_{ij} \dot{r}_{ij}$
$\ddot{\tau}_{ij}$	$\ddot{\tau}_{ij} = r_{ij} \ddot{r}_{ij} + \dot{r}_{ij}^2$

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