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.).

---

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 ($\ddot{\tau}_{ij}$)	$\ddot{\tau}_{ij} \doteq \vec{a}_{ij} \cdot \vec{r}_{ij} + \vec{v}_{ij} \cdot \vec{v}_{ij}$	$\ddot{\tau}_{ij} \doteq \frac{d^2(\tau_{ij})}{dt^2}$

---

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:

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

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

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

Where $\vec{r}_i$, $\vec{v}_i$, and $\vec{a}_i$ are the vectorial position, velocity, and acceleration of particle $i$ with respect to 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:

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

---

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.

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

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

$\text{Since } (\vec{r}_i - \vec{r}_j) = (\vec{r}'_i - \vec{r}'_j)$

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

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

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

---

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.

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

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

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

$\text{Since } (-\ \vec{\omega} \times (\vec{r}_i - \vec{r}_j)) \cdot (\vec{r}_i - \vec{r}_j) = 0 \text{ (The rotational term is orthogonal to the relative position vector.)}$

$\text{Because } (\vec{A} \times \vec{B}) \cdot \vec{B} = 0 \text{ (Property of the Scalar Triple Product)}$

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

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

---

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.

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

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

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

$\text{Since } - (\vec{\alpha} \times (\vec{r}_i - \vec{r}_j)) \cdot (\vec{r}_i - \vec{r}_j) = 0 \text{ (Angular acceleration term cancels)}$

$\text{Because } (\vec{A} \times \vec{B}) \cdot \vec{B} = 0 \text{ (Property of the Scalar Triple Product)}$

$\text{Since } - 2 (\vec{\omega} \times (\vec{v}_i - \vec{v}_j)) \cdot (\vec{r}_i - \vec{r}_j) - 2 (\vec{v}_i - \vec{v}_j) \cdot (\vec{\omega} \times (\vec{r}_i - \vec{r}_j)) = 0 \text{ (Coriolis terms cancel)}$

$\text{Because } (\vec{A} \times \vec{B}) \cdot \vec{C} = \vec{A} \cdot (\vec{B} \times \vec{C}) \text{ (Cyclic Permutation Property of the Scalar Triple Product)}$

$\text{Since } + (\vec{\omega} \times (\vec{\omega} \times (\vec{r}_i - \vec{r}_j))) \cdot (\vec{r}_i - \vec{r}_j) + (\vec{\omega} \times (\vec{r}_i - \vec{r}_j)) \cdot (\vec{\omega} \times (\vec{r}_i - \vec{r}_j)) = 0 \text{ (Centrifugal terms cancel)}$

$\text{Since } + \vec{P} \cdot (\vec{r}_i - \vec{r}_j) + E = 0$

$\text{Because } \vec{P} = \vec{A} \times (\vec{B} \times \vec{C}) = (\vec{A} \cdot \vec{C}) \ \vec{B} - (\vec{A} \cdot \vec{B}) \ \vec{C} \text{ (Vector Triple Product)}$

$\text{Because } E = (\vec{A} \times \vec{B}) \cdot (\vec{A} \times \vec{B}) = (\vec{A} \cdot \vec{A}) \ (\vec{B} \cdot \vec{B}) - (\vec{A} \cdot \vec{B})^2 \text{ (Lagrange's Identity)}$

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

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

---

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

---

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