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<!DOCTYPE html>
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    <title>Relational Magnitudes</title>
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        <h1>Relational Magnitudes</h1>

        <blockquote>
            Relational Magnitudes are invariant vectorial quantities that conserve their value and form under transformations of translation and rotation.
        </blockquote>

        <hr>

        <h2>I. Definitions I (Relational Magnitudes)</h2>

        <p>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:</p>

        <div class="math-block">
            $\mathbf{r}_i \doteq \vec{r}_i$
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        <div class="math-block">
            $\mathbf{v}_i \doteq \dfrac{d(\vec{r}_i)}{dt} = \vec{v}_i$
        </div>
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            $\mathbf{a}_i \doteq \dfrac{d^2(\vec{r}_i)}{dt^2} = \vec{a}_i$
        </div>

        <p>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.</p>

        <p><strong>Note</strong></p>
        <p>The Relational (Vectorial) Magnitudes are always the same as the Ordinary (Vectorial) Magnitudes in the Auxiliary Reference Frame.</p>

        <hr>

        <h2>II. Definitions II (Relational Magnitudes)</h2>

        <p>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:</p>

        <div class="math-block">
            $\mathbf{r}_i \doteq \vec{r}_i - \vec{R}$
        </div>
        <div class="math-block">
            $\mathbf{v}_i \doteq (\vec{v}_i - \vec{V}) - \vec{\omega} \times (\vec{r}_i - \vec{R})$
        </div>
        <div class="math-block">
            $\mathbf{a}_i \doteq (\vec{a}_i - \vec{A}) - 2\vec{\omega} \times (\vec{v}_i - \vec{V}) + \vec{\omega} \times [\ \vec{\omega} \times (\vec{r}_i - \vec{R})\ ] - \vec{\alpha} \times (\vec{r}_i - \vec{R})$
        </div>

        <p>Where:</p>
        <ul>
            <li>$\vec{r}_i, \vec{v}_i, \vec{a}_i$ are the ordinary vectorial position, velocity, and acceleration of particle $i$ with respect to the Frame $S$.</li>
            <li>$\vec{R}, \vec{V}, \vec{A}$ are the position, velocity, and acceleration of the Auxiliary Frame's origin with respect to $S$.</li>
            <li>$\vec{\omega}$ and $\vec{\alpha}$ are the angular velocity and angular acceleration of the Auxiliary Frame with respect to $S$.</li>
        </ul>

        <hr>

        <h2>III. Transformations (Invarianza$\cdot$Relations)</h2>

        <p>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:</p>

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            $\mathbf{r}_i \doteq (\vec{r}_i - \vec{R}) = \mathbf{r}'_i$
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            $\mathbf{r}'_i \doteq (\vec{r}'_i - \vec{R}') = \mathbf{r}_i$
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            $\mathbf{v}_i \doteq (\vec{v}_i - \vec{V}) - \vec{\omega} \times (\vec{r}_i - \vec{R}) = \mathbf{v}'_i$
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            $\mathbf{v}'_i \doteq (\vec{v}'_i - \vec{V}') - \vec{\omega}' \times (\vec{r}'_i - \vec{R}') = \mathbf{v}_i$
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            $\mathbf{a}_i \doteq (\vec{a}_i - \vec{A}) - 2\vec{\omega} \times (\vec{v}_i - \vec{V}) + \vec{\omega} \times [\ \vec{\omega} \times (\vec{r}_i - \vec{R})\ ] - \vec{\alpha} \times (\vec{r}_i - \vec{R}) = \mathbf{a}'_i$
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            $\mathbf{a}'_i \doteq (\vec{a}'_i - \vec{A}') - 2\vec{\omega}' \times (\vec{v}'_i - \vec{V}') + \vec{\omega}' \times [\ \vec{\omega}' \times (\vec{r}'_i - \vec{R}')\ ] - \vec{\alpha}' \times (\vec{r}'_i - \vec{R}') = \mathbf{a}_i$
        </div>

        <hr>

        <h2>IV. Bibliography</h2>

        <ol>
            <li>A. Blatter, <em>A Reformulation of Classical Mechanics</em> (2015). <a href="https://atorassa.github.io/physics-authors/blatter/english/pdf/09.pdf">[PDF]</a></li>
            <li>A. Tobla, <em>A Reformulation of Classical Mechanics</em> (2024). <a href="https://atorassa.github.io/physics-authors/tobla/english/pdf/02.pdf">[PDF]</a></li>
        </ol>

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