739: Motion

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definition of motion

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of normed vectors space.

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Target Context

• The reader will have a definition of motion.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$F_1$: $\in \{\mathbb{R},\mathbb{C}\}$, with the canonical field structure
$F_2$: $\in \{\mathbb{R},\mathbb{C}\}$, with the canonical field structure
$V_1$: $\in \{\text{ the normed }{F}_1\text{ vectors spaces }\}$
$V_2$: $\in \{\text{ the normed }{F}_2\text{ vectors spaces }\}$
$\ast f$: $:V_1\to V_2$
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Conditions:
$\mathrm{\forall }v,v^{\prime }\in V_1(\Vert v-v^{\prime }\Vert =\Vert f(v)-f(v^{\prime })\Vert )$
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2: Natural Language Description

For any $F_1\in \{\mathbb{R},\mathbb{C}\}$, with the canonical field structure, any $F_2\in \{\mathbb{R},\mathbb{C}\}$, with the canonical field structure, any normed $F_1$ vectors space, $V_1$, and any normed $F_2$ vectors space, $V_2$, any map, $f:V_1\to V_2$, such that $\mathrm{\forall }v,v^{\prime }\in V_1(\Vert v-v^{\prime }\Vert =\Vert f(v)-f(v^{\prime })\Vert )$

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3: Note

$f$ is not linear in general. For example, $V_1$ and $V_2$ are the Euclidean normed vectors spaces, ${\mathbb{R}}^n$, and the translation, $f:v\mapsto v+v_0$, where $v_0\ne 0$, is a motion, which is not linear, because $f(0)=v_0$, so, $f(r0)=f(0)=v_0\ne r{v}_0=rf(0)$.

$F_1$ and $F_2$ do not need to be the same, because $f$ is not supposed to be linear.

It is $\Vert f(v)-f(v^{\prime })\Vert$ not $\Vert f(v-v^{\prime })\Vert$: they can be different, because $f$ is not linear in general. For example, $V_1$ and $V_2$ are the Euclidean normed vectors spaces, ${\mathbb{R}}^n$, and for the translation, $f:v\mapsto v+v_0$, where $v_0\ne 0$, $\Vert v-v\Vert =\Vert 0\Vert =0\ne \Vert v_0\Vert =\Vert f(0)\Vert =\Vert f(v-v)\Vert$.

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References

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