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Hallo Rafael,<br>
Sorry for my delayed answer! Your mail and your next one ("Bitte")
accidently have gone into my recycle bin - I still have to find out
why because in the case of your mail this false allocation is really
annoying.<br>
<br>
Your approach seems to be correct. I have done a couple of
simulations with results that do not arouse suspicion anythng could
be wrong. From the point of kinetics I find "<span lang="ES"><span>-</span><span
class="SpellE"><span>fc</span></span><span>*</span><span>u</span><span>/</span><b><span>m</span></b>
<span>+</span> <span>Ft</span><span>/</span><b><span>m</span></b><span>;"</span></span>
plausible. Probably your simulation model is appropriate and cannot
be falsified. <br>
Looking for sources I only find Lagrangian and Hamiltonian
approaches in sperical coordinates - the results of which often
cause numerical problems. <br>
<br>
Did you find the cartesian model anywhere in literature or web or
did it just come from your inspiration and command on the subject of
classical mechanics?<br>
<br>
And thank you very much for your response again and sorry for the
delay too.<br>
<br>
Best regards<br>
Jens <br>
----------------------------------------------------------------<br>
<br>
<br>
<div class="moz-cite-prefix">Am 11.08.2018 02:24, schrieb Rafael
Guerra:<br>
</div>
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<p class="MsoNormal"><span>Hi Jens,</span></p>
<p class="MsoNormal"><span> </span></p>
<p class="MsoNormal"><span>I have got the particle moving on the
sphere by writing the total acceleration as a sum of a
centripetal component and a tangential component:</span></p>
<p class="MsoNormal"><span> </span></p>
<pre><i><span>// START OF CODE</span></i><span></span></pre>
<pre><i><span>// Particle motion on a sphere using cartesian coordinates</span></i><span></span></pre>
<pre><span> </span></pre>
<pre><span class="SpellE"><span lang="FR">function</span></span><span lang="FR"> <b><span>dz</span></b><span>=</span><span class="SpellE"><u><span>EoM</span></u></span><span>(</span><b><span>t</span></b><span>, </span><b><span>z</span></b><span>, </span><b><span>m</span></b><span>, </span><b><span>f</span></b><span>)</span><i><span>//z=[x; y; z;<span> </span>vx; <span class="SpellE">vy</span>; <span class="SpellE">vz</span>]<span> </span>(6 x 1)</span></i></span></pre>
<pre><span lang="FR"><span> </span></span><span class="SpellE"><b><span lang="ES">dz</span></b></span><span lang="ES">(</span><span lang="ES">1</span><span lang="ES">:</span><span lang="ES">3</span><span lang="ES">)</span><span lang="ES">=</span><span lang="ES"> <b><span>z</span></b><span>(</span><span>4</span><span>:</span><span>6</span><span>)</span><span>;</span></span></pre>
<pre><span lang="ES"><span> </span><span class="SpellE"><span>nz</span></span> <span>=</span> <span class="SpellE"><span>norm</span></span><span>(</span><b><span>z</span></b><span>(</span><span>1</span><span>:</span><span>3</span><span>))</span><span>;</span></span></pre>
<pre><span lang="ES"><span> </span><span>u</span> <span>=</span> <b><span>z</span></b><span>(</span><span>1</span><span>:</span><span>3</span><span>)</span><span>/</span><span class="SpellE"><span>nz</span></span><span>;</span></span></pre>
<pre><span lang="ES"><span> </span></span><span>fc</span><span>=</span><span> <b><span>m</span></b><span>*</span><span>norm</span><span>(</span><b><span>z</span></b><span>(</span><span>4</span><span>:</span><span>6</span><span>))</span><span>^</span><span>2</span><span>/</span><span class="SpellE"><span>nz</span></span><span>;</span><span> </span><i><span>// centripetal force</span></i></span></pre>
<pre><span lang="ES"><span> </span><span>Ft</span><span>=</span> <b><span>f</span></b> <span>-</span> <span>(</span><b><span>f</span></b><span>'*</span><span>u</span><span>)</span><span>*</span><span>u;</span><span> </span><i><span>// <span class="SpellE">tangential</span> <span class="SpellE">force</span></span></i></span></pre>
<pre><span lang="ES"><span> </span><span class="SpellE"><b><span>dz</span></b></span><span>(</span><span>4</span><span>:</span><span>6</span><span>)</span><span>=</span> <span>-</span><span class="SpellE"><span>fc</span></span><span>*</span><span>u</span><span>/</span><b><span>m</span></b> <span>+</span> <span>Ft</span><span>/</span><b><span>m</span></b><span>;</span></span></pre>
<pre><span class="SpellE"><span>endfunction</span></span><span></span></pre>
<pre><span> </span></pre>
<pre><span>R</span><span>=</span><span>2</span><span>;</span><span><span> </span><i><span>// sphere radius</span></i></span></pre>
<pre><span>r0</span><span> <span>=</span> <span>[</span><span>0</span><span>;</span> <span>0</span><span>;</span> <span>R</span><span>]</span><span>;</span><span> </span><i><span>// initial position</span></i></span></pre>
<pre><span>v0</span><span> <span>=</span> <span>[</span><span>-</span><span>1</span><span>;</span> <span>3</span><span>;</span> <span>0</span><span>]</span><span>;</span> <i><span>// must be tangent to the sphere at r0</span></i></span></pre>
<pre><span lang="FR">z0</span><span lang="FR">=</span><span lang="FR">[</span><span lang="FR">r0;v0</span><span lang="FR">]</span><span lang="FR">;</span><span lang="FR"></span></pre>
<pre><span lang="FR">t0</span><span lang="FR">=</span><span lang="FR">0</span><span lang="FR">;</span><span lang="FR"></span></pre>
<pre><span class="SpellE"><span lang="FR">dt</span></span><span lang="FR">=</span><span lang="FR">0.05</span><span lang="FR">;</span><span lang="FR"></span></pre>
<pre><span class="SpellE"><span lang="FR">tmax</span></span><span lang="FR"> <span>=</span> <span>50</span><span>;</span></span></pre>
<pre><span lang="FR">t</span><span lang="FR">=</span><span class="SpellE"><span lang="FR">dt</span><span lang="FR">:</span><span lang="FR">dt</span><span lang="FR">:</span><span lang="FR">tmax</span></span><span lang="FR">;</span><span lang="FR"></span></pre>
<pre><span lang="FR">m</span><span lang="FR"> <span>=</span> <span>1</span><span>;</span><span> </span><i><span>// mass</span></i></span></pre>
<pre><span lang="FR">f</span><span lang="FR">=</span><span lang="FR"> <span>[</span><span>-</span><span>1</span><span>;</span><span>1</span><span>;</span><span>0.5</span><span>]</span><span>;</span> <i><span>// <span class="SpellE">external</span> force</span></i></span></pre>
<pre><span lang="FR">z</span><span lang="FR"> <span>=</span> <span>ode</span><span>(</span><span>z0,t0,t,<u>EoM</u></span><span>)</span></span></pre>
<pre><span class="SpellE"><span>clf</span></span><span>;</span><span></span></pre>
<pre><span>scatter3</span><span>(</span><span>z</span><span>(</span><span>1</span><span>,</span><span>:</span><span>)</span><span>,z</span><span>(</span><span>2</span><span>,</span><span>:</span><span>)</span><span>,z</span><span>(</span><span>3</span><span>,</span><span>:</span><span>))</span><span></span></pre>
<pre><span class="SpellE"><span>gca</span></span><span>()</span><span>.</span><span class="SpellE"><span>axes_reverse</span></span><span> <span>=</span> <span>[</span><span>"on"</span><span>,</span> <span>"off"</span><span>,</span> <span>"off"</span><span>]</span><span>;</span></span></pre>
<pre><span class="SpellE"><span>isoview</span></span><span> <span>on</span></span></pre>
<pre><i><span>// END OF CODE</span></i><span></span></pre>
<p class="MsoNormal"><span> </span></p>
<p class="MsoNormal"><span>Let me know if you see any flaw in
the physics.</span></p>
<p class="MsoNormal"><span> </span></p>
<p class="MsoNormal"><span>Regards,</span></p>
<p class="MsoNormal"><span>Rafael</span></p>
<p class="MsoNormal"><span> </span></p>
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