o001Explode.fodt

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   <text:h text:style-name="Heading_20_3" text:outline-level="3">Partial Derivative of<draw:frame draw:style-name="fr1" draw:name="Object18" text:anchor-type="as-char" svg:width="0.372in" svg:height="0.1937in" draw:z-index="7"><draw:object>
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   <text:h text:style-name="Heading_20_2" text:outline-level="2">Relative Position</text:h>
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{bold nitalic r}_j
-
{bold nitalic r}_i
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   <text:h text:style-name="Heading_20_3" text:outline-level="3">Partial Derivative of<draw:frame draw:style-name="fr1" draw:name="Object23" text:anchor-type="as-char" svg:width="0.211in" svg:height="0.2201in" draw:z-index="15"><draw:object>
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    </draw:frame></text:p>
   <text:h text:style-name="Heading_20_2" text:outline-level="2">Relative Distance</text:h>
   <text:p text:style-name="Standard">Refer to C:\askoh\mbdtheory\MbDsxwXEVOme\2020kineij.odt</text:p>
   <text:p text:style-name="equation"><draw:frame draw:style-name="fr1" draw:name="Object9" text:anchor-type="as-char" svg:width="0.7528in" svg:height="0.2626in" draw:z-index="11"><draw:object>
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          <mi>r</mi>
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            <mi mathvariant="italic">ij</mi>
           </msub>
          </mrow>
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        </mrow>
        <annotation encoding="StarMath 5.0">r_ij
=
sqrt {
{bold nitalic r}^T_ij
{bold nitalic r}_ij
}
</annotation>
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      </office:binary-data>
     </draw:image>
     <svg:desc>formula</svg:desc>
    </draw:frame></text:p>
   <text:h text:style-name="Heading_20_3" text:outline-level="3">Partial Derivative of<draw:frame draw:style-name="fr1" draw:name="Object32" text:anchor-type="as-char" svg:width="0.2126in" svg:height="0.2173in" draw:z-index="25"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <msub>
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        </msub>
        <annotation encoding="StarMath 5.0">r_ij</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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      </office:binary-data>
     </draw:image>
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       <semantics>
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        </mstyle>
        <annotation encoding="StarMath 5.0">{bold nitalic q}</annotation>
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      </math>
     </draw:object><draw:image>
      <office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAb4BAADr
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        </mrow>
        <annotation encoding="StarMath 5.0">{partial r_ij}
over
{partial {bold nitalic %alpha}}
=
1 over {r_ij}
{partial {bold nitalic r}^T_ij}
over
{partial {bold nitalic %alpha}}
{bold nitalic r}_ij
</annotation>
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         </mrow>
        </mrow>
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over
{partial {bold nitalic q}_i}
=
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</annotation>
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{{bold nitalic r}^T_ij}
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     <svg:desc>formula</svg:desc>
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-
{bold nitalic r}_0i
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over
{partial {bold nitalic q}_i}
=
{{bold nitalic r}_{0i i}} over {r_{0i i}}


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   <text:h text:style-name="Heading_20_2" text:outline-level="2">Potential Energy of an assembly of parts is</text:h>
   <text:p text:style-name="Standard">https://en.wikipedia.org/wiki/Elastic_energy</text:p>
   <text:p text:style-name="Standard"><text:a xlink:type="simple" xlink:href="https://en.wikipedia.org/wiki/Electric_potential_energy" text:style-name="Internet_20_link" text:visited-style-name="Visited_20_Internet_20_Link">https://en.wikipedia.org/wiki/Electric_potential_energy</text:a></text:p>
   <text:p text:style-name="Standard">https://en.wikipedia.org/wiki/Lagrange_multiplier</text:p>
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=
{1 over 2}k
sum from i=1 to n
m_{i}
r^2_{0i i}
+
{1 over 2}c
sum from i=1 to n
sum from j=1 to {n(j neq i)}
left ({
m_{i}
m_{j}}
over
{
r_ij
}
right)
+
{1 over 2}
sum from i=1 to n
%lambda_i
G_i</annotation>
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   <text:h text:style-name="Heading_20_2" text:outline-level="2">Minimizing Potential Energy</text:h>
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   <text:h text:style-name="Heading_20_2" text:outline-level="2">Newton-Raphson’s method.</text:h>
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matrix
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over
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left lbrace
matrix
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right rbrace_i


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{partial {bold nitalic q}_{i} partial {bold nitalic q}_{i}}

=
km_{i} left[{bold nitalic I}right]
+
{1 over 2}c
sum from j=1 to {n(j neq i)}
left(
-3
{m_{i}m_{j} {bold nitalic r}_{ij}}
over
{r^4_ij}
{partial {r_ij}}
over
{partial {bold nitalic q}_{i}}
+
{m_{i}m_{j}}
over
{r^3_ij}
{partial {bold nitalic r}_{ij}}
over
{partial {bold nitalic q}_{i}}
right)
+
%lambda_i 
{partial {bold nitalic r}_{i}}
over
{partial {bold nitalic q}_{i}}




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left(
3
{m_{i}m_{j} {bold nitalic r}_{ij} {bold nitalic r}^T_{ij}}
over
{r^5_ij}
-
{m_{i}m_{j}}
over
{r^3_ij}
left[{bold nitalic I}right]
right)
+
%lambda_i 
left[{bold nitalic I}right]




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over
{partial {bold nitalic q}_{i} partial {bold nitalic q}_{j}}
=
{1 over 2}c
left(
-3
{m_{i}m_{j} {bold nitalic r}_{ij} {bold nitalic r}^T_{ij}}
over
{r^5_ij}
+
{m_{i}m_{j}}
over
{r^3_ij}
left[{bold nitalic I}right]
right)




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over
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{
{{bold nitalic r}_i}
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        <annotation encoding="StarMath 5.0">{partial^2 V}
over
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=
{
{{bold nitalic r}^T_i}
}
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        <mrow>
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         </mstyle>
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        <annotation encoding="StarMath 5.0">{partial^2 V}
over
{partial {bold nitalic %lambda} partial {bold nitalic %lambda}}
=
{
{{bold nitalic 0}}
}
</annotation>
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     </draw:image>
     <svg:desc>formula</svg:desc>
    </draw:frame></text:p>
   <text:h text:style-name="Heading_20_2" text:outline-level="2">Example with three parts.</text:h>
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       <semantics>
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         <mi>k</mi>
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          <mrow>
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           <mrow>
            <mrow>
             <msub>
              <mi>m</mi>
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             </msub>
             <mrow>
              <msubsup>
               <mi>r</mi>
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                <mn>01</mn>
                <mn>1</mn>
               </mrow>
               <mn>2</mn>
              </msubsup>
              <mo stretchy="false">+</mo>
              <msub>
               <mi>m</mi>
               <mn>2</mn>
              </msub>
             </mrow>
             <mrow>
              <msubsup>
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        </mrow>
        <annotation encoding="StarMath 5.0">V
=
{1 over 2}k
left (
m_{1} r^2_{01 1}
+
m_{2} r^2_{02 2}
+
m_{3} r^2_{03 3}
right)
+
{1 over 2}c
left (
{m_{1}m_{2}} over {r_{01 2}}
+
{m_{1}m_{3}} over {r_{01 3}}
+
{m_{2}m_{1}} over {r_{02 1}}
+
{m_{2}m_{3}} over {r_{02 3}}
+
{m_{3}m_{1}} over {r_{03 1}}
+
{m_{3}m_{2}} over {r_{03 2}}
right)
+
{1 over 2}
left(%lambda_1 G_1 + %lambda_2 G_2 + %lambda_3 G_3 right)


</annotation>
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