m000OneDimension.fodt

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   <text:h text:style-name="P3" text:outline-level="1">One Dimensional Lagrange Equations of the First Kind (alternative)</text:h>
   <text:p text:style-name="P1">The basis of this multibody dynamics formulation is the Lagrange Equations of the first kind:</text:p>
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left(
{partial T} 
over 
{partial dot q}_{iq}  
right)
-
{partial T} 
over 
{partial q}_{iq}
-
Q_{iq}
-
sum from{i bar G=1} to{n bar G}
{
{partial {bar G}_{i bar G}} 
over 
{partial dot q}_{iq}
{bar %lambda}_{i bar G}
}
-
sum from{iG=1} to{n_G}
{
{partial { G}_{iG}} 
over 
{partial q}_{iq}
{%lambda}_{iG}
}
=
 0
 ~~~~~~~~
 {iq}=1,dotsaxis, nq</annotation>
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      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mrow>
          <mrow>
           <mfrac>
            <msub>
             <mi mathvariant="italic">dp</mi>
             <mi mathvariant="italic">iq</mi>
            </msub>
            <mi mathvariant="italic">dt</mi>
           </mfrac>
           <mo stretchy="false">−</mo>
           <mfrac>
            <mrow>
             <mo stretchy="false">∂</mo>
             <mi>T</mi>
            </mrow>
            <msub>
             <mrow>
              <mo stretchy="false">∂</mo>
              <mi>q</mi>
             </mrow>
             <mi mathvariant="italic">iq</mi>
            </msub>
           </mfrac>
           <mo stretchy="false">−</mo>
           <msub>
            <mi>Q</mi>
            <msub>
             <mi>i</mi>
             <mi>q</mi>
            </msub>
           </msub>
           <mo stretchy="false">−</mo>
           <mrow>
            <munderover>
             <mo stretchy="false">∑</mo>
             <mrow>
              <mi>i</mi>
              <mrow>
               <mover accent="true">
                <mi>G</mi>
                <mo stretchy="false">¯</mo>
               </mover>
               <mo stretchy="false">=</mo>
               <mn>1</mn>
              </mrow>
             </mrow>
             <mrow>
              <mi>n</mi>
              <mover accent="true">
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              </mover>
             </mrow>
            </munderover>
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              <mrow>
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               <msub>
                <mover accent="true">
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                </mover>
                <mrow>
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                 <mover accent="true">
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                 </mover>
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               </msub>
              </mrow>
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                <mover accent="true">
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                </mover>
               </mrow>
               <mi mathvariant="italic">iq</mi>
              </msub>
             </mfrac>
             <msub>
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               <mo stretchy="false">¯</mo>
              </mover>
              <mrow>
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                <mo stretchy="false">¯</mo>
               </mover>
              </mrow>
             </msub>
            </mrow>
           </mrow>
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           <mrow>
            <munderover>
             <mo stretchy="false">∑</mo>
             <mrow>
              <mi mathvariant="italic">iG</mi>
              <mo stretchy="false">=</mo>
              <mn>1</mn>
             </mrow>
             <msub>
              <mi>n</mi>
              <mi>G</mi>
             </msub>
            </munderover>
            <mrow>
             <mfrac>
              <mrow>
               <mo stretchy="false">∂</mo>
               <msub>
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                <mi mathvariant="italic">iG</mi>
               </msub>
              </mrow>
              <msub>
               <mrow>
                <mo stretchy="false">∂</mo>
                <mi>q</mi>
               </mrow>
               <mi mathvariant="italic">iq</mi>
              </msub>
             </mfrac>
             <msub>
              <mi>λ</mi>
              <mi mathvariant="italic">iG</mi>
             </msub>
            </mrow>
           </mrow>
          </mrow>
          <mo stretchy="false">=</mo>
          <mn>0</mn>
         </mrow>
         <mspace width="16em"/>
         <mrow>
          <mi mathvariant="italic">iq</mi>
          <mo stretchy="false">=</mo>
          <mn>1</mn>
         </mrow>
         <mi>,</mi>
         <mo stretchy="false">⋯</mo>
         <mi>,</mi>
         <mi mathvariant="italic">nq</mi>
        </mrow>
        <annotation encoding="StarMath 5.0">{dp_{iq} over dt}
-
{partial T} 
over 
{partial q}_{iq}
-
Q_{i_q}
-
sum from{i bar G=1} to{n bar G}
{
{partial {bar G}_{i bar G}} 
over 
{partial dot q}_{iq}
{bar %lambda}_{i bar G}
}
-
sum from{iG=1} to{n_G}
{
{partial { G}_{iG}} 
over 
{partial q}_{iq}
{%lambda}_{iG}
}
=
 0
 ~~~~~~~~
 {iq}=1,dotsaxis, nq</annotation>
       </semantics>
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%delta W
=
left(
sum from if to nf 
{bold nitalic F}^T_{Oif}
{partial {bold nitalic r}_{OifO}}
over
{partial {bold nitalic q}}
right)
%delta {bold nitalic q}


newline
&quot;&quot;~~~~
=
{bold nitalic Q}^T_{q}
%delta {bold nitalic q}
</annotation>
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   <text:p text:style-name="P2">Note that<draw:frame draw:style-name="fr1" draw:name="Object35" text:anchor-type="as-char" svg:width="0.2311in" svg:height="0.2201in" draw:z-index="16"><draw:object>
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   <text:h text:style-name="P3" text:outline-level="1">Generalized Force,<draw:frame draw:style-name="fr1" draw:name="Object40" text:anchor-type="as-char" svg:width="0.2118in" svg:height="0.1929in" draw:z-index="20"><draw:object>
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   <text:p text:style-name="P2"><draw:frame draw:style-name="fr1" draw:name="Object41" text:anchor-type="as-char" svg:width="1.1929in" svg:height="0.461in" draw:z-index="21"><draw:object>
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        <annotation encoding="StarMath 5.0">{bold nitalic Q}_{IX}
=
sum from if to {force on Ip}
{
{bold nitalic F}_{Oif}
}</annotation>
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=
sum from if to {force on Ip}
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over
{partial {bold nitalic v}}
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   <text:h text:style-name="P4" text:outline-level="2">Partial Derivative of<draw:frame draw:style-name="fr1" draw:name="Object51" text:anchor-type="as-char" svg:width="0.2043in" svg:height="0.1898in" draw:z-index="27"><draw:object>
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   <text:h text:style-name="P3" text:outline-level="1">Generalized Force due to Reaction,<draw:frame draw:style-name="fr1" draw:name="Object67" text:anchor-type="as-char" svg:width="0.2925in" svg:height="0.2311in" draw:z-index="29"><draw:object>
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{partial {bold nitalic Q}_{reJE}}
over
{partial {bold nitalic q}_{JE}}

=
{partial {bold nitalic r}^T_JpIeJp}
over
{partial {bold nitalic q}_{JE}}
{partial left [{bold nitalic A} right]^T_OJp}
over
{partial {bold nitalic q}_{JE}}
{bold nitalic F}_{JeO}

+
{bold nitalic r}^T_JpIeJp
{partial left [{bold nitalic A} right]^T_OJp}
over
{partial {bold nitalic q}_{JE}}
{partial {bold nitalic F}_{JeO}}
over
{partial {bold nitalic q}_{JE}}

newline
&quot;&quot;~~~~~~~~~~
+
{partial^2 {bold nitalic %omega}^T_{OJeO}}
over
{partial {bold nitalic v}_{JE} partial {bold nitalic q}_{JE}}
{bold nitalic T}_{JeO}
+
{partial {bold nitalic %omega}^T_{OJeO}}
over
{partial {bold nitalic v}_{JE}}
{partial {bold nitalic T}_{JeO}}
over
{partial {bold nitalic q}_{JE}}

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