4200ic2.fodt

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left(
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   <text:p text:style-name="equation">Taking partial derivatives wrt<draw:frame draw:style-name="fr1" draw:name="Object34" text:anchor-type="as-char" svg:width="0.1701in" svg:height="0.1929in" draw:z-index="10"><draw:object>
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   <text:p text:style-name="Standard">Store user input of initial conditions for repeatability or restart.</text:p>
   <text:p text:style-name="Standard">Use mass and moment of inertias as weights for least squares improvement to initial conditions.</text:p>
   <text:p text:style-name="Standard">If an EXACT coordinate is requested, increase its weight by 1000 times.</text:p>
   <text:p text:style-name="Standard">Remove redundant constraints.</text:p>
   <text:h text:style-name="Heading_20_2" text:outline-level="2">Velocity Initial Conditions</text:h>
   <text:p text:style-name="Standard">Similarly, the initial state velocities must satisfy</text:p>
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   <text:h text:style-name="Heading_20_2" text:outline-level="2">Acceleration and Lambda Initial Conditions</text:h>
   <text:p text:style-name="Standard">Although not strictly necessary, the initial state accelerations and lambdas facilitates the accurate starting of the integration method. The initial state accelerations must satisfy</text:p>
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{partial {bold nitalic dot G}}
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{partial {bold nitalic dot y}}
{bold nitalic ddot y}
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{partial {bold nitalic dot G}}
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{partial t}

newline
&quot;&quot;~~~~~~~~~~~~~
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{partial {bold nitalic G}}
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{partial {bold nitalic y}}
{bold nitalic ddot y}
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   <text:p text:style-name="equation">The lambdas can be found from the definition of the generalized momenta and the substituted Lagrange’s Equations.</text:p>
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   <text:p text:style-name="equation">Differentiating wrt time</text:p>
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   <text:p text:style-name="equation">Substitute into Lagrange’s Equations</text:p>
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    </draw:frame></text:p>
   <text:p text:style-name="equation">Note that the equations are linear if there is not friction.</text:p>
   <text:p text:style-name="Standard">The intial value of<draw:frame draw:style-name="fr1" draw:name="Object23" text:anchor-type="as-char" svg:width="0.1783in" svg:height="0.1929in" draw:z-index="30"><draw:object>
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       <semantics>
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        <annotation encoding="StarMath 5.0">{bold nitalic dot p}</annotation>
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     </draw:object><draw:image>
      <office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAcUBAADr
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       AP//AAAAAAgAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAIgAAQACAAAAAQCHAAEABQAAAP////8A
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       AAByAAIAFwAAAGoAAACNAQAAAQBw7QAAAAAA//8BAHAAjAABAAAAAACVAAEABAAAAAAAAACW
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    </draw:frame>can be found by substituting<draw:frame draw:style-name="fr1" draw:name="Object24" text:anchor-type="as-char" svg:width="0.1752in" svg:height="0.1929in" draw:z-index="31"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mstyle mathvariant="bold">
         <mover accent="true">
          <mi mathvariant="normal">q</mi>
          <mo stretchy="false">¨</mo>
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        <annotation encoding="StarMath 5.0">{bold nitalic ddot q}</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
      <office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAb0BAADr
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       AAEAAgAAAAkAiwABAAIAAAAfAIoAAQA6AAAAAgA0AAAACgBTdGFyU3ltYm9sAAAAAAAApwEA
       AP//AAAAAAgAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAIgAAQACAAAAAQCHAAEABQAAAP////8A
       hgABAAQAAAAAAAAAcgACABoAAACfAAAAjQEAAAEAAACoAHcAAAAAAP//AQCoAIwAAQAAAAAA
       iwABAAIAAAAfAIoAAQA5AAAAAgAzAAAACQBUaG9ybmRhbGUAAAAAAACnAQAAAAAAAAAACAAA
       AAAAAAD/AwAAAAAAAAAAAP8DAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAA
       AAByAAIAFwAAAGoAAACNAQAAAQBx7QAAAAAA//8BAHEAjAABAAAAAACVAAEABAAAAAAAAACW
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     </draw:image>
    </draw:frame>in an equation above.</text:p>
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  </office:text>
 </office:body>
</office:document>