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\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
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|
| 75 |
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Newtonian Mechanics suffers from two important limitations: motions |
| 76 |
< |
can only be described in cartesian coordinate systems. Moreover, It |
| 77 |
< |
become impossible to predict analytically the properties of the |
| 76 |
> |
can only be described in cartesian coordinate systems. Moreover, it |
| 77 |
> |
becomes impossible to predict analytically the properties of the |
| 78 |
|
system even if we know all of the details of the interaction. In |
| 79 |
|
order to overcome some of the practical difficulties which arise in |
| 80 |
|
attempts to apply Newton's equation to complex system, approximate |
| 1312 |
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Using \ref{introEqaution:RBMotionPI}, one can construct a skew |
| 1313 |
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matrix, |
| 1314 |
|
|
| 1315 |
< |
\begin{eqnarry*} |
| 1315 |
> |
\begin{eqnarray*} |
| 1316 |
|
(\dot \Pi - \dot \Pi ^T ){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ |
| 1317 |
|
}}(J^{ - 1} \Pi + \Pi J^{ - 1} ) + \sum\limits_i {[Q^T F_i |
| 1318 |
|
(r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - (\Lambda - \Lambda ^T ). |
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The equations of motion corresponding to potential energy and |
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kinetic energy are listed in the below table, |
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\begin{table} |
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< |
\caption{Equations of motion due to Potential and Kinetic Energies} |
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> |
\caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES} |
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\begin{center} |
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\begin{tabular}{|l|l|} |
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\hline |
