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\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
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Newtonian Mechanics suffers from two important limitations: motions |
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can only be described in cartesian coordinate systems. Moreover, It |
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become impossible to predict analytically the properties of the |
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can only be described in cartesian coordinate systems. Moreover, it |
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becomes impossible to predict analytically the properties of the |
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system even if we know all of the details of the interaction. In |
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order to overcome some of the practical difficulties which arise in |
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attempts to apply Newton's equation to complex system, approximate |
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Using \ref{introEqaution:RBMotionPI}, one can construct a skew |
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matrix, |
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|
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\begin{eqnarry*} |
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\begin{eqnarray*} |
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(\dot \Pi - \dot \Pi ^T ){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ |
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}}(J^{ - 1} \Pi + \Pi J^{ - 1} ) + \sum\limits_i {[Q^T F_i |
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(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 |