Transcription of Lecture L20 - Energy Methods: Lagrange’s
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Lecture L20 - Energy Methods: Lagrange’s
S. Widnall Dynamics Fall 2009 Version Lecture L20 - Energy Methods: Lagrange s Equations The motion of particles and rigid bodies is governed by Newton s law. In this section, we will derive an alternate approach, placing Newton s law into a form particularly convenient for multiple degree of freedom systems or systems in complex coordinate systems. This approach results in a set of equations called Lagrange s equations. They are the beginning of a complex, more mathematical approach to mechanics called analytical dynamics. In this course we will only deal with this method at an elementary level. Even at this simplified level, it is clear that considerable simplification occurs in deriving the equations of motion for complex systems.
dt Equating like terms from our manipulations on kinetic energy and the potential of a conservative force field, we write d ∂T ∂V dt ∂x˙ i = − ∂x i (6) Now we make use of the fact that ∂T = 0 (7) ∂x i and ∂V = 0 (8) ∂x˙ i Using these results, we can rewrite Equation (6) as dt d ∂(T ∂x − ˙ i V ) − ∂(T ∂x − i V )
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