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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. These two approaches Newton s Law and lagrange s equations are totally compatible.

Simple Pendulum by Lagrange’s Equations We first apply Lagrange’s equation to derive the equations of motion of a simple pendulum in polar coor­ dinates. This is a one degree of freedom system. However, it is convenient for later analysis of the double pendulum, to begin by describing the position of the mass point m 1 with cartesian ...

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