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lecture_17

# lecture_17 - Lecture 17 Lagrangian Mechanics We use the...

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Lecture 17: Lagrangian Mechanics We use the approach of Lagrangian mechanics to provide an easy way of deriving equations of motions for complicated interconnected systems. All we need to know to use this approach is the kinetic energy T of the system and its potential energy V . Once we know these, we form L = T - V and take some derivatives in a way defined by the so-called Euler-Lagrange equations to obtain equations of motion for the system. Next we describe how those equations are derived. The coordinates used for the description of motion of the system, that are not the regular Cartesian coordinates x, y, z in the three- dimensional space are referred to as generalized coordinates . One exam- ple is the polar coordinates that you have used in description of many systems that rotate around an axis. The systems for which internal Figure 1: coordinates (think angle θ describing rotation of a disk) completely de- termine the position of the system in space. A disk restricted to rotate around a pin is a holonomic system. A disk rotating on a floor is not. For holonomic systems: the number of generalized coordinates is equal to the number of Cartesian coordinates minus the number of constraint equations.

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