Transcription of Lagrangian Mechanics - Physics Courses
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Lagrangian Mechanics - Physics Courses
Chapter 6 Lagrangian Generalized CoordinatesA set ofgeneralized coordinatesq1, .. , qncompletely describes the positions of all particlesin a mechanical system. In a system withdfdegrees of freedom andkconstraints,n=df kindependent generalized coordinates are needed to completely specify all the positions. Aconstraint is a relation among coordinates, such asx2+y2+z2=a2for a particle movingon a sphere of radiusa. In this case,df= 3 andk= 1. In this case, we could eliminatezin favor ofxandy, writingz= a2 x2 y2, or we could choose as coordinatesthe polar and azimuthal angles and .For the moment we will assume thatn=df k, and that the generalized coordinates areindependent, satisfying no additional constraints among them. Later on we will learn howto deal with any remaining constraints among the{q1.}
2 CHAPTER 6. LAGRANGIAN MECHANICS 6.2 Hamilton’s Principle The equations of motion of classical mechanics are embodied in a variational principle, called Hamilton’s principle. Hamilton’s principle states that the motion of a system is such that the action functional S q(t) = Zt2 t1 dtL(q,q,t˙ ) (6.2) is an extremum, i.e. δS = 0.
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