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.
of the calculus of variations, momentum conservation is what follows when the integrand of a functional is independent of the independent variable. 6.3.2 Energy conservation When the integrand of a functional is independent of the dependent variable, another con-servation law follows. For Lagrangian mechanics, consider the expression H(q,q,t ...
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