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, .. , qn}.The generalized coordinates may have units of length, or angle, or perhaps something totallydifferent. In the theory of small oscillations, the normal coordinates are conventionallychosen to have units of (mass)1/2 (length).
As another example, consider a particle moving in the (x,y) plane under the influence of a potential U(x,y) = U p x2 +y2 which depends only on the particle’s distance from the origin ρ = p x2 +y2. The Lagrangian, expressed in two-dimensional polar coordinates (ρ,φ), is L = 1 2m ρ˙2 +ρ2φ˙2 −U(ρ) . (6.24) We see that L is cyclic in ...
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