Mathematical Methods for Physics and Engineering - Matematica.NET

NORMAL MODESP P Pθ 1θ 2θ 2lθ 1θ 1θ 2(a) (b) (c)lFigure 9.1 A uniform rod of length l attached to the fixed point P by a lightstring of the same length: (a) the general coordinate system; (b) approximationto the normal mode with lower frequency; (c) approximation to the mode withhigher frequency.With these expressions for T and V we now apply the conservation of energy,d(T + V )=0, (9.6)dtassuming that there are no external forces other than gravity. In matrix form(9.6) becomesddt (˙qT A˙q + q T Bq) =¨q T A˙q + ˙q T A¨q + ˙q T Bq + q T B˙q =0,which, using A = A T and B = B T , gives2˙q T (A¨q + Bq) =0.We will assume, although it is not clear that this gives the only possible solution,that the above equation implies that the coefficient of each ˙q i is separately zero.HenceA¨q + Bq = 0. (9.7)For a rigorous derivation Lagrange’s equations should be used, as in chapter 22.Nowwesearchforsetsofcoordinatesq that all oscillate with the same period,i.e. the total motion repeats itself exactly after a finite interval. Solutions of thisform will satisfyq = x cos ωt; (9.8)the relative values of the elements of x in such a solution will indicate how each318