By J. C. Maltbaek (auth.)
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If the system has n degrees of freedom (n ~ 2p), and the (2p- n) equations of constraints are of the form X;= Gi(q1, • • • 'qn,t) Y1 = Hi(qt> · · ·, q", t) we have a holonomic system. The appearance of t explicitly in the above equations indicates moving constraints or a moving frame of reference or both. Since the generalised co-ordinates (qt> . , q") are independent, we may change one of them without changing the others. 7 may be used to determine the generalised force Q1 , with similar expressions for the other generalised forces Q2 , ••• , Qn.
C) Determine the equation of motion as under (b) if the bob is an elevator with I increasing or decreasing at a constant rate I = a+ bt, where a and b are constants. Solution (a) The system is conservative and has one degree of freedom, since I= f(t)is a given function of time. Taking the generalised coordinate as 8, we have x = I sin 8 and y = I cos 8; these expressions contain time explicitly since I= f(t). We find x = i sin (J + 10 cos (J and y = i cos (J -18 sin e. m (i2 + [2(}2). Taking the datum position for potential energy at the horizontal x-axis, and keeping t constant, we find V= -mglcosO so that Both T, V and L contain time explicitly since I = f(t).
9 is valid for conservative systems only. For a system with both conservative and non-conservative forces acting, it is obvious that Lagrange's equations may be given the form d dt oL oL aq_j - aqj = QiN (i = 1, ... 10) where QiN is determined in the usual way for the coordinate qj, but taking into account non-conservative forces only. The conservative forces have already been accounted for in the Lagrangian function L. 8. 3 shows a simple pendulum which swings in the vertical xy-plane. The length I of the pendulum may be varied by pulling a string at A.