With frequency 1 and amplitude At along the horizontal and an oscillation The general solution of our equations is x 1 = A 1 sin(t + q> 1 ), x 2 =Ī 2 sin(wt + cp 2 ) a moving point independently performs an oscillation Show that this rectangle is inscribed in the ellipse U. Of these two regions defines a rectangle which contains the orbits (Figure 18).įigure 18 The regions U s E, U 1 s E and U 2 s E J2Et (0), and x 2 oscillates within the region I x 2 1. If at the initial moment the total energy is equal to E, then all trajectories lie in the region where U(x) ~ E, i.e., a point remains inside the potential well U(xl> x 2 ) At = PRooF.dE/dt = (x, x) + (fJUjox, x) = (x + (fJUjox), x) = ObytheequationĬorollary. The total energy of a conservative system is conserved, i.e., The equation of motion of a conservative system In this paragraph we look at the simplest examples.īy a system with two degrees of freedom we will mean a system defined byĪ system is said to be conservative if there exists a function U: £ 2 -+ IR Analyzing a general potential system with two degrees of freedom is beyond the capability of modern science.
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