6 1. Geometric Approach to Differential Equations for ω = k /m, which is preserved when b = 0. For the system with b 0 and so c 0, d dt E(x, v) = v ˙ + ω2 x ˙ = v(−2cv ω2x) + ω2xv = −2cv2 0. This shows that the energy is nonincreasing. A simple argument, which we give in the section on Lyapunov functions, shows that all of the solutions go to the fixed point at the origin. The use of this real-valued function E(x, v) is a way to show that the origin is attracting without using the explicit representation of the solutions. The system of equations (1.2) are linear, and most of the equations we consider are nonlinear. A simple nonlinear example is given by the pendulum mL ¨ = −mg sin(θ). Setting x = θ and v = ˙, we get the system ˙ = v, (1.4) ˙ = g L sin(x). It is difficult to get explicit solutions for this nonlinear system of equations. How- ever, the “energy method” just discussed can be used to find important properties of the solutions. By a derivation similar to the foregoing, we see that E(x, v) = v2 2 + 1 g L cos(x) is constant along solutions. Thus, as in the linear case, the solution moves on a level set of E, so the level sets determine the path of the solution. The level sets are given in Figure 3 without justification, but we shall return to this example in Section 5.2 and give more details. See Example 5.3. v x Figure 3. Level sets of energy for the pendulum There are fixed points at (x, v) = (0, 0), (±π, 0). The solutions near the origin are periodic, but those farther away have x either monotonically increasing or
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