4 1. Geometric Approach to Differential Equations or (1.3) m λ2 + b λ + k = 0, which is called the characteristic equation. When b = 0, the solution of the characteristic equation is λ2 = −k /m or λ = ±i ω, where ω = k /m. Since eiω t = cos(ω t) + i sin(ω t), the real and imaginary parts, cos(ω t) and sin(ω t), are each solutions. Linear combinations are also solutions, so the general solution is x(t) = A cos(ωt) + B sin(ωt), v(t) = −ωA sin(ωt) + ωB cos(ωt), where A and B are arbitrary constants. These solutions are both periodic, with the same period 2π/ω. See Figure 1. x t ω 4 2 0 −2 −4 Figure 1. Solutions for the linear harmonic oscillator: x as a function of t Another way to understand the solutions when b = 0 is to find the energy that is preserved by this system. Multiplying the equation ˙ + ω2 x = 0 by v = ˙, we get v ˙ + ω2 x ˙ = 0. The left-hand side of this equation is the derivative with respect to t of E(x, v) = 1 2 v2 + ω2 2 x2, so this function E(x, v) is a constant along a solution of the equation. This integral of motion shows clearly that the solutions move on ellipses in the (x, v)-plane given by level sets of E. There is a fixed point or equilibrium at (x, v) = (0, 0) and other solutions travel on periodic orbits in the shape of ellipses around the origin. For this linear equation, all the orbits have the same shape and period which is independent of size: We say that the local and global behavior is the same. See Figure 2. For future reference, a point x∗ is called a fixed point of a system of differential equations ˙ = F(x) provided F(x∗) = 0. The solution starting at a fixed point has zero velocity, so it just stays there. Therefore, if x(t) is a solution with x(0) = x∗, then x(t) = x∗ for all t. Traditionally, such a point was called an equilibrium point because the forces are in balance and the point does not move.
Previous Page Next Page