7.4 Hill’s equation
The second-order equation \[ y'' + q(t)y = 0, \qquad q(t + T) = q(t), \] is Hill’s equation, introduced in the study of the Moon’s orbit. As a system, \(\tr A = 0\), so \(\rho _1\rho _2 = 1\). Let \(y_1\), \(y_2\) be the solutions with \(y_1(0) = 1\), \(y_1'(0) = 0\), \(y_2(0) = 0\), \(y_2'(0) = 1\); then \[ M = \begin {bmatrix} y_1(T) & y_2(T)\\ y_1'(T) & y_2'(T)\end {bmatrix}, \qquad \rho ^2 - \Delta \rho + 1 = 0, \qquad \Delta = \tr M = y_1(T) + y_2'(T) . \]
Theorem 7.4.1. For Hill’s equation, with \(\Delta \) the discriminant:
- (i)
- if \(|\Delta | < 2\), the multipliers are \(e^{\pm i\theta }\) with \(\cos \theta = \Delta /2\), \(0 < \theta < \pi \), and every solution is bounded (stable);
- (ii)
- if \(|\Delta | > 2\), the multipliers are real, \(\rho \) and \(1/\rho \) with \(|\rho | > 1\), and there are unbounded solutions (unstable);
- (iii)
- if \(\Delta = 2\) there is a \(T\)-periodic solution, and if \(\Delta = -2\) a \(2T\)-periodic one.
Proof. The roots of \(\rho ^2 - \Delta \rho + 1\) are \(\frac 12(\Delta \pm \sqrt {\Delta ^2 - 4})\), with product \(1\). (i) If \(|\Delta | < 2\) they are complex conjugates of modulus \(1\), and distinct, so \(M\) is diagonalisable and Theorem 7.3.2(ii) gives stability. (ii) If \(|\Delta | > 2\) they are real with product \(1\), so one has modulus greater than \(1\). (iii) If \(\Delta = \pm 2\), \(\rho = \pm 1\) is a double root, and Theorem 7.3.2(iii) applies. □
The boundaries \(|\Delta | = 2\) between stability and instability are exactly the parameter values with periodic solutions of period \(T\) or \(2T\), a fact used to compute them.
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