Lagrange’s quintic and Descartes’ rule

Do fifth degree polynomial equations come up in applications? Yes, and this post will give an example.

In general the three-body problem, describing the motion of three objects interacting under gravity, does not have a closed-form solution. However, Euler and Lagrange discovered a few special cases that do have closed-form solutions. We will look at Lagrange’s quintic, an equation that came out of Lagrange’s elaboration on Euler’s solution involving three masses moving so that they remain colinear.

Lagrange’s quintic equation is

\begin{align*} (m_1 + m_2)x^5 &+ (3m_1 + 2m_2)x^4 + (3m_1 + m_2)x^3 \\ &- (m_2 + 3m_3)x^2 - (2m_2 + 3m_3)x - (m_2+m_3) = 0 \end{align*}

where m1, m2, and m3 are the masses of the three bodies and x represents the ratio of the distances from the second body to the other two bodies.

Descartes’ rule of signs says that the equation above has exactly one positive root. This is because the signs of the coefficients only change once: the first three coefficients are positive and the next three are negative. Only positive roots are physically meaningful since x represents a ratio of (unsigned) distances, so Laplace’s quintic has a unique meaningful solution. Note that this argument places no restrictions on the relative masses of the three objects.

We can set x = −w and apply Descartes’ rule to the polynomial equation in w. This tells us that the equation has 0, 2, or 4 solutions for positive w, i.e. for negative x. But this doesn’t tell us anything we wouldn’t know from more general principles. Fifth degree polynomials have five roots, and complex roots to equations with real coefficients come in conjugate pairs, so Lagrange equation either has two complex roots (and thus two negative real roots) or four complex roots (and no negative real roots).

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