By Mark Levi
Figure 1 depicts a point mass \(m=1\) moving in a plane, pulled into the origin by a Hookean spring and subject to friction that is linear in velocity. Position \(z\in {\mathbb R} ^2\) obeys Newton’s second law,
\[ \ddot z = - q z - p \dot z. \tag1 \]
Figure 1. Both coordinates of \(z\) satisfy \((3)\). Here, \(p=p(t), q=q(t)\). Figure courtesy of Mark Levi.
We allow \(q\) and \(p\) to depend on time (so that “Hooke’s constant” \(q=q(t)\) is constant only in \(z\) but not necessarily in \(t\)).
On the one hand, taking the two-dimensional cross product with \(z\) yields the evolution of the angular momentum \(L= z\times \dot z\):
\[ \frac{d}{dt} ( z\times \dot z) = - p \;(z \times \dot z). \tag2 \]
On the other hand, the coordinates of \(z=(x,y)\) satisfy the same ordinary differential equation
\[\ddot u + p \dot u + q u =0, \tag3 \]
and we recognize the angular momentum \(L = z \times \dot z = x \dot y - \dot x y \stackrel{def}{=} W[x,y]\) as these solutions’ Wronskian! Newton’s law \((2)\) thus becomes
\[ \frac{d}{dt} W = - p W, \tag4 \]
which is Abel’s formula for the Wronskian. This concludes the justification of the claims made in the article’s title, where “” stands for “a special case of.” Complexification—i.e., going from \((3)\) to \((1)\)—revealed something not seen in one space dimension.
A Logical Question
An entirely different way to understand \((4)\) is to observe that the divergence of any linear vector field is the logarithmic derivative of the area of a region carried by the field (there is no need to consider infinitesimal areas for linear flows). But the divergence of the vector field in the phase plane of \((3)\) is \(-p\), and thus the area \(W\) of the parallelogram generated by two solution vectors satisfies \(\dot W/ W = - p\), as in \((4)\).
The two aforementioned arguments, which are entirely different, lead to the same conclusion \((4)\). Are these arguments homotopic? This is an interesting question for logicians.
A Hidden Symmetry
Here is another puzzling connection between Abel’s theorem and Noether’s theorem on conserved quantities. Note that \((1)\) is invariant under rotations; Noether’s theorem applies in the conservative case \((p\equiv 0)\) and guarantees conservation of the angular momentum \(L = {\rm const.}\), thus implying a special case of Liouville’s theorem: \(W = {\rm const.}\) One could thus say that the cause is the symmetry hidden in \((3)\) but revealed in \((1)\).
Mark Levi ([email protected]) is a professor of mathematics at the Pennsylvania State University.