Download The search for equilibrium between forces: Lagrange

The search for equilibrium between
forces: Lagrange points
Alejandro Vidal Crespo
Director Market Strategies
MONTHLY STRATEGY REPORT
February 2016
Monthly Strategy Report. February 2016
Afther the Star of Bethlehem
We live in turbulent times, with major forces seeking to obtain balance, distance, and
position within a system that must be stable. Something similar occurs in orbital systems,
where smaller bodies can coexist in stability. This is nothing new; we have known about
these equilibria since the 18th century when the French mathematician, Jean Louis Lagrange,
described the ideal points at which to position satellites 200 years before their existence
In the late 18th century, the scientific community was still assimilating the implications of Newtonian
mechanics and the laws of gravitation into its observations of the universe. While Newton’s theory
perfectly described the motion of a system consisting of two bodies, the situation became increasingly
more complicated when considering how three or more objects in motion formed a stable system based
on their initial mass, velocity, and position, in what is known as the three-body problem.
This was the mission of Jean Louis Lagrange, who theorised about how to stabilise systems in which
more than two bodies moved without descending into chaos. Part of this chaos was prompted by a third
law formulated by another luminary of science, Johannes Kepler: no two planets rotate synchronously
around the same star at different distances, if we only consider the interaction between each planet and
the star. The greater the distance, the lower the velocity; otherwise the numbers don’t add up.
Lagrange made considerable progress in solving the problem (he ultimately passed the baton to the
illustrious Henri Poincaré before finding a solution) by developing Lagrangian mechanics, the physical
explanation and multiple applications of which will not be addressed in this article, but rather one of its
most natural and immediate consequences: let us set aside Kepler, and focus on an object that moves
in sync with the Earth in a different orbit, to find points where the gravitational pull between our planet
and the Sun is offset. In fact, there are five theoretical points where this occurs. See the diagram below:
All are theoretically possible (L3 is, in fact, purely theoretical), but
let us focus on L1, L4 and L5.
The L1 point is perhaps the most intuitive. Somewhere between
the two bodies, the gravitational attraction of the Earth and Sun
are equal, and an object is stabilised between the two. This point
actually exists and is used to position solar observation satellites
like SOHO (Solar and Heliospheric Observatory). It is ideal for
these purposes because it is beyond the orbit of the Moon and is
therefore never eclipsed. With a constant view of the sun, these
satellites can carry out their activity from a privileged perspective.
It may be the most intuitive, but L1 is not the most stable. The most stable are L4 and L5, which lie at
the vertices of the equilateral triangle that forms the system. And, in fact, at these two points, curious
natural phenomena occur.
For example, it was discovered some time ago that two groups of asteroids orbit around L4 and L5 of
the Sun-Jupiter system. Those at L4, that precede Jupiter in its orbit, are known as Greeks, while those
at L5, that trail Jupiter, are known as Trojans.
Moreover, the giant impact hypothesis suggests that a planetary body formed at the Sun-Earth L4 or L5
points and crashed into Earth after its orbit destabilised, forming the Moon.
Now imagine another kind of unstable system like, for example, a Parliament, searching for a new
equilibrium to avoid a situation bordering on disorder. Like the master, Lagrange, we should begin by
Monthly Strategy Report. February 2016
determining the current mass, velocity, and position, on the basis of which, the smaller body may opt
for one of two stable solutions:
To position itself between the two major forces and try to offset the strong tidal effects in either
direction; ultimately balance is achieved because the body is pulled with monumental force (the smaller
the distance, the greater the force) in opposing directions. This presupposes that the body in question is
strong enough to remain intact (hence, there are so few naturally occurring orbits around L1) and that
it accurately measures the actual distance between the two bodies flanking it. The force exerted upon
it is the inverse of the distance squared: each approach of the larger objects is squared in terms of the
force brought to bear on the median body.
Another option is to find a triangulation that enables it to rest at an equilibrium between the two main
objects, without aligning itself between them. This presents two sub-alternatives: to lead like Jupiter’s
Greeks, or to trail like the Trojans. Initiative or control, in more social terms. In either case, this position
–further from the tidal forces– is what nature uses to stabilise itself.
And the cosmos has plenty of experience in this respect.