The five Lagrange points in the Earth-Sun system (not to scale obviously). I can only understand L1
So sometimes I just want to understand things – and not just advocate for female domination. For example, what exactly are Lagrange points, why are they important, and who was Lagrange, when he wasn’t Laplace?
First the easy stuff. Joseph-Louis Lagrange (1736-1813) was an Italian-born French naturalist (mathematician/astronomer/physicist). He also has an Italian name, and note that Italy wasn’t a country in his day, and France had quite flexible boundaries. In fact he was born in Turin, which then belonged to the kingdom of Sardinia. Most of his best work was produced in a Prussian city called Berlin. So much for the enduring permanence of nations.
The list of Lagrange’s mathematical contributions is long, and my general mathematical understanding is minuscule, but my fascination with the very sensible notion that there should be a point or region between two massive, gravitationally attracting bodies, such as, say, two planets, where an object would be ‘suspended’ between those two bodies, as their opposite forces (but gravity isn’t a force, they keep telling me), are counter-balanced – that fascination has brought me to attempt to understand, to know more…
So here’s a Wikipedia quote on Lagrange:
He studied the three-body problem for the Earth, Sun and Moon (1764) and the movement of Jupiter’s satellites (1766), and in 1772 found the special-case solutions to this problem that yield what are now known as Lagrangian points.
I’m thinking maybe that my description of a body in a space between two other bodies exerting a more or less equal and opposite gravitational attraction upon it has something to do with this ‘three body problem’ that I’ve heard about only recently. And again, looking at Wikipedia, that magical resource, this seems to be the case:
In celestial mechanics, the Lagrange points… also Lagrangian points or libration points) are points of equilibrium for small-mass objects under the gravitational influence of two massive orbiting bodies. Mathematically, this involves the solution of the restricted three-body problem.
And this is where it gets very complicated, at least for me. The restricted three-body problem seems to be, in essence, a two-body problem, due to the third body’s mass being negligible in the Newtonian scheme of things, such as in the case of a satellite or small ‘planetoid’. In such a situation, at such points, the two large gravitational forces and the centrifugal force are in balance. The centrifugal force is a type of inertial force in Newtonian mechanics. But how can a force be inert? When it’s not a force, obviously. It’s also called a fictitious or pseudo force, but such forces appear to act when viewed in a ‘rotating frame of reference’. And it must be hard to dismiss such rotating frames when we consider that our Earth rotates on its axis, our solar system rotates around its sun and our galaxy rotates around its black hole. And maybe our universe rotates around its centre, if it has one.
But I’m only writing this to avoid the mathematics. Anyway the point about rotating frames of reference is that, if that frame is regular or constant, as is the Earth’s rotation, it will appear to be stationary, and ‘the standard’, which can lead to confusion about other observable bodies, a confusion that lasted for millennia before the likes of Galileo and Newton began to question what had hitherto seemed obvious.
So, Newton’s second law of motion can’t be avoided. I’ll first state it in English words, then… I’m not sure how much further I’ll get:
At any instant of time, the net force on a body is equal to the body’s acceleration multiplied by its mass or, equivalently, the rate at which the body’s momentum is changing with time.
Apparently the dummy’s version of this is F = ma (force equals mass times acceleration), and the more sciencey versions are:
F = m.dv/dt = ma
F = d/dt.(mv)… where d stands for derivative, v for velocity and t for time.
And there are other versions, I think. It’s this second law that has proved the most controversial and it seems the most fruitful for further research and analysis. But don’t trust me on any of this. What is most interesting is that this classical description of forces has been fruitful enough for later (but not much later!) physicists like Lagrange to work out mathematically certain points in space where satellites and telescopes can hover or circulate well beyond Earth’s atmosphere. We now know of five Lagrange points within the Earth-Sun gravitational system, and another five within the Earth-Moon system. To explain why there are so many would be beyond my current level of competence, but I intend to try an online course in classical mechanics, to get me up to speed, or up to equilibrium.
References
https://en.wikipedia.org/wiki/Lagrange_point
https://en.wikipedia.org/wiki/Joseph-Louis_Lagrange