The three-body problem

The inclusion of solar perturbations of the motion of the Moon results in a "three-body problem" (Earth-Moon-Sun), which is the simplest complication of the completely solvable two-body problem discussed above. When Earth, the Moon, and the Sun are considered to be point masses, this particular three-body problem is called "the main problem of the lunar theory," which has been studied extensively with a variety of methods beginning with Newton. Although the three-body problem has no complete analytic solution in closed form, various series solutions by successive approximations achieve such accuracy that complete theories of the lunar motion must include the effects of the nonspherical mass distributions of both Earth and the Moon as well as the effects of the planets if the precision of the predicted positions is to approach that of the observations. Most of the schemes for the main problem are partially numerical and therefore apply only to the lunar motion. An exception is the completely analytic work of the French astronomer Charles-Eugéne Delaunay (1816–72), who exploited and developed the most elegant techniques of classical mechanics pioneered by his contemporary, the Irish astronomer and mathematician William R. Hamilton (1805–65). Delaunay could predict the position of the Moon to within its own diameter over a 20-year time span. Since his development was entirely analytic, the work was applicable to the motions of satellites about other planets where the series expansions converged much more rapidly than they did for the application to the lunar motion.

Delaunay's work on the lunar theory demonstrates some of the influence that celestial mechanics has had on the development of the techniques of classical mechanics. This close link between the development of classical mechanics and its application to celestial mechanics was probably no better demonstrated than in the work of the French mathematician Henri Poincaré (1854–1912). Poincaré, along with other great mathematicians such as George D. Birkhoff (1884–1944), Aurel Wintner (1903–58), and Andrey N. Kolmogorov (1903–87), placed celestial mechanics on a more sound mathematical basis and was less concerned with quantitatively accurate prediction of celestial body motion. Poincaré demonstrated that the series solutions in use in celestial mechanics for so long generally did not converge but that they could give accurate descriptions of the motion for significant periods of time in truncated form. The elaborate theoretical developments in celestial and classical mechanics have received more attention recently with the realization that a large class of motions are of an irregular or chaotic nature and require fundamentally different approaches for their description.

Physicists Discover a Whopping 13 New Solutions to Three-Body Problem

Shape up. Solutions to the three-body problem, such as the "figure eight" and "yarn," can be viewed on an abstract shape-sphere (top) or in real space (bottom).

Milovan Šuvakov and Veljko Dmitrašinovic, Phys. Rev. Lett., (2013); Milovan Šuvakov and Veljko Dmitrašinovic /Institute of Physics Belgrade.

It's the sort of abstract puzzle that keeps a scientist awake at night: Can you predict how three objects will orbit each other in a repeating pattern? In the 300 years since this "three-body problem" was first recognized, just three families of solutions have been found. Now, two physicists have discovered 13 new families. It's quite a feat in mathematical physics, and it could conceivably help astrophysicists understand new planetary systems.

The trove of new solutions has researchers jazzed. "I love these things," says Robert Vanderbei a mathematician at Princeton University who was not involved in the work. He says he, in fact, spent all night thinking about the work.

The three-body problem dates back to the 1680s. Isaac Newton had already shown that his new law of gravity could always predict the orbit of two bodies held together by gravity—such as a star and a planet—with complete accuracy. The orbit is basically always an ellipse. However, Newton couldn't come up with a similar solution for the case of three bodies orbiting one another. For 2 centuries, scientists tried different tacks until the German mathematician Heinrich Bruns pointed out that the search for a general solution for the three-body problem was futile, and that only specific solutions—one-offs that work under particular conditions—were possible. Generally, the motion of three bodies is now known to be nonrepeating.

Specific repeating solutions have been hard to come by, however. The famed mathematicians Joseph-Louis Lagrange and Leonhard Euler had come up with some in the 18th century, but it wasn't until the 1970s, with a little help from modern computing, that U.S. mathematician Roger Broucke and French astronomer Michel Hénon discovered more. Until now, specific solutions could be sorted into just three families: the Lagrange-Euler family, the Broucke-Hénon family, and the figure-eight family, the last of which was discovered in 1993 by physicist Cristopher Moore at the Santa Fe Institute.

The figure-eight family is so called because it describes three objects chasing one another in a figure eight shape. The Lagrange-Euler solutions are simpler, with the equally spaced bodies going around in a circle like horses on a merry-go-round. The Broucke-Hénon solutions are the most complex: Two objects dash back and forth on the inside, while the third object orbits around the outside.

The discovery of 13 new families, made by physicists Milovan Šuvakov and Veljko Dmitrašinovic at the Institute of Physics Belgrade, brings the new total to 16. "The results are beautiful, and beautifully presented," says Richard Montgomery, a mathematician at the University of California, Santa Cruz, who was not involved with the discovery.

Finding any solution is a daunting prospect. Three objects in space can be set off in infinite ways. Somehow, initial conditions—starting points, velocities, and so on—must be found that bring the objects back to those conditions so the whole dance can start over again. Šuvakov and Dmitrašinovic's method was to start with an existing solution on a computer simulation and then tweak its initial conditions until a new type of orbit materialized, as they report in a paper in press at Physical Review Letters. "What we did was the most simple-minded thing that you could do," Dmitrašinovic says. "We were shocked when we discovered all these things, and we were even more shocked when we discovered that they had not been discovered before us."

Faced with so many new solutions, the Belgrade physicists invented a new classification system. They used an abstract space called a "shape-sphere," which describes the shape of the orbits in terms of the relative distances between the objects. Three spots around the sphere's equator mark where two of the particles would collide, and a line drawn over the ball, which must avoid those spots, maps how near the objects get to each other. The simplest solution on the shape-sphere is of the Lagrange-Euler family. It is just a single point, because it corresponds to all three objects maintaining the same distance apart by orbiting (in real space) in a circle.

Others solutions are much more intricate, however. The researchers' new "yarn" solution looks like a ball of yarn on the shape-sphere, although its real space orbit looks even more complex—rather like a huge mass of spaghetti. (All the solutions can be viewed online.) Šuvakov and Dmitrašinovic separated all the orbits—including the previously known ones—into 16 families based on their shapes on the shape sphere. They then sorted these families into four classes based on symmetry and other properties of the shapes, the first which included all the previous solutions.

The next step for the Belgrade physicists is to see how many of their new solutions are stable and will stay on track if perturbed a little. If some of the solutions are stable, then they might even be glimpsed in real life. Now, the only widely accepted three-body solution observed in space is a system made up of the sun, Jupiter, and one of the nearby Trojan asteroids, which make up a Lagrange-Euler type circle. But if we look at enough "exoplanet" systems beyond our own, we might see some of the new, more complex solutions, too, such as a solar system that looks like a heap of spaghetti.

"Observational astronomy is moving very quickly forward," Dmitrašinovic says. But he adds that spotting one of the solutions will still be very difficult. "It will take a while. I'm pretty sure it won't happen tomorrow."

Kepler's laws of planetary motion

In astronomy and classical physics, (these are) laws describing the motions of the planets in the solar system. They were derived by the German astronomer Johannes Kepler, whose analysis of the observations of the 16th-century Danish astronomer Tycho Brahe enabled him to announce his first two laws in the year 1609 and a third law nearly a decade later, in 1618. Kepler himself never numbered these laws or specially distinguished them from his other discoveries.

Kepler's three laws of planetary motion can be stated as follows:

1. All planets move about the Sun in elliptical orbits, having the Sun as one of the foci.
2. A radius vector joining any planet to the Sun sweeps out equal areas in equal lengths of time.
3. The squares of the sidereal periods (of revolution) of the planets are directly proportional to the cubes of their mean distances from the Sun.

Knowledge of these laws, especially the second (the law of areas), proved crucial to Sir Isaac Newton in 1684–85, when he formulated his famous law of gravitation between Earth and the Moon and between the Sun and the planets, postulated by him to have validity for all objects anywhere in the universe. Newton showed that the motion of bodies subject to central gravitational force need not always follow the elliptical orbits specified by the first law of Kepler but can take paths defined by other, open conic curves; the motion can be in parabolic or hyperbolic orbits, depending on the total energy of the body. Thus, an object of sufficient energy—e.g., a comet—can enter the solar system and leave again without returning. From Kepler's second law, it may be observed further that the angular momentum of any planet about an axis through the Sun and perpendicular to the orbital plane is also unchanging.

The usefulness of Kepler's laws extends to the motions of natural and artificial satellites as well as to unpowered spacecraft in orbit in stellar systems or near planets. As formulated by Kepler, the laws do not, of course, take into account the gravitational interactions (as perturbing effects) of the various planets on each other. The general problem of accurately predicting the motions of more than two bodies under their mutual attractions is quite complicated; analytical solutions of the three-body problem are unobtainable except for some special cases. It may be noted that Kepler's laws apply not only to gravitational but also to all other inverse-square-law forces and, if due allowance is made for relativistic and quantum effects, to the electromagnetic forces within the atom.

Newton's Laws of Motion

(Newton's laws explain) relations between the forces acting on a body and the motion of the body, first formulated by Isaac Newton.

Newton's first law states that, if a body is at rest or moving at a constant speed in a straight line, it will remain at rest or keep moving in a straight line at constant speed unless it is acted upon by a force. This postulate is known as the law of inertia. The law of inertia was first formulated by Galileo Galilei for horizontal motion on Earth and was later generalized by René Descartes. Before Galileo it had been thought that all horizontal motion required a direct cause, but Galileo deduced from his experiments that a body in motion would remain in motion unless a force (such as friction) caused it to come to rest.

Newton's second law is a quantitative description of the changes that a force can produce on the motion of a body. It states that the time rate of change of the momentum of a body is equal in both magnitude and direction to the force imposed on it. The momentum of a body is equal to the product of its mass and its velocity. Momentum, like velocity, is a vector quantity, having both magnitude and direction. A force applied to a body can change the magnitude of the momentum, or its direction, or both. Newton's second law is one of the most important in all of physics. For a body whose mass m is constant, it can be written in the form F = ma, where F (force) and a (acceleration) are both vector quantities. If a body has a net force acting on it, it is accelerated in accordance with the equation. Conversely, if a body is not accelerated, there is no net force acting on it.

Newton's third law states that when two bodies interact, they apply forces to one another that are equal in magnitude and opposite in direction. The third law is also known as the law of action and reaction. This law is important in analyzing problems of static equilibrium, where all forces are balanced, but it also applies to bodies in uniform or accelerated motion. The forces it describes are real ones, not mere bookkeeping devices. For example, a book resting on a table applies a downward force equal to its weight on the table. According to the third law, the table applies an equal and opposite force to the book. This force occurs because the weight of the book causes the table to deform slightly so that it pushes back on the book like a coiled spring.

Newton's laws first appeared in his masterpiece, Philosophiae Naturalis Principia Mathematica (1687), commonly known as the Principia. In 1543 Nicolaus Copernicus suggested that the Sun, rather than the Earth, might be at the centre of the universe. In the intervening years Galileo, Johannes Kepler, and Descartes laid the foundations of a new science that would both replace the Aristotelian worldview, inherited from the ancient Greeks, and explain the workings of a heliocentric universe. In the Principia Newton created that new science. He developed his three laws in order to explain why the orbits of the planets are ellipses rather than circles, at which he succeeded, but it turned out that he explained much more. The series of events from Copernicus to Newton is known collectively as the scientific revolution.

In the 20th century Newton's laws were replaced by quantum mechanics and relativity as the most fundamental laws of physics. Nevertheless, Newton's laws continue to give an accurate account of nature, except for very small bodies such as electrons or for bodies moving close to the speed of light. Quantum mechanics and relativity reduce to Newton's laws for larger bodies or for bodies moving more slowly.

Three classical problems:

Although Euclid solves more than 100 construction problems in the Elements, many more were posed whose solutions required more than just compass and straight edge. Three such problems stimulated so much interest among later geometers that they have come to be known as the "classical problems":

1. Doubling the cube (i.e., constructing a cube whose volume is twice that of a given cube)
2. Trisecting the angle
3. Squaring the circle

Even in the pre-Euclidean period the effort to construct a square equal in area to a given circle had begun. Some related results came from Hippocrates. others were reported from Antiphon and Bryson; and Euclid's theorem on the circle in Elements, Book XII, proposition 2, which states that circles are in the ratio of the squares of their diameters, was important for this search. But the first actual constructions (not, it must be noted, by means of the Euclidean tools, for this is impossible) came only in the 3rd century BC. The early history of angle trisection is obscure. Presumably, it was attempted in the pre-Euclidean period, although solutions are known only from the 3rd century or later.