Perturbation theory in the context of "Airy wave theory"

Self-organization, also called spontaneous order in the social sciences, is a process where some form of overall order arises from local interactions between parts of an initially disordered system. The process can be spontaneous when sufficient energy is available, not needing control by any external agent. It is often triggered by seemingly random fluctuations, amplified by positive feedback. The resulting organization is wholly decentralized, distributed over all the components of the system. As such, the organization is typically robust and able to survive or self-repair substantial perturbation. Chaos theory discusses self-organization in terms of islands of predictability in a sea of chaotic unpredictability.

Self-organization occurs in many physical, chemical, biological, robotic, and cognitive systems. Examples of self-organization include crystallization, thermal convection of fluids, chemical oscillation, animal swarming, neural circuits, and black markets.

Celestial mechanics is the branch of astronomy that deals with the motions and gravitational interactions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to produce ephemeris data. The computation of the motion of the bodies through orbital mechanics can be simplified by using an appropriate inertial frame of reference. This leads to the use of various different coordinate systems, such as the Heliocentric (Sun-centered) coordinate system.

In a binary system of objects interacting through gravity, Newtonian mechanics can used to produce a set of orbital elements that will predict with reasonable accuracy the future position of the two bodies. This method demonstrates the correctness of Kepler's laws of planetary motion. Where one of the bodies is sufficiently massive, general relativity must be included to predict apsidal precession. The problem becomes more complicated when another body is added, creating a three-body problem that can not be solved exactly. Perturbation theory is used to find an approximate solution to this problem.

In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions P_ε, where ε is a small number, or a vector of small quantities. The infinitesimal conditions are the result of applying the approach of differential calculus to solving a problem with constraints. The name is an analogy to non-rigid structures that deform slightly to accommodate external forces.

Some characteristic phenomena are: the derivation of first-order equations by treating the ε quantities as having negligible squares; the possibility of isolated solutions, in that varying a solution may not be possible, or does not bring anything new; and the question of whether the infinitesimal constraints actually 'integrate', so that their solution does provide small variations. In some form these considerations have a history of centuries in mathematics, but also in physics and engineering. For example, in the geometry of numbers a class of results called isolation theorems was recognised, with the topological interpretation of an open orbit (of a group action) around a given solution. Perturbation theory also looks at deformations, in general of operators.

A trans-lunar injection (TLI) is a propulsive maneuver, which is used to send a spacecraft to the Moon. Typical lunar transfer trajectories approximate Hohmann transfers, although low-energy transfers have also been used in some cases, as with the Hiten probe. For short duration missions without significant perturbations from sources outside the Earth-Moon system, a fast Hohmann transfer is typically more practical.

A spacecraft performs TLI to begin a lunar transfer from a low circular parking orbit around Earth. The large TLI burn, usually performed by a chemical rocket engine, increases the spacecraft's velocity, changing its orbit from a circular low Earth orbit to a highly eccentric orbit. The mission phase following TLI – while the spacecraft is flying passively towards the moon under its own momentum and influenced by terrestrial and lunar gravity – is called translunar coast. As the spacecraft begins coasting on the lunar transfer arc, its trajectory approximates an elliptical orbit about the Earth with an apogee near to the radius of the Moon's orbit. The TLI burn is sized and timed to precisely target the Moon as it revolves around the Earth. The burn is timed so that the spacecraft nears apogee as the Moon approaches. Finally, the spacecraft enters the Moon's sphere of influence, making a hyperbolic lunar swingby.

Fermi liquid theory (also known as Landau's Fermi-liquid theory) is a theoretical model of interacting fermions that describes the normal state of the conduction electrons in most metals at sufficiently low temperatures. The theory describes the behavior of many-body systems of particles in which the interactions between particles may be strong. The phenomenological theory of Fermi liquids was introduced by the Soviet physicist Lev Davidovich Landau in 1956, and later developed by Alexei Abrikosov and Isaak Khalatnikov using diagrammatic perturbation theory. The theory explains why some of the properties of an interacting fermion system are very similar to those of the ideal Fermi gas (collection of non-interacting fermions), and why other properties differ.

Fermi liquid theory applies most notably to conduction electrons in normal (non-superconducting) metals, and to liquid helium-3. Liquid helium-3 is a Fermi liquid at low temperatures (but not low enough to be in its superfluid phase). An atom of helium-3 has two protons, one neutron and two electrons, giving an odd number of fermions, so the atom itself is a fermion. Fermi liquid theory also describes the low-temperature behavior of electrons in heavy fermion materials, which are metallic rare-earth alloys having partially filled f orbitals. The effective mass of electrons in these materials is much larger than the free-electron mass because of interactions with other electrons, so these systems are known as heavy Fermi liquids. Strontium ruthenate displays some key properties of Fermi liquids, despite being a strongly correlated material that is similar to high temperature superconductors such as the cuprates. The low-momentum interactions of nucleons (protons and neutrons) in atomic nuclei are also described by Fermi liquid theory.