The hydrogen atom has special significance in quantum mechanics and quantum field theory as a simple two-body problem physical system which has yielded many simple analytical solutions in closed-form.

In the case that n = 2 ( two-body problem ), the situation is much simpler than for larger n. Various explicit formulas apply, where in the more general case typically only numerical solutions are possible.

Given the orbital eccentricity of this object, different epochs can generate quite different heliocentric unperturbed two-body best-fit solutions to the aphelion distance ( maximum distance ) of this object.

Analytical solutions ( mathematical expressions to predict the positions and motions at any future time ) for the two-body and three-body problems exist ; none has been found for the n-body problem except for certain special cases.

This is a quantity which allows the two-body problem to be solved as if it were a one-body problem.

One illustrative example of a two-body interaction where this form would not apply is for electrostatic potentials due to charged particles, because they certainly do interact with each other by the coulomb interaction ( electrostatic force ), shown below.

To this Newtonian approximation, for a system of two point masses or spherical bodies, only influenced by their mutual gravitation ( the two-body problem ), the orbits can be exactly calculated.

Note that that while bound orbits around a point mass or around a spherical body with an Newtonian gravitational field are closed ellipses, which repeat the same path exactly and indefinitely, any non-spherical or non-Newtonian effects ( as caused, for example, by the slight oblateness of the Earth, or by relativistic effects, changing the gravitational field's behavior with distance ) will cause the orbit's shape to depart from the closed ellipses characteristic of Newtonian two-body motion.

The elements are more descriptive of the size, shape and orientation of an orbit, and the elements may be used to quickly and easily estimate the object's state at any arbitrary time provided its motion is accurately modeled by the two-body problem with only small perturbations.

In the gravitational two-body problem, the specific orbital energy ( or vis-viva energy ) of two orbiting bodies is the constant sum of their mutual potential energy () and their total kinetic energy (), divided by the reduced mass.

A key insight applied by Boltzmann was to determine the collision term resulting solely from two-body collisions between particles that are assumed to be uncorrelated prior to the collision.

In the gravitational two-body problem, the orbits of the two bodies are described by two overlapping conic sections each with one of their foci being coincident at the center of mass ( barycenter ).

Some applications are conveniently treated by perturbation theory, in which the system is considered as a two-body problem plus additional forces causing deviations from a hypothetical unperturbed two-body trajectory.

Since the osculating orbit is easily calculated by two-body methods, and are accounted for and can be solved.

In general, if the behaviour of a system of more than two objects cannot be described by the two-body interactions between all possible pairs, as a first approximation, the deviation is mainly due to a three-body force.

where is the two-body phase-space factor, is the nuclear matrix element, and m < sub > ββ </ sub > is the so called effective Majorana neutrino mass given by

In classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force F that varies in strength as the inverse square of the distance r between them.

Centrifugal force arises in the analysis of orbital motion and, more generally, of motion in a central-force field: in the case of a two-body problem, it is easy to convert to an equivalent one-body problem with force directed to or from an origin, and motion in a plane, so we consider only that.

Again the Dirac equation may be solved analytically in the special case of a two-body system, such as the hydrogen atom.

A contour plot of the effective potential due to gravity and the centrifugal force of a two-body system in a rotating frame of reference.

The stable orbits that arise in a two-body approximation ignore the influence of other bodies.

Another situation in which parabolae may arise in nature is in two-body orbits, for example, of a small planetoid or other object under the influence of the gravitation of the sun.

The assumption of conservation of momentum as well as the conservation of kinetic energy makes possible the calculation of the final velocities in two-body collisions.

We also note that a two-body Dirac equation composed of a Dirac operator for each of the two point particles interacting via the Coulomb interaction can be exactly separated in the ( relativistic ) center of momentum frame and the resulting ground state eigenvalue has been obtained very accurately using the Finite element methods of J. Shertzer.

The vibrations of a diatomic molecule are an example of a two-body version of the quantum harmonic oscillator.

) The equation below holds true for the two-body ( Body A, Body B ) system collision in the example above.

In physics, the Reduced mass is the " effective " inertial mass appearing in the two-body problem of Newtonian mechanics.

Thus we have reduced the two-body problem to that of one body.

Reduced mass occurs in a multitude of two-body problems, where classical mechanics is applicable.

In celestial mechanics, this is usually a Keplerian ellipse, which is correct when there are only two gravitating bodies ( say, the Earth and the Moon ), or a circular orbit, which is only correct in special cases of two-body motion, but is often close enough for practical use.

* Orbital elements are the parameters needed to specify a Newtonian two-body orbit uniquely.

Simple two-body problems, for example, can be solved analytically.

Then Milankovich treated the two-body and the many-body problems of celestial mechanics.

His policies had resolved the conflicts that threatened to ignite the cold war and workable solutions were beginning to take shape.

The solutions were not arrived at by any theoreticians of the Karl Marx stripe but by men of government -- lawyers, most of them -- and men of business.

These cells were thawed at 37-degrees-C for 30 min and were deglycerolized by alternately centrifuging and mixing with descending concentrations of glycerol solutions ( 20, 18, 10, 8, 4 and 2% ).

Two solutions with additional hardware were provided.

Some were little more than stopgap solutions, mounting an anti-tank gun on a tracked vehicle to give mobility, while others were more sophisticated designs.

In this period more general black hole solutions were found.

At first, it was suspected that the strange features of the black hole solutions were pathological artifacts from the symmetry conditions imposed, and that the singularities would not appear in generic situations.

The cosmological solutions of general relativity were found by Alexander Friedmann in the early 1920s.

Previously these tools were generally limited to simple CRM solutions which focused on monitoring and recording interactions and communications.

These solutions brought security and economic prosperity to Israel for a time, but did not bring peace with the Israelite prophets, who were interested in a strict deuteronomic interpretation of Mosaic law.

The first solutions were provided by Franz Nauck in 1850.

Numerous solutions were tried.

Approximate solutions may also be found by perturbation theories such as linearized gravity and its generalization, the post-Newtonian expansion, both of which were developed by Einstein.

At the same time environmental organizations and the political opposition were demanding " solutions that contrasted with the government's ".

Since black holes are exact solutions of Einstein's equations, they were thought not to have any entropy either.

After learning the rules, they discovered Judit was able to find solutions to the problems they were studying and she began to be invited into the group.

Many solutions were proposed for how to determine longitude at the end of an exploratory sea voyage and hence the longitude of the place that was visited ( in case one would want to revisit it, place it on a map, or more urgently, avoid known marine hazards ).

The tactical strike aircraft programs were meant to serve as interim solutions until the next generation of aircraft arrived.

Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as exemplified by Galois through the algebraic notion of a group ; geometric theory and the explicit solutions of differential equations of mechanics, worked out by Poisson and Jacobi ; and the new understanding of geometry that emerged in the works of Plücker, Möbius, Grassmann and others, and culminated in Riemann's revolutionary vision of the subject.

Laplace also recognised that Joseph Fourier's method of Fourier series for solving the diffusion equation could only apply to a limited region of space as the solutions were periodic.

He studied chemistry under Henry Edward Armstrong, an English chemist whose interests were primarily in organic chemistry but also included the nature of ions in aqueous solutions.

Some IBM PC clone vendors offered somewhat customised hardware solutions that were delivered running NeXTSTEP on Intel, such as the Elonex NextStation and the Canon object. station 41.