**Tisserand's criterion** is used to determine whether or not an observed orbiting body, such as a comet or an asteroid, is the same as a previously observed orbiting body.^{ [1] }^{ [2] }

While all the orbital parameters of an object orbiting the Sun during the close encounter with another massive body (e.g. Jupiter) can be changed dramatically, the value of a function of these parameters, called Tisserand's relation (due to Félix Tisserand) is approximately conserved, making it possible to recognize the orbit after the encounter.

Tisserand's criterion is computed in a circular restricted three-body system. In a circular restricted three-body system, one of the masses is assumed to be much smaller than the other two. The other two masses are assumed to be in a circular orbit about the system's center of mass. In addition, Tisserand's criterion also relies on the assumptions that a) one of the two larger masses is much smaller than the other large mass and b) the comet or asteroid has not had a close approach to any other large mass.

Two observed orbiting bodies are possibly the same if they satisfy or nearly satisfy Tisserand's criterion:^{ [1] }^{ [2] }^{ [3] }

where a is the semimajor axis (in units of Jupiters semimajor axis), e is the eccentricity, and i is the inclination of the body's orbit.

In other words, if a function of the orbital elements (named Tisserand's parameter) of the first observed body (nearly) equals the same function calculated with the orbital elements of the second observed body, the two bodies might be the same.

The relation defines a function of orbital parameters, conserved approximately when the third body is far from the second (perturbing) mass.^{ [3] }

The relation is derived from the Jacobi constant selecting a suitable unit system and using some approximations. Traditionally, the units are chosen in order to make **μ _{1}** and the (constant) distance from

In addition, given the very large mass of **μ _{1}** compared

These conditions are satisfied for example for the Sun–Jupiter system with a comet or a spacecraft being the third mass.

The Jacobi constant, a function of coordinates ξ, η, ζ, (distances r_{1}, r_{2} from the two masses) and the velocities remains the constant of motion through the encounter.

The goal is to express the constant using orbital parameters.

It is assumed, that far from the mass **μ _{2}**, the test particle (comet, spacecraft) is on an orbit around

Second, observing that the component of the angular momentum (per unit mass) is

where is the mutual inclination of the orbits of **μ _{3}** and

Substituting these into the Jacobi constant C_{J}, ignoring the term with **μ _{2}**<<1 and replacing r

In physics, a **Langevin equation** is a stochastic differential equation describing how a system evolves when subjected to a combination of deterministic and fluctuating ("random") forces. The dependent variables in a Langevin equation typically are collective (macroscopic) variables changing only slowly in comparison to the other (microscopic) variables of the system. The fast (microscopic) variables are responsible for the stochastic nature of the Langevin equation. One application is to Brownian motion, which models the fluctuating motion of a small particle in a fluid.

In mathematical physics, *n*-dimensional **de Sitter space** is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an *n*-sphere.

In classical mechanics, the **Laplace–Runge–Lenz (LRL) vector** is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit; equivalently, the LRL vector is said to be *conserved*. More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems.

In physics, the **Hamilton–Jacobi equation**, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics.

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In physics, precisely in the study of the theory of general relativity and many alternatives to it, the **post-Newtonian formalism** is a calculational tool that expresses Einstein's (nonlinear) equations of gravity in terms of the lowest-order deviations from Newton's law of universal gravitation. This allows approximations to Einstein's equations to be made in the case of weak fields. Higher-order terms can be added to increase accuracy, but for strong fields, it may be preferable to solve the complete equations numerically. Some of these post-Newtonian approximations are expansions in a small parameter, which is the ratio of the velocity of the matter forming the gravitational field to the speed of light, which in this case is better called the speed of gravity. In the limit, when the fundamental speed of gravity becomes infinite, the post-Newtonian expansion reduces to Newton's law of gravity.

In theoretical physics, **massive gravity** is a theory of gravity that modifies general relativity by endowing the graviton with a nonzero mass. In the classical theory, this means that gravitational waves obey a massive wave equation and hence travel at speeds below the speed of light.

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In fluid dynamics, a **cnoidal wave** is a nonlinear and exact periodic wave solution of the Korteweg–de Vries equation. These solutions are in terms of the Jacobi elliptic function *cn*, which is why they are coined *cn*oidal waves. They are used to describe surface gravity waves of fairly long wavelength, as compared to the water depth.

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**Accelerations in special relativity** (SR) follow, as in Newtonian Mechanics, by differentiation of velocity with respect to time. Because of the Lorentz transformation and time dilation, the concepts of time and distance become more complex, which also leads to more complex definitions of "acceleration". SR as the theory of flat Minkowski spacetime remains valid in the presence of accelerations, because general relativity (GR) is only required when there is curvature of spacetime caused by the energy–momentum tensor. However, since the amount of spacetime curvature is not particularly high on Earth or its vicinity, SR remains valid for most practical purposes, such as experiments in particle accelerators.

A **proper reference frame** in the theory of relativity is a particular form of accelerated reference frame, that is, a reference frame in which an accelerated observer can be considered as being at rest. It can describe phenomena in curved spacetime, as well as in "flat" Minkowski spacetime in which the spacetime curvature caused by the energy–momentum tensor can be disregarded. Since this article considers only flat spacetime—and uses the definition that special relativity is the theory of flat spacetime while general relativity is a theory of gravitation in terms of curved spacetime—it is consequently concerned with accelerated frames in special relativity.

In physics and mathematics, the **Klein–Kramers equation** or sometimes referred as **Kramers–Chandrasekhar equation** is a partial differential equation that describes the probability density function *f* of a Brownian particle in phase space (**r**, **p**). It is a special case of the Fokker–Planck equation.

- 1 2 Roy, John A.E. (Dec 31, 2004).
*Orbital Motion*(4th ed.). CRC Press. p. 121. ISBN 9781420056884. - 1 2 Gurzadyan, Grigor A. (Oct 21, 1996).
*Theory of Interplanetary Flights*. CRC Press. p. 192. ISBN 9782919875153. - 1 2 Danby, John M.A. (1992).
*Fundamentals of Celestial Mechanics*(2nd ed.). Willman-Bell Inc. pp. 253–254. ISBN 9780943396200.

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