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Encyclopedia > Two body problem

In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, two stars orbiting each other (a binary star), and a classical electron orbiting an atomic nucleus. Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... An Earth observation satellite, ERS 2 For other uses, see Satellite (disambiguation). ... For the astrological concept, see Planets in astrology. ... For the astrological concept, see Planets in astrology. ... This article is about the astronomical object. ... This article is about the astronomical object. ... This article is about the astronomical phenomenon. ... The electron is a fundamental subatomic particle that carries an electric charge. ... A semi-accurate depiction of the helium atom. ...


The two-body problem can be re-formulated as two independent one-body problems, which involve solving for the motion of one particle in an external potential. Since many one-body problems can be solved exactly, the corresponding two-body problem can also be solved. By contrast, the three-body problem (and, more generally, the n-body problem for ) cannot be solved, except in special cases. It has been suggested that this article or section be merged with Scalar potential. ...

Two bodies with similar mass orbiting around a common barycenter with elliptic orbits.
Two bodies with similar mass orbiting around a common barycenter with elliptic orbits.
Two bodies with a slight difference in mass orbiting around a common barycenter. The sizes, and this particular type of orbit are similar to the Pluto-Charon system.
Two bodies with a slight difference in mass orbiting around a common barycenter. The sizes, and this particular type of orbit are similar to the Pluto-Charon system.

Contents

Image File history File links Two bodies with similar mass orbiting around a common barycenter (red cross) with elliptic orbits. ... Image File history File links Two bodies with similar mass orbiting around a common barycenter (red cross) with elliptic orbits. ... It has been suggested that Center of gravity be merged into this article or section. ... Image File history File links Two bodies with a slight difference in mass orbiting around a common barycenter (red cross) with circular orbits. ... Image File history File links Two bodies with a slight difference in mass orbiting around a common barycenter (red cross) with circular orbits. ... Unsolved problems in physics: What causes anything to have mass? Mass is a property of a physical object that quantifies the amount of matter and energy it is equivalent to. ... It has been suggested that Center of gravity be merged into this article or section. ... Adjective Plutonian Atmospheric characteristics Atmospheric pressure 0. ... Media:Example. ...

Statement of problem

Let and be the positions of the two bodies, and m1 and m2 be their masses. The goal is to determine the trajectories and for all times t, given the initial positions

and

and the initial velocities

and .

When applied to the two masses, Newton's second law states that Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ...

where

is the force on mass 1 due to its interactions with mass 2, and
is the force on mass 2 due to its interactions with mass 1.

Adding and subtracting these two equations decouples them into two one-body problems, which can be solved independently. Adding equations (1) and (2) results in an equation describing the center of mass (barycenter) motion. By contrast, subtracting equation (2) from equation (1) results in an equation that describes how the vector between the masses changes with time. The solutions of these independent one-body problems can be combined to obtain the solutions for the trajectories and . In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ... It has been suggested that Center of gravity be merged into this article or section. ...


Center of mass motion (1st one-body problem)

Addition of the force equations (1) and (2) yields

where we have used Newton's third law and where Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ...

is the position of the center of mass (barycenter) of the system. The resulting equation In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ... It has been suggested that Center of gravity be merged into this article or section. ...

shows that the velocity of the center of mass is constant, from which follows that the total momentum is also constant (conservation of momentum). Hence, the position and velocity of the center of mass can be determined at all times from the initial positions and velocities. In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ...


Displacement vector motion (2nd one-body problem)

Subtracting force equation (2) from force equation (1) and rearranging gives the equation

where we have again used Newton's third law and where (defined above) is the displacement vector from mass 2 to mass 1. Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... In Newtonian mechanics, displacement is the vector that specifies the position of a point or a particle in reference to an origin or to a previous position. ...


The force between the two objects should only be a function of and not of their absolute positions and ; otherwise, physics would not have translational symmetry, i.e., the laws of physics would change from place to place. Therefore, the subtracted equation can be written Physics (from the Greek, (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space, and time. ... A translation slides an object by a vector a: Ta(p) = p + a. ...

where μ is the reduced mass Reduced mass is an algebraic term of the form that simplifies an equation of the form The reduced mass is typically used as a relationship between two system elements in parallel, such as resistors; whether these be in the electrical, thermal, hydraulic, or mechanical domains. ...

Once we have solved for and , the original trajectories may be obtained from the equations

as may be verified by substituting into the defining equations for and .


Two-body motion is planar

Remarkably, the motion of two bodies always lies in a plane. Let us define the linear momentum and the angular momentum In physics, momentum is a physical quantity related to the velocity and mass of an object. ... Gyroscope. ...

The rate of change of the angular momentum equals the net torque It has been suggested that this article or section be merged with Moment (physics). ...

However, Newton's strong third law of motion holds for most physical forces, and says that the force between two particles acts along the line between their positions, i.e., . Therefore, and angular momentum is conserved. Therefore, the displacement vector and its velocity are always in the plane perpendicular to the constant vector . Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... In physics, angular momentum intuitively measures how much the linear momentum is directed around a certain point called the origin; the moment of momentum. ... Fig. ...


General solution for distance-dependent forces

It is often useful to switch to polar coordinates, since the motion is planar and, for many physical problems, the force is a function only of the radius r (a central force). Since the r-component of acceleration is , the r-component of the displacement vector equation can be written This article describes some of the common coordinate systems that appear in elementary mathematics. ... A central force acting on an object is one whose magnitude depends only on the scalar distance r of the object from the origin and whose direction is along the position vector from the origin to the object. ...

where and the angular momentum L = μr2ω is conserved. The conservation of angular momentum allows us to solve for the trajectory r(θ) by making a change of independent variable from t to θ Gyroscope. ... In physics, angular momentum intuitively measures how much the linear momentum is directed around a certain point called the origin; the moment of momentum. ...

giving the new equation of motion

This equation becomes quasilinear on making the change of variables and multiplying both sides by

Application to inverse-square force laws

If F is an inverse-square law central force such as gravity or electrostatics in classical physics A central force acting on an object is one whose magnitude depends only on the scalar distance r of the object from the origin and whose direction is along the position vector from the origin to the object. ... Gravity is a force of attraction that acts between bodies that have mass. ... Electrostatics is the branch of physics that deals with the forces exerted by a static (i. ... Classical physics is physics based on principles developed before the rise of quantum theory, usually including the special theory of relativity and general theory of relativity. ...

for some constant α, the trajectory equation becomes linear

The solution of this equation is

where A > 0 and θ0 are constants. This solution shows that the orbit is a conic section, i.e., an ellipse, a hyperbola or parabola, depending on whether A is less than, greater than, or equal to . This special case of the two-body problem is called the Kepler problem. Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ... The ellipse and some of its mathematical properties. ... A graph of a hyperbola. ... For other uses, see Parabola (disambiguation). ... In mathematics, the Kepler conjecture is a conjecture about sphere packing in three dimensional Euclidean space. ...


Examples

Any classical system of two particles is, by definition, a two-body problem. In many cases, however, one particle is significantly heavier than the other, e.g., the Earth and the Sun. In such cases, the heavier particle is approximately the center of mass, and the reduced mass is approximately the lighter mass. Hence, the heavier mass may be treated roughly as a fixed center of force, and the motion of the lighter mass may be solved for directly by one-body methods. Earth (IPA: , often referred to as the Earth, Terra, the World or Planet Earth) is the third planet in the solar system in terms of distance from the Sun, and the fifth largest. ... The Sun is the star at the centre of the Solar System. ...


In other cases, however, the masses of the two bodies are roughly equal, so that neither of them can be approximated as being at rest. Astronomical examples include:

This article is about the astronomical phenomenon. ... Alpha Centauri (α Cen / α Centauri) is the brightest star system in the southern constellation of Centaurus. ... Pluto and Charon are sometimes informally considered to be a double (dwarf) planet. ... Atmospheric characteristics Atmospheric pressure 0. ... Media:Example. ... The term binary asteroid refers to a system in which two asteroids orbit their common centre of gravity, in analogy with binary stars. ... 90 Antiope (an-tye-a-pee) is an asteroid discovered on October 1, 1866 by Robert Luther. ...

Other usages

The phrase two body problem is also used to refer to the problem of two academics (usually married) finding jobs at the same university.


See also

Johannes Keplers primary contributions to astronomy/astrophysics were his three laws of planetary motion. ... In physics, the virial theorem states that the average kinetic energy of a system of particles whose motions are bounded is given by where ri and Fi are the position and force vectors on the i th particle respectively. ... The n-body problem is the problem of finding, given the initial positions, masses, and velocities of n bodies, their subsequent motions as determined by classical mechanics, i. ... In classical mechanics, Bertrands theorem states that only two types of potentials produce stable, closed orbits: an inverse-square force such as the gravitational or electrostatic potential and the radial harmonic oscillator potential // General Preliminaries All attractive central forces can produce circular orbits, which are naturally closed orbits. ... In mathematics, the Kepler conjecture is a conjecture about sphere packing in three dimensional Euclidean space. ...

References

  • Lev D. Landau and E. M. Lifshitz, (1976) Mechanics, 3rd. ed., Pergamon Press. ISBN 0-08-021022-8 (hardcover) and ISBN 0-08-029141-4 (softcover).
  • H. Goldstein, (1980) Classical Mechanics, 2nd. ed., Addison-Wesley. ISBN 0-201-02918-9

 
 

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