two-body motion
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In
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
, the two-body problem is to predict the motion of two massive objects which are abstractly viewed as
point particle A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take up ...
s. The problem assumes that the two objects interact only with one another; the only force affecting each object arises from the other one, and all other objects are ignored. The most prominent case of the classical two-body problem is the gravitational case (see also Kepler problem), arising in astronomy for predicting the orbits (or escapes from orbit) of objects such as
satellite A satellite or artificial satellite is an object intentionally placed into orbit in outer space. Except for passive satellites, most satellites have an electricity generation system for equipment on board, such as solar panels or radioi ...
s,
planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
s, and
stars A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth ma ...
. A two-point-particle model of such a system nearly always describes its behavior well enough to provide useful insights and predictions. A simpler "one body" model, the " central-force problem", treats one object as the immobile source of a force acting on the other. One then seeks to predict the motion of the single remaining mobile object. Such an approximation can give useful results when one object is much more massive than the other (as with a light planet orbiting a heavy star, where the star can be treated as essentially stationary). However, the one-body approximation is usually unnecessary except as a stepping stone. For many forces, including gravitational ones, the general version of the two-body problem can be reduced to a pair of one-body problems, allowing it to be solved completely, and giving a solution simple enough to be used effectively. By contrast, the
three-body problem In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...
(and, more generally, the ''n''-body problem for ''n'' ≥ 3) cannot be solved in terms of first integrals, except in special cases.


Results for prominent cases


Gravitation and other inverse-square examples

The two-body problem is interesting in astronomy because pairs of astronomical objects are often moving rapidly in arbitrary directions (so their motions become interesting), widely separated from one another (so they will not collide) and even more widely separated from other objects (so outside influences will be small enough to be ignored safely). Under the force of
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
, each member of a pair of such objects will orbit their mutual center of mass in an elliptical pattern, unless they are moving fast enough to escape one another entirely, in which case their paths will diverge along other planar
conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...
s. If one object is very much heavier than the other, it will move far less than the other with reference to the shared center of mass. The mutual center of mass may even be inside the larger object. For the derivation of the solutions to the problem, see Classical central-force problem or Kepler problem. In principle, the same solutions apply to macroscopic problems involving objects interacting not only through gravity, but through any other attractive scalar force field obeying an inverse-square law, with
electrostatic attraction Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
being the obvious physical example. In practice, such problems rarely arise. Except perhaps in experimental apparatus or other specialized equipment, we rarely encounter electrostatically interacting objects which are moving fast enough, and in such a direction, as to avoid colliding, and/or which are isolated enough from their surroundings. The dynamical system of a two-body system under the influence of torque turns out to be a Sturm-Liouville equation.


Inapplicability to atoms and subatomic particles

Although the two-body model treats the objects as point particles, classical mechanics only apply to systems of macroscopic scale. Most behavior of subatomic particles ''cannot'' be predicted under the classical assumptions underlying this article or using the mathematics here.
Electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
s in an atom are sometimes described as "orbiting" its
nucleus Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to: *Atomic nucleus, the very dense central region of an atom * Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA Nucl ...
, following an early conjecture of
Niels Bohr Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922 ...
(this is the source of the term " orbital"). However, electrons don't actually orbit nuclei in any meaningful sense, and
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
are necessary for any useful understanding of the electron's real behavior. Solving the classical two-body problem for an electron orbiting an atomic nucleus is misleading and does not produce many useful insights.


Reduction to two independent, one-body problems

The complete two-body problem can be solved by re-formulating it as two one-body problems: a trivial one and one that involves solving for the motion of one particle in an external
potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
. Since many one-body problems can be solved exactly, the corresponding two-body problem can also be solved. Let and be the vector positions of the two bodies, and ''m''1 and ''m''2 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 Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in moti ...
states that where F12 is the force on mass 1 due to its interactions with mass 2, and F21 is the force on mass 2 due to its interactions with mass 1. The two dots on top of the x position vectors denote their second derivative with respect to time, or their acceleration vectors. Adding and subtracting these two equations decouples them into two one-body problems, which can be solved independently. ''Adding'' equations (1) and () results in an equation describing the center of mass (
barycenter In astronomy, the barycenter (or barycentre; ) is the center of mass of two or more bodies that orbit one another and is the point about which the bodies orbit. A barycenter is a dynamical point, not a physical object. It is an important con ...
) 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 .


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

Let \mathbf be the position of the center of mass (
barycenter In astronomy, the barycenter (or barycentre; ) is the center of mass of two or more bodies that orbit one another and is the point about which the bodies orbit. A barycenter is a dynamical point, not a physical object. It is an important con ...
) of the system. Addition of the force equations (1) and (2) yields m_1 \ddot_1 + m_2 \ddot_2 = (m_1 + m_2)\ddot = \mathbf_ + \mathbf_ = 0 where we have used
Newton's third law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in moti ...
and where \ddot \equiv \frac. The resulting equation: \ddot = 0 shows that the velocity \mathbf = \frac of the center of mass is constant, from which follows that the total momentum is also constant (
conservation of momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
). Hence, the position of the center of mass can be determined at all times from the initial positions and velocities.


Displacement vector motion (2nd one-body problem)

Dividing both force equations by the respective masses, subtracting the second equation from the first, and rearranging gives the equation \ddot = \ddot_ - \ddot_ = \left( \frac - \frac \right) = \left(\frac + \frac \right)\mathbf_ where we have again used
Newton's third law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in moti ...
and where is the
displacement vector In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along a ...
from mass 2 to mass 1, as defined above. The force between the two objects, which originates in the two objects, should only be a function of their separation and not of their absolute positions and ; otherwise, there would not be
translational symmetry In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equati ...
, and the laws of physics would have to change from place to place. The subtracted equation can therefore be written: \mu \ddot = \mathbf_(\mathbf_,\mathbf_) = \mathbf(\mathbf) where \mu is the
reduced mass In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, however, that the mass ...
\mu = \frac = \frac. Solving the equation for is the key to the two-body problem. The solution depends on the specific force between the bodies, which is defined by \mathbf(\mathbf). For the case where \mathbf(\mathbf) follows an inverse-square law, see the Kepler problem. Once and have been determined, the original trajectories may be obtained \mathbf_1(t) = \mathbf (t) + \frac \mathbf(t) \mathbf_2(t) = \mathbf (t) - \frac \mathbf(t) as may be verified by substituting the definitions of R and r into the right-hand sides of these two equations.


Two-body motion is planar

The motion of two bodies with respect to each other always lies in a plane (in the center of mass frame). Proof: Defining the
linear momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass a ...
and the
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
of the system, with respect to the center of mass, by the equations \mathbf = \mathbf \times \mathbf = \mathbf \times \mu \frac, where is the
reduced mass In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, however, that the mass ...
and is the relative position (with these written taking the center of mass as the origin, and thus both parallel to ) the rate of change of the angular momentum equals the net
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
\mathbf = \frac = \dot \times \mu\dot + \mathbf \times \mu\ddot \ , and using the property of the
vector cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is d ...
that for any vectors and pointing in the same direction, \mathbf \ = \ \frac = \mathbf \times \mathbf \ , with . Introducing the assumption (true of most physical forces, as they obey Newton's strong third law of motion) that the force between two particles acts along the line between their positions, it follows that and the angular momentum vector is constant (conserved). Therefore, the displacement vector and its velocity are always in the plane
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It ca ...
to the constant vector .


Energy of the two-body system

If the force is
conservative Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization in ...
then the system has a potential energy , so the total
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
can be written as E_\text = \frac m_1 \dot_1^2 + \frac m_2 \dot_2^2 + U(\mathbf) = \frac (m_1 + m_2) \dot^2 + \mu \dot^2 + U(\mathbf) In the center of mass frame the
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...
is the lowest and the total energy becomes E = \frac \mu \dot^2 + U(\mathbf) The coordinates and can be expressed as \mathbf_1 = \frac \mathbf \mathbf_2 = - \frac \mathbf and in a similar way the energy ''E'' is related to the energies and that separately contain the kinetic energy of each body: \begin E_1 & = \frac E = \frac m_1 \dot_1^2 + \frac U(\mathbf) \\ ptE_2 & = \frac E = \frac m_2 \dot_2^2 + \frac U(\mathbf) \\ ptE_\text & = E_1 + E_2 \end


Central forces

For many physical problems, the force is a
central force In classical mechanics, a central force on an object is a force that is directed towards or away from a point called center of force. : \vec = \mathbf(\mathbf) = \left\vert F( \mathbf ) \right\vert \hat where \vec F is the force, F is a vecto ...
, i.e., it is of the form \mathbf(\mathbf) = F(r)\hat where and is the corresponding
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
. We now have: \mu \ddot = (r) \hat \ , where ''F''(''r'') is negative in the case of an attractive force.


See also

* Energy drift *
Equation of the center In two-body, Keplerian orbital mechanics, the equation of the center is the angular difference between the actual position of a body in its elliptical orbit and the position it would occupy if its motion were uniform, in a circular orbit of the ...
* Euler's three-body problem *
Kepler orbit Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...
* Kepler problem * ''n''-body problem *
Virial theorem In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by potential forces, with that of the total potential energy of the system. ...


References


Bibliography

* *


External links


Two-body problem
at Eric Weisstein's World of Physics {{DEFAULTSORT:Two-Body Problem Orbits Dynamical systems