140 relations: Adaptive mesh refinement, Angular momentum, Annals of Mathematics, Astronomical object, Évariste Galois, Barnes–Hut simulation, Barycenter, Black hole, Boltzmann equation, Celestial mechanics, Celestial Mechanics and Dynamical Astronomy, Center of mass, Chaos theory, Charles-Eugène Delaunay, Circle, Collinearity, Conic section, Conservation of energy, Constant of motion, Cross product, Determinism, Differential equation, Donald G. Saari, Double planet, Earth, Einstein field equations, Einstein–Infeld–Hoffmann equations, Electric potential, Electrostatics, Ellipse, Equations of motion, Equilateral triangle, Equilibrium point, Euler's three-body problem, Event horizon, Ewald summation, Fast Fourier transform, Fast multipole method, Few-body systems, Florian Cajori, Focus (geometry), Forest Ray Moulton, Gösta Mittag-Leffler, General relativity, Globular cluster, Gravitational constant, Gravitational two-body problem, Gravity, Hamiltonian mechanics, Henri Poincaré, ..., Hill sphere, Homogeneous function, Hyperbola, Inertial frame of reference, Initial condition, Invariant manifold, Isaac Newton, Isometry, Jacobi coordinates, Jacobi integral, Jean le Rond d'Alembert, Johann Bernoulli, Johannes Kepler, John Flamsteed, Joseph-Louis Lagrange, Jupiter, Karl F. Sundman, Kepler problem, Kepler's laws of planetary motion, Kernel density estimation, Kernel method, Kinetic energy, Kolmogorov–Arnold–Moser theorem, Lagrangian point, Lebesgue measure, Lemniscate, Leonhard Euler, Loss function, Lunar theory, Machine learning, Mass, Mean field theory, Moment of inertia, Moon, Multigrid method, Multipole expansion, Natural units, Newton's law of universal gravitation, Newton's laws of motion, Niels Henrik Abel, Nonlinear dimensionality reduction, Numerical integration, Numerical model of the Solar System, Numerical relativity, Orbital resonance, Oscar II of Sweden, P3M, Painlevé conjecture, Parabola, Parameterized post-Newtonian formalism, Paul Painlevé, Periodic boundary conditions, Perturbation theory, Philosophiæ Naturalis Principia Mathematica, Physics, Planet, Poisson's equation, Protein, Qiudong Wang, Quasiperiodic motion, Quintic function, Relative velocity, Rings of Saturn, Robert Hooke, Roche lobe, Rotational symmetry, Saturn, Scale invariance, Scholarpedia, Series (mathematics), Smoothed-particle hydrodynamics, Stability of the Solar System, Star, Statistics, Stephen Hawkins, Structural biology, Sun, Symplectic integrator, T-symmetry, Taylor series, Three-body problem, Time complexity, Topology, Translational symmetry, Trojan (astronomy), Two-body problem, Two-body problem in general relativity, Tycho Brahe, Virial theorem, Vladimir Arnold. Expand index (90 more) » « Shrink index
In numerical analysis, adaptive mesh refinement, or AMR, is a method of adapting the accuracy of a solution within certain sensitive or turbulent regions of simulation, dynamically and during the time the solution is being calculated.
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of linear momentum.
The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study.
An astronomical object or celestial object is a naturally occurring physical entity, association, or structure that exists in the observable universe.
Évariste Galois (25 October 1811 – 31 May 1832) was a French mathematician.
The Barnes–Hut simulation (Josh Barnes and Piet Hut) is an approximation algorithm for performing an ''n''-body simulation.
The barycenter (or barycentre; from the Ancient Greek βαρύς heavy + κέντρον centre) is the center of mass of two or more bodies that are orbiting each other, which is the point around which they both orbit.
A black hole is a region of spacetime exhibiting such strong gravitational effects that nothing—not even particles and electromagnetic radiation such as light—can escape from inside it.
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.
Celestial mechanics is the branch of astronomy that deals with the motions of celestial objects.
Celestial Mechanics and Dynamical Astronomy is a scientific journal covering the fields of astronomy and astrophysics.
In physics, the center of mass of a distribution of mass in space is the unique point where the weighted relative position of the distributed mass sums to zero, or the point where if a force is applied it moves in the direction of the force without rotating.
Chaos theory is a branch of mathematics focusing on the behavior of dynamical systems that are highly sensitive to initial conditions.
Charles-Eugène Delaunay (9 April 1816 – 5 August 1872) was a French astronomer and mathematician.
A circle is a simple closed shape.
In geometry, collinearity of a set of points is the property of their lying on a single line.
In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane.
In physics, the law of conservation of energy states that the total energy of an isolated system remains constant, it is said to be ''conserved'' over time.
In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion.
In mathematics and vector algebra, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space \left(\mathbb^3\right) and is denoted by the symbol \times.
Determinism is the philosophical theory that all events, including moral choices, are completely determined by previously existing causes.
A differential equation is a mathematical equation that relates some function with its derivatives.
Donald Gene Saari (born March 1940) is an American mathematician, the Distinguished Professor of Mathematics and Economics and director of the Institute for Mathematical Behavioral Sciences at the University of California, Irvine.
In astronomy, a double planet (also binary planet) is a binary system where both objects are of planetary mass.
Earth is the third planet from the Sun and the only astronomical object known to harbor life.
The Einstein field equations (EFE; also known as Einstein's equations) comprise the set of 10 equations in Albert Einstein's general theory of relativity that describe the fundamental interaction of gravitation as a result of spacetime being curved by mass and energy.
The Einstein–Infeld–Hoffmann equations of motion, jointly derived by Albert Einstein, Leopold Infeld and Banesh Hoffmann, are the differential equations of motion describing the approximate dynamics of a system of point-like masses due to their mutual gravitational interactions, including general relativistic effects.
An electric potential (also called the electric field potential, potential drop or the electrostatic potential) is the amount of work needed to move a unit positive charge from a reference point to a specific point inside the field without producing any acceleration.
Electrostatics is a branch of physics that studies electric charges at rest.
In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.
In geometry, an equilateral triangle is a triangle in which all three sides are equal.
In mathematics, specifically in differential equations, an equilibrium point is a constant solution to a differential equation.
In physics and astronomy, Euler's three-body problem is to solve for the motion of a particle that is acted upon by the gravitational field of two other point masses that are fixed in space.
In general relativity, an event horizon is a region in spacetime beyond which events cannot affect an outside observer.
Ewald summation, named after Paul Peter Ewald, is a method for computing long-range interactions (e.g., electrostatic interactions) in periodic systems.
A fast Fourier transform (FFT) is an algorithm that samples a signal over a period of time (or space) and divides it into its frequency components.
The fast multipole method (FMM) is a numerical technique that was developed to speed up the calculation of long-ranged forces in the ''n''-body problem.
In mechanics, a few-body system consists of a small number of well-defined structures or point particles.
Florian Cajori (February 28, 1859 – August 14 or 15, 1930) was a Swiss-American historian of mathematics.
In geometry, focuses or foci, singular focus, are special points with reference to which any of a variety of curves is constructed.
Forest Ray Moulton (April 29, 1872 – December 7, 1952) was an American astronomer.
Magnus Gustaf (Gösta) Mittag-Leffler (16 March 1846 – 7 July 1927) was a Swedish mathematician.
General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics.
A globular cluster is a spherical collection of stars that orbits a galactic core as a satellite.
The gravitational constant (also known as the "universal gravitational constant", the "Newtonian constant of gravitation", or the "Cavendish gravitational constant"), denoted by the letter, is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general theory of relativity.
←For further relevant mathematical developments see also Two-body problem, also Kepler orbit, and Kepler problem, and Equation of the center – Analytical expansions The gravitational two-body problem concerns the motion of two point particles that interact only with each other, due to gravity.
Gravity, or gravitation, is a natural phenomenon by which all things with mass or energy—including planets, stars, galaxies, and even light—are brought toward (or gravitate toward) one another.
Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics and predicts the same outcomes as non-Hamiltonian classical mechanics.
Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science.
An astronomical body's Hill sphere is the region in which it dominates the attraction of satellites.
In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor.
In mathematics, a hyperbola (plural hyperbolas or hyperbolae) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set.
An inertial frame of reference in classical physics and special relativity is a frame of reference in which a body with zero net force acting upon it is not accelerating; that is, such a body is at rest or it is moving at a constant speed in a straight line.
In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t.
In dynamical systems, a branch of mathematics, an invariant manifold is a topological manifold that is invariant under the action of the dynamical system.
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, astronomer, theologian, author and physicist (described in his own day as a "natural philosopher") who is widely recognised as one of the most influential scientists of all time, and a key figure in the scientific revolution.
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.
In the theory of many-particle systems, Jacobi coordinates often are used to simplify the mathematical formulation.
In celestial mechanics, Jacobi's integral (also known as the Jacobi integral or Jacobi constant) is the only known conserved quantity for the circular restricted three-body problem.
Jean-Baptiste le Rond d'Alembert (16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist.
Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family.
Johannes Kepler (December 27, 1571 – November 15, 1630) was a German mathematician, astronomer, and astrologer.
John Flamsteed FRS (19 August 1646 – 31 December 1719) was an English astronomer and the first Astronomer Royal.
Joseph-Louis Lagrange (or;; born Giuseppe Lodovico Lagrangia, Encyclopædia Britannica or Giuseppe Ludovico De la Grange Tournier, Turin, 25 January 1736 – Paris, 10 April 1813; also reported as Giuseppe Luigi Lagrange or Lagrangia) was an Italian Enlightenment Era mathematician and astronomer.
Jupiter is the fifth planet from the Sun and the largest in the Solar System.
Karl Frithiof Sundman (28 October 1873, Kaskinen28 September 1949, Helsinki) was a Finnish mathematician who used analytic methods to prove the existence of a convergent infinite series solution to the three-body problem in 1906 and 1909.
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.
In astronomy, Kepler's laws of planetary motion are three scientific laws describing the motion of planets around the Sun.
In statistics, kernel density estimation (KDE) is a non-parametric way to estimate the probability density function of a random variable.
In machine learning, kernel methods are a class of algorithms for pattern analysis, whose best known member is the support vector machine (SVM).
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
The Kolmogorov–Arnold–Moser theorem (KAM theorem) is a result in dynamical systems about the persistence of quasiperiodic motions under small perturbations.
In celestial mechanics, the Lagrangian points (also Lagrange points, L-points, or libration points) are positions in an orbital configuration of two large bodies, wherein a small object, affected only by the gravitational forces from the two larger objects, will maintain its position relative to them.
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space.
In algebraic geometry, a lemniscate is any of several figure-eight or -shaped curves.
Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.
In mathematical optimization, statistics, econometrics, decision theory, machine learning and computational neuroscience, a loss function or cost function is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event.
Lunar theory attempts to account for the motions of the Moon.
Machine learning is a subset of artificial intelligence in the field of computer science that often uses statistical techniques to give computers the ability to "learn" (i.e., progressively improve performance on a specific task) with data, without being explicitly programmed.
Mass is both a property of a physical body and a measure of its resistance to acceleration (a change in its state of motion) when a net force is applied.
In physics and probability theory, mean field theory (MFT also known as self-consistent field theory) studies the behavior of large and complex stochastic models by studying a simpler model.
The moment of inertia, otherwise known as the angular mass or rotational inertia, of a rigid body is a tensor that determines the torque needed for a desired angular acceleration about a rotational axis; similar to how mass determines the force needed for a desired acceleration.
The Moon is an astronomical body that orbits planet Earth and is Earth's only permanent natural satellite.
Multigrid (MG) methods in numerical analysis are algorithms for solving differential equations using a hierarchy of discretizations.
A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles on a sphere.
In physics, natural units are physical units of measurement based only on universal physical constants.
Newton's law of universal gravitation states that a particle attracts every other particle in the universe with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
Newton's laws of motion are three physical laws that, together, laid the foundation for classical mechanics.
Niels Henrik Abel (5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields.
High-dimensional data, meaning data that requires more than two or three dimensions to represent, can be difficult to interpret.
In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations.
A numerical model of the Solar System is a set of mathematical equations, which, when solved, give the approximate positions of the planets as a function of time.
Numerical relativity is one of the branches of general relativity that uses numerical methods and algorithms to solve and analyze problems.
In celestial mechanics, an orbital resonance occurs when orbiting bodies exert a regular, periodic gravitational influence on each other, usually because their orbital periods are related by a ratio of small integers.
Oscar II (Oscar Fredrik; 21 January 1829 – 8 December 1907) was King of Sweden from 1872 until his death, and the last Bernadotte King of Norway from 1872 until his dethronement in 1905.
Particle–Particle-Particle–Mesh (P3M) is a Fourier-based Ewald summation method to calculate potentials in N-body simulations.
In physics, the Painlevé conjecture is a conjecture about singularities among the solutions to the ''n''-body problem: there are noncollision singularities for n ≥ 4.
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped.
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.
Paul Painlevé (5 December 1863 – 29 October 1933) was a French mathematician and statesman.
Periodic boundary conditions (PBCs) are a set of boundary conditions which are often chosen for approximating a large (infinite) system by using a small part called a unit cell.
Perturbation theory comprises mathematical methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem.
Philosophiæ Naturalis Principia Mathematica (Latin for Mathematical Principles of Natural Philosophy), often referred to as simply the Principia, is a work in three books by Isaac Newton, in Latin, first published 5 July 1687.
Physics (from knowledge of nature, from φύσις phýsis "nature") is the natural science that studies matterAt the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept: "If, in some cataclysm, all scientific knowledge were to be destroyed one sentence what statement would contain the most information in the fewest words? I believe it is that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another..." and its motion and behavior through space and time and that studies the related entities of energy and force."Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves."Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena.""Physics is the study of your world and the world and universe around you." Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics often enable advances in new technologies. For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.
A planet is an astronomical body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.
In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics.
Proteins are large biomolecules, or macromolecules, consisting of one or more long chains of amino acid residues.
Qiudong Wang is a Professor at the Department of Mathematics, the University of Arizona.
In mathematics and theoretical physics, quasiperiodic motion is in rough terms the type of motion executed by a dynamical system containing a finite number (two or more) of incommensurable frequencies.
In algebra, a quintic function is a function of the form where,,,, and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero.
The relative velocity \vec_ (also \vec_ or \vec_) is the velocity of an object or observer B in the rest frame of another object or observer A.
The rings of Saturn are the most extensive ring system of any planet in the Solar System.
Robert Hooke FRS (– 3 March 1703) was an English natural philosopher, architect and polymath.
The Roche lobe (or Roche limit) is the region around a star in a binary system within which orbiting material is gravitationally bound to that star.
Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn.
Saturn is the sixth planet from the Sun and the second-largest in the Solar System, after Jupiter.
In physics, mathematics, statistics, and economics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, thus represent a universality.
Scholarpedia is an English-language online wiki-based encyclopedia with features commonly associated with open-access online academic journals, which aims to have quality content.
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.
Smoothed-particle hydrodynamics (SPH) is a computational method used for simulating the mechanics of continuum media, such as solid mechanics and fluid flows.
The stability of the Solar System is a subject of much inquiry in astronomy.
A star is type of astronomical object consisting of a luminous spheroid of plasma held together by its own gravity.
Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data.
Stephen Mark Hawkins OAM (born 14 January 1971) is an Australian former national champion, World Champion and Olympic gold medal winning lightweight rower.
Structural biology is a branch of molecular biology, biochemistry, and biophysics concerned with the molecular structure of biological macromolecules (especially proteins, made up of amino acids, and RNA or DNA, made up of nucleic acids), how they acquire the structures they have, and how alterations in their structures affect their function.
The Sun is the star at the center of the Solar System.
In mathematics, a symplectic integrator (SI) is a numerical integration scheme for Hamiltonian systems.
T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal: T-symmetry can be shown to be equivalent to the conservation of entropy, by Noether's Theorem.
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.
In physics and classical mechanics, the three-body problem is the problem of taking an initial set of data that specifies the positions, masses, and velocities of three bodies for some particular point in time and then determining the motions of the three bodies, in accordance with Newton's laws of motion and of universal gravitation, which are the laws of classical mechanics.
In computer science, the time complexity is the computational complexity that describes the amount of time it takes to run an algorithm.
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.
In geometry, a translation "slides" a thing by a: Ta(p).
In astronomy, a trojan is a minor planet or moon that shares the orbit of a planet or larger moon, wherein the trojan remains in the same, stable position relative to the larger object.
In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other.
The two-body problem (or Kepler problem) in general relativity is the determination of the motion and gravitational field of two bodies as described by the field equations of general relativity.
Tycho Brahe (born Tyge Ottesen Brahe;. He adopted the Latinized form "Tycho Brahe" (sometimes written Tÿcho) at around age fifteen. The name Tycho comes from Tyche (Τύχη, meaning "luck" in Greek, Roman equivalent: Fortuna), a tutelary deity of fortune and prosperity of ancient Greek city cults. He is now generally referred to as "Tycho," as was common in Scandinavia in his time, rather than by his surname "Brahe" (a spurious appellative form of his name, Tycho de Brahe, only appears much later). 14 December 154624 October 1601) was a Danish nobleman, astronomer, and writer known for his accurate and comprehensive astronomical and planetary observations.
In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy, \left\langle T \right\rangle, of a stable system consisting of N particles, bound by potential forces, with that of the total potential energy, \left\langle V_\text \right\rangle, where angle brackets represent the average over time of the enclosed quantity.
Vladimir Igorevich Arnold (alternative spelling Arnol'd, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician.