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Binet Equation
The Binet equation, derived by Jacques Philippe Marie Binet, provides the form of a central force given the shape of the orbital motion in plane polar coordinates. The equation can also be used to derive the shape of the orbit for a given force law, but this usually involves the solution to a second order nonlinear ordinary differential equation. A unique solution is impossible in the case of circular motion about the center of force. Equation The shape of an orbit is often conveniently described in terms of relative distance r as a function of angle \theta . For the Binet equation, the orbital shape is instead more concisely described by the reciprocal u=1/r as a function of \theta. Define the specific angular momentum as h=L/m where L is the angular momentum and m is the mass. The Binet equation, derived in the next section, gives the force in terms of the function u(\theta) : :F(^)=-mh^u^\left(\frac+u\right). Derivation Newton's Second Law for a purely central force is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacques Philippe Marie Binet
Jacques Philippe Marie Binet (; 2 February 1786 – 12 May 1856) was a French mathematician, physicist and astronomer born in Rennes; he died in Paris, France, in 1856. He made significant contributions to number theory, and the mathematical foundations of matrix algebra which would later lead to important contributions by Cayley and others. In his memoir on the theory of the conjugate axis and of the moment of inertia of bodies he enumerated the principle now known as ''Binet's theorem''. He is also recognized as the first to describe the rule for multiplying matrices in 1812, and ''Binet's formula'' expressing Fibonacci numbers in closed form is named in his honour, although the same result was known to Abraham de Moivre a century earlier. Career Binet graduated from l'École Polytechnique in 1806, and returned as a teacher in 1807. He advanced in position until 1816 when he became an inspector of studies at l'École. He held this post until 13 November 1830, when he was dism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwarzschild Coordinates
In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of ''nested round spheres''. In such a spacetime, a particularly important kind of coordinate chart is the Schwarzschild chart, a kind of polar spherical coordinate chart on a static and spherically symmetric spacetime, which is ''adapted'' to these nested round spheres. The defining characteristic of Schwarzschild chart is that the radial coordinate possesses a natural geometric interpretation in terms of the surface area and Gaussian curvature of each sphere. However, radial distances and angles are not accurately represented. These charts have many applications in metric theories of gravitation such as general relativity. They are most often used in static spherically symmetric spacetimes. In the case of general relativity, Birkhoff's theorem states that every ''isolated'' spherically symmetric vacuum or electrovacuum solution of the Einstein field equation is static, but this i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epispiral
The epispiral is a plane curve with polar equation :\ r=a \sec. There are ''n'' sections if ''n'' is odd and 2''n'' if ''n'' is even. It is the polar or circle inversive geometry, inversion of the rose (mathematics), rose curve. In astronomy the epispiral is Newton%27s_theorem_of_revolving_orbits#Illustrative_example:_Cotes's_spirals, related to the equations that explain planet, planets' orbits. See also *Logarithmic spiral *Rose (mathematics) References * * https://www.mathcurve.com/courbes2d.gb/epi/epi.shtml {{geometry-stub Plane curves ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cotes Spiral
Introduction In physics and in the mathematics of plane curves, a Cotes's spiral (also written Cotes' spiral and Cotes spiral) is one of a family of spirals classified by Roger Cotes. Cotes introduces his analysis of these curves as follows: “It is proposed to list the different types of trajectories which bodies can move along when acted on by centripetal forces in the inverse ratio of the cubes of their distances, proceeding from a given place, with given speed, and direction.” (N. b. he does not describe them as spirals). The shape of spirals in the family depends on the parameters. The curves in polar coordinates, (''r'', ''θ''), ''r'' > 0 are defined by one of the following five equations: : \frac = \begin A \cosh(k\theta + \varepsilon) \\ A \exp(k\theta + \varepsilon) \\ A \sinh(k\theta + \varepsilon) \\ A (k\theta + \varepsilon) \\ A \cos(k\theta + \varepsilon) \\ \end ''A'' > 0, ''k'' > 0 and ''ε'' are arbitrary real number constants. ''A'' determines th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time or four-dimensional spacetime. In particular, the ' is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of second order partial differential equations. Newton's law of universal gravitation, which describes classical gravity, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond Newton's law of universal gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parameterized Post-Newtonian Formalism
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. The parameterized post-Newtonian formalism or PPN formalism, is a version of this form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coulomb's Law
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 conventionally called ''electrostatic force'' or Coulomb force. Although the law was known earlier, it was first published in 1785 by French physicist Charles-Augustin de Coulomb, hence the name. Coulomb's law was essential to the development of the theory of electromagnetism, maybe even its starting point, as it made it possible to discuss the quantity of electric charge in a meaningful way. The law states that the magnitude of the electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them. Coulomb studied the repulsive force between bodies having electrical charges of the same sign: Coulomb al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton's Law Of Universal Gravitation
Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers.It was shown separately that separated spherically symmetrical masses attract and are attracted as if all their mass were concentrated at their centers. The publication of the law has become known as the " first great unification", as it marked the unification of the previously described phenomena of gravity on Earth with known astronomical behaviors. This is a general physical law derived from empirical observations by what Isaac Newton called inductive reasoning. It is a part of classical mechanics and was formulated in Newton's work '' Philosophiæ Naturalis Principia Mathematica'' ("the ''Principia''"), first published on 5 July 1687. When Newton presented Book 1 of the unpublished text in April 1686 to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Focus (geometry)
In geometry, focuses or foci (), singular focus, are special points with reference to which any of a variety of curves is constructed. For example, one or two foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola. In addition, two foci are used to define the Cassini oval and the Cartesian oval, and more than two foci are used in defining an ''n''-ellipse. Conic sections Defining conics in terms of two foci An ellipse can be defined as the locus of points for which the sum of the distances to two given foci is constant. A circle is the special case of an ellipse in which the two foci coincide with each other. Thus, a circle can be more simply defined as the locus of points each of which is a fixed distance from a single given focus. A circle can also be defined as the circle of Apollonius, in terms of two different foci, as the locus of points having a fixed ratio of distances to the two foci. A parabola i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptical Orbit
In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1 (thus excluding the circular orbit). In a wider sense, it is a Kepler's orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1. In a gravitational two-body problem with negative energy, both bodies follow similar elliptic orbits with the same orbital period around their common barycenter. Also the relative position of one body with respect to the other follows an elliptic orbit. Examples of elliptic orbits include: Hohmann transfer orbit, Molniya orbit, and tundra orbit. Velocity Under standard assumptions, no other forces acting except two spherically symmetrical bodies m1 and m2, the orbital speed (v\,) of one bod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vacuum Permittivity
Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric constant, or the distributed capacitance of the vacuum. It is an ideal (baseline) physical constant. Its CODATA value is: : (farads per meter), with a relative uncertainty of It is a measure of how dense of an electric field is "permitted" to form in response to electric charges, and relates the units for electric charge to mechanical quantities such as length and force. For example, the force between two separated electric charges with spherical symmetry (in the vacuum of classical electromagnetism) is given by Coulomb's law: :F_\text = \frac \frac Here, ''q''1 and ''q''2 are the charges, ''r'' is the distance between their centres, and the value of the constant fraction 1/4 \pi \varepsilon_0 (known as the Coulomb constant, ''k''e) is ap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electric Charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respectively). Like charges repel each other and unlike charges attract each other. An object with an absence of net charge is referred to as neutral. Early knowledge of how charged substances interact is now called classical electrodynamics, and is still accurate for problems that do not require consideration of quantum effects. Electric charge is a conserved property; the net charge of an isolated system, the amount of positive charge minus the amount of negative charge, cannot change. Electric charge is carried by subatomic particles. In ordinary matter, negative charge is carried by electrons, and positive charge is carried by the protons in the nuclei of atoms. If there are more electrons than protons in a piece of matter, it will ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
