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Cotes's Spiral
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. Description 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 coordinate system, 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 const ... [...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. Alternative definition There is another definition of the epispiral that has to do with tangents to circles: Begin with a circle. Rotate some single point on the circle around the circle by some angle \theta and at the same time by an angle in constant proportion to \theta, say c\theta for some constant c. The intersections of the tangent lines to the circle at these new points rotated from that single point for every \theta would trace out an epispiral. The polar equation can be derived through simple geometry as follows: To determine the polar coordinates (\rho,\phi) of the int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and 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." It is one of the most fundamental scientific disciplines. "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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. \mathbf(\mathbf) = F( \mathbf ) where F is a force vector, ''F'' is a scalar valued force function (whose absolute value gives the magnitude of the force and is positive if the force is outward and negative if the force is inward), r is the position vector, , , r, , is its length, and \hat = \mathbf r / \, \mathbf r\, is the corresponding unit vector. Not all central force fields are conservative or spherically symmetric. However, a central force is conservative if and only if it is spherically symmetric or rotationally invariant. Examples of spherically symmetric central forces include the Coulomb force and the force of gravity. Properties Central forces that are conservative can always be expressed as the negative gradient of a potential energy: \mathbf(\mathbf) = - \mathbf V(\mathbf) \; \text V(\mathbf) = \int_^ F(r)\,\mathrmr (the upper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A Treatise On The Analytical Dynamics Of Particles And Rigid Bodies
''A Treatise on the Analytical Dynamics of Particles and Rigid Bodies'' is a treatise and textbook on analytical dynamics by British mathematician Sir Edmund Taylor Whittaker. Initially published in 1904 by the Cambridge University Press, the book focuses heavily on the three-body problem and has since gone through four editions and has been translated to German and Russian. Considered a landmark book in English mathematics and physics, the treatise presented what was the state-of-the-art at the time of publication and, remaining in print for more than a hundred years, it is considered a classic textbook in the subject. Section 1 ''Introduction'' In addition to the original editions published in 1904, 1917, 1927, and 1937, a reprint of the fourth edition was released in 1989 with a new foreword by William Hunter McCrea. The book was very successful and received many positive reviews. A 2014 "biography" of the book's development wrote that it had "remarkable longevity" and note ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Samuel Earnshaw
Samuel Earnshaw (1 February 1805, Sheffield, Yorkshire – 6 December 1888, Sheffield, Yorkshire) was an English clergyman and mathematician and physicist, noted for his contributions to theoretical physics, especially for proving Earnshaw's theorem. Earnshaw was born in Sheffield and entered St John's College, Cambridge, graduating Senior Wrangler and Smith's Prizeman in 1831. From 1831 to 1847 Earnshaw worked in Cambridge as tripos coach, and in 1846 was appointed to the parish church St. Michael, Cambridge. For a time he acted as curate to the Revd Charles Simeon. In 1847 his health broke down and he returned to Sheffield working as a chaplain and teacher. Earnshaw published several mathematical and physical articles and books. His most famous contribution, Earnshaw's theorem, shows the impossibility of stability of matter under purely electrostatic forces: other topics included optics, waves, dynamics and acoustics in physics, calculus, trigonometry and partial di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isaac Newton
Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed. His book (''Mathematical Principles of Natural Philosophy''), first published in 1687, achieved the Unification of theories in physics#Unification of gravity and astronomy, first great unification in physics and established classical mechanics. Newton also made seminal contributions to optics, and Leibniz–Newton calculus controversy, shares credit with German mathematician Gottfried Wilhelm Leibniz for formulating calculus, infinitesimal calculus, though he developed calculus years before Leibniz. Newton contributed to and refined the scientific method, and his work is considered the most influential in bringing forth modern science. In the , Newton formulated the Newton's laws of motion, laws of motion and Newton's law of universal g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philosophiæ Naturalis Principia Mathematica
(English: ''The Mathematical Principles of Natural Philosophy''), often referred to as simply the (), is a book by Isaac Newton that expounds Newton's laws of motion and his law of universal gravitation. The ''Principia'' is written in Latin and comprises three volumes, and was authorized, imprimatur, by Samuel Pepys, then-President of the Royal Society on 5 July 1686 and first published in 1687. The is considered one of the most important works in the history of science. The French mathematical physicist Alexis Clairaut assessed it in 1747: "The famous book of ''Mathematical Principles of Natural Philosophy'' marked the epoch of a great revolution in physics. The method followed by its illustrious author Sir Newton ... spread the light of mathematics on a science which up to then had remained in the darkness of conjectures and hypotheses." The French scientist Joseph-Louis Lagrange described it as "the greatest production of the human mind". French polymath Pierre-Simon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lituus (mathematics)
300px, Branch for positive The lituus spiral () is a spiral in which the angle is inversely proportional to the square of the radius . This spiral, which has two branches depending on the sign of , is asymptotic to the axis. Its points of inflexion are at : (\theta, r) = \left(\tfrac12, \pm\sqrt\right). The curve was named for the ancient Roman lituus by Roger Cotes in a collection of papers entitled ''Harmonia Mensurarum'' (1722), which was published six years after his death. Coordinate representations Polar coordinates The representations of the lituus spiral in polar coordinates is given by the equation : r = \frac, where and . Cartesian coordinates The lituus spiral with the polar coordinates can be converted to Cartesian coordinates like any other spiral with the relationships and . With this conversion we get the parametric representations of the curve: : \begin x &= \frac \cos\theta, \\ y &= \frac \sin\theta. \\ \end These equations can in turn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytical Dynamics Of Particles And Rigid Bodies
''A Treatise on the Analytical Dynamics of Particles and Rigid Bodies'' is a treatise and textbook on analytical dynamics by British mathematician Sir Edmund Taylor Whittaker. Initially published in 1904 by the Cambridge University Press, the book focuses heavily on the three-body problem and has since gone through four editions and has been translated to German and Russian. Considered a landmark book in English mathematics and physics, the treatise presented what was the state-of-the-art at the time of publication and, remaining in print for more than a hundred years, it is considered a classic textbook in the subject. Section 1 ''Introduction'' In addition to the original editions published in 1904, 1917, 1927, and 1937, a reprint of the fourth edition was released in 1989 with a new foreword by William Hunter McCrea. The book was very successful and received many positive reviews. A 2014 "biography" of the book's development wrote that it had "remarkable longevity" and note ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cosine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle \theta, the sine and cosine functions are denoted as \sin(\theta) and \cos(\theta). The definitions of sine and cosine have been extended to any real number, real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as Series (mathematics), infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. The sine and cosine functions are commonly used to model periodic function, periodic pheno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Specific Relative Angular Momentum
In celestial mechanics, the specific relative angular momentum (often denoted \vec or \mathbf) of a body is the angular momentum of that body divided by its mass. In the case of two orbiting bodies it is the vector product of their relative position and relative linear momentum, divided by the mass of the body in question. Specific relative angular momentum plays a pivotal role in the analysis of the two-body problem, as it remains constant for a given orbit under ideal conditions. " Specific" in this context indicates angular momentum per unit mass. The SI unit for specific relative angular momentum is square meter per second. Definition The specific relative angular momentum is defined as the cross product of the relative position vector \mathbf and the relative velocity vector \mathbf . \mathbf = \mathbf\times \mathbf = \frac where \mathbf is the angular momentum vector, defined as \mathbf \times m \mathbf. The \mathbf vector is always perpendicular to the instant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
