|
Hénon–Heiles System
While at Princeton University in 1962, Michel Hénon and Carl Heiles worked on the non-linear motion of a star around a galactic center with the motion restricted to a plane. In 1964 they published an article titled "The applicability of the third integral of motion: Some numerical experiments". Their original idea was to find a third integral of motion in a galactic dynamics. For that purpose they took a simplified two-dimensional nonlinear rotational symmetric potential and found that the third integral existed only for a limited number of initial conditions. In the modern perspective the initial conditions that do not have the third integral of motion are called chaotic orbits. Introduction The Hénon–Heiles potential can be expressed as : V(x,y) = \frac(x^2 + y^2) + \lambda \left(x^2y - \frac\right). The Hénon–Heiles Hamiltonian can be written as : H = \frac12 (p_x^2 + p_y^2) + \frac(x^2 + y^2) + \lambda \left(x^2y - \frac\right). The Hénon–Heiles system (HHS) is de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henon Heiles Potential
''Phyllostachys nigra'', commonly known as black bamboo or purple bamboo ( zh, 紫竹), is a species of bamboo, native to Hunan Province of China, and is widely cultivated elsewhere. Growing up to tall by broad, it forms clumps of slender arching canes which turn black after two or three seasons. The abundant lance-shaped leaves are long. Numerous forms and cultivars are available for garden use. The species and the form ''P. nigra'' f. ''henonis'' have both gained the Royal Horticultural Society's Award of Garden Merit. The form ''henonis'' is also known as Henon bamboo and as cultivar 'Henon'. Life cycle Like many species of bamboo, black bamboo synchronizes its flowering, with flowering events happening every 40-120 years. According to one source, it has bloomed every 120 years "since records have been kept". It is monocarpic, that is, after flowering, the plants die. Henon bamboo flowers every 120 years and is predicted to flower in the 2020s. Since it is widely di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Princeton University
Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial Colleges, fourth-oldest institution of higher education in the United States and one of the nine colonial colleges chartered before the American Revolution. The institution moved to Newark, New Jersey, Newark in 1747 and then to its Mercer County, New Jersey, Mercer County campus in Princeton nine years later. It officially became a university in 1896 and was subsequently renamed Princeton University. The university is governed by the Trustees of Princeton University and has an endowment of $37.7 billion, the largest List of colleges and universities in the United States by endowment, endowment per student in the United States. Princeton provides undergraduate education, undergraduate and graduate education, graduate instruction in the hu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michel Hénon
Michel Hénon (; 23 July 1931, Paris – 7 April 2013, Nice) was a French mathematician and astronomer. He worked for a long time at the Nice Observatory. In astronomy, Hénon is well known for his contributions to stellar dynamics. In the late 1960s and early 1970s he made important contributions on the dynamical evolution of star clusters, in particular globular clusters. He developed a numerical technique using Monte Carlo methods to follow the dynamical evolution of a spherical star cluster much faster than the so-called ''n''-body methods. In mathematics, he is well known for the Hénon map, a simple discrete dynamical system that exhibits chaotic behavior. He published a two-volume work on the restricted three-body problem. In 1978 he was awarded the '' Prix Jean Ricard''. See also * N-body units References External links Hénon's publications(a partial list from NASA Astrophysics Data System). A discussionof Hénon's equation, contains further links. Simulationof ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Heiles
Carl Eugene Heiles (born 1939) is an American astrophysicist noted for his contributions to the understanding of diffuse interstellar matter through observational radio astronomy. Biography Heiles was born in Toledo, Ohio. He did his undergraduate work at Cornell University, receiving a degree in engineering physics, and then received his doctorate under George B. Field in 1966 from Princeton University in astrophysical sciences. He has worked at the University of California, Berkeley, since then and is currently a professor of astronomy. Research While Heiles was still a graduate student at Princeton, he wrote a paper with Michel Hénon about a third integral of motion in axisymmetric potentials, from which the Hénon-Heiles equation is derived. Though his paper with Hénon has been cited more than all but one of his other papers, most of Heiles's work has been in the field of radio astronomy. Heiles was part of the team which discovered the first millisecond pulsar, PSR B1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Of Motion
In mechanics, a constant of motion is a physical quantity conserved throughout the motion, imposing in effect a constraint on the motion. However, it is a ''mathematical'' constraint, the natural consequence of the equations of motion, rather than a ''physical'' constraint (which would require extra constraint forces). Common examples include energy, linear momentum, angular momentum and the Laplace–Runge–Lenz vector (for inverse-square force laws). Applications Constants of motion are useful because they allow properties of the motion to be derived without solving the equations of motion. In fortunate cases, even the trajectory of the motion can be derived as the intersection of isosurfaces corresponding to the constants of motion. For example, Poinsot's construction shows that the torque-free rotation of a rigid body is the intersection of a sphere (conservation of total angular momentum) and an ellipsoid (conservation of energy), a trajectory that might be otherwise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Function
In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta''. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics. Overview Phase space coordinates (''p'', ''q'') and Hamiltonian ''H'' Let (M, \mathcal L) be a mechanical system with configuration space M and smooth Lagrangian \mathcal L. Select a standard coordinate system (\boldsymbol,\boldsymbol) on M. The quantities \textstyle p_i(\boldsymbol,\boldsymbol,t) ~\stackrel~ / are called ''momenta''. (Also ''generalized momenta'', ''conjugate momenta'', and ''canonical momenta''). For a time instant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Painlevé Transcendents
In mathematics, Painlevé transcendents are solutions to certain nonlinear second-order ordinary differential equations in the complex plane with the Painlevé property (the only movable singularities are poles), but which are not generally solvable in terms of elementary functions. They were discovered by , , , and . History Origins Painlevé transcendents have their origin in the study of special functions, which often arise as solutions of differential equations, as well as in the study of isomonodromic deformations of linear differential equations. One of the most useful classes of special functions are the elliptic functions. They are defined by second-order ordinary differential equations whose singularities have the Painlevé property: the only movable singularities are simple poles. This property is rare in nonlinear equations. Poincaré and Lazarus Fuchs showed that any first order equation (that is, an ODE involving only up to the first derivative) with the Painlev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian (quantum Mechanics)
In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's ''energy spectrum'' or its set of ''energy eigenvalues'', is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum and time-evolution of a system, it is of fundamental importance in most formulations of quantum theory. The Hamiltonian is named after William Rowan Hamilton, who developed a revolutionary reformulation of Newtonian mechanics, known as Hamiltonian mechanics, which was historically important to the development of quantum physics. Similar to vector notation, it is typically denoted by \hat, where the hat indicates that it is an operator. It can also be written as H or \check. Introduction The Hamiltonian of a system represents the total energy of the system; that is, the sum of the kine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schrödinger Equation
The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after Erwin Schrödinger, an Austrian physicist, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of the wave function, the quantum-mechanical characterization of an isolated physical system. The equation was postulated by Schrödinger based on a postulate of Louis de Broglie that all matter has an associated matter wave. The equati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wada Property
In mathematics, the are three disjoint connected open sets of the plane or open unit square with the counterintuitive property that they all have the same boundary. In other words, for any point selected on the boundary of ''one'' of the lakes, the other two lakes' boundaries also contain that point. More than two sets with the same boundary are said to have the Wada property; examples include Wada basins in dynamical systems. This property is rare in real-world systems. The lakes of Wada were introduced by , who credited the discovery to Takeo Wada. His construction is similar to the construction by of an indecomposable continuum, and in fact it is possible for the common boundary of the three sets to be an indecomposable continuum. Construction of the lakes of Wada The Lakes of Wada are formed by starting with a closed unit square of dry land, and then digging 3 lakes according to the following rule: *On day ''n'' = 1, 2, 3,... extend lake ''n'' mod 3 (= 0, 1, 2) so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian System
A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can be studied in both Hamiltonian mechanics and dynamical systems theory. Overview Informally, a Hamiltonian system is a mathematical formalism developed by William Rowan Hamilton, Hamilton to describe the evolution equation, evolution equations of a physical system. The advantage of this description is that it gives important insights into the dynamics, even if the initial value problem cannot be solved analytically. One example is the Three-body problem, planetary movement of three bodies: while there is no closed-form solution to the general problem, Henri Poincaré, Poincaré showed for the first time that it exhibits deterministic chaos. Formally, a Hamiltonian system is a dynamical system characterised by the scalar function H(\bol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Miguel Angel Fernández Sanjuán
--> Miguel is a given name and surname, the Portuguese and Spanish form of the Hebrew name Michael. It may refer to: Places * Pedro Miguel, a parish in the municipality of Horta and the island of Faial in the Azores Islands *São Miguel (other), various locations in Azores, Portugal, Brazil and Cape Verde People * Miguel (surname) Arts, entertainment, and media *Miguel (singer) (born 1985), Miguel Jontel Pimentel, American recording artist *Miguel Bosé (born 1956), Spanish pop new wave musician and actor * Miguel Calderón (born 1971), artist and writer *Miguel Cancel (born 1968), former American singer *Miguel Córcega (1929–2008), Mexican actor and director *Miguel de Cervantes (1547–1616), Spanish author *Miguel Delibes (1920–2010), Spanish novelist *Miguel Ferrer (1955–2017), American actor * Miguel Galván (1957–2008), Mexican actor *Miguel Gómez (photographer) (born 1974), Colombian / American photographer. *Miguel Ángel Landa (born 1936), Venezuela ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
