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Non-squeezing Theorem
The non-squeezing theorem, also called ''Gromov's non-squeezing theorem'', is one of the most important theorems in symplectic geometry. It was first proven in 1985 by Mikhail Gromov. The theorem states that one cannot embed a ball into a cylinder via a symplectic map unless the radius of the ball is less than or equal to the radius of the cylinder. The theorem is important because formerly very little was known about the geometry behind symplectic maps. One easy consequence of a transformation being symplectic is that it preserves volume. One can easily embed a ball of any radius into a cylinder of any other radius by a volume-preserving transformation: just picture squeezing the ball into the cylinder (hence, the name non-squeezing theorem). Thus, the non-squeezing theorem tells us that, although symplectic transformations are volume-preserving, it is much more restrictive for a transformation to be symplectic than it is to be volume-preserving. Background and statement W ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Geometry
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed differential form, closed, nondegenerate form, nondegenerate differential form, 2-form. Symplectic geometry has its origins in the Hamiltonian mechanics, Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold. The term "symplectic", introduced by Weyl, is a calque of "complex"; previously, the "symplectic group" had been called the "line complex group". "Complex" comes from the Latin ''com-plexus'', meaning "braided together" (co- + plexus), while symplectic comes from the corresponding Greek ''sym-plektikos'' (συμπλεκτικός); in both cases the stem comes from the Indo-European root wiktionary:Reconstruction:Proto-Indo-European/pleḱ-, *pleḱ- The name reflects the deep connections between complex and sym ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maurice A
Maurice may refer to: People *Saint Maurice (died 287), Roman legionary and Christian martyr *Maurice (emperor) or Flavius Mauricius Tiberius Augustus (539–602), Byzantine emperor *Maurice (bishop of London) (died 1107), Lord Chancellor and Lord Keeper of England *Maurice of Carnoet (1117–1191), Breton abbot and saint *Maurice, Count of Oldenburg (fl. 1169–1211) *Maurice of Inchaffray (14th century), Scottish cleric who became a bishop *Maurice, Elector of Saxony (1521–1553), German Saxon nobleman *Maurice, Duke of Saxe-Lauenburg (1551–1612) *Maurice of Nassau, Prince of Orange (1567–1625), stadtholder of the Netherlands *Maurice, Landgrave of Hesse-Kassel or Maurice the Learned (1572–1632) *Maurice of Savoy (1593–1657), prince of Savoy and a cardinal *Maurice, Duke of Saxe-Zeitz (1619–1681) *Maurice of the Palatinate (1620–1652), Count Palatine of the Rhine *Maurice of the Netherlands (1843–1850), prince of Orange-Nassau *Maurice Chevalier (1888–1972), Fre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dusa McDuff
Dusa McDuff FRS CorrFRSE (born 18 October 1945) is an English mathematician who works on symplectic geometry. She was the first recipient of the Ruth Lyttle Satter Prize in Mathematics, was a Noether Lecturer, and is a Fellow of the Royal Society. She is currently the Helen Lyttle Kimmel '42 Professor of Mathematics at Barnard College. Personal life and education Margaret Dusa Waddington was born in London, England, on 18 October 1945 to Edinburgh architect Margaret Justin Blanco White, second wife of biologist Conrad Hal Waddington, her father. Her sister is the anthropologist Caroline Humphrey, and she has an elder half-brother C. Jake Waddington by her father's first marriage. Her mother was the daughter of Amber Reeves, the noted feminist, author and lover of H. G. Wells. McDuff grew up in Scotland where her father was Professor of Genetics at the University of Edinburgh. McDuff was educated at St George's School for Girls in Edinburgh and, although the standard was lowe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Coordinates
In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of classical mechanics. A closely related concept also appears in quantum mechanics; see the Stone–von Neumann theorem and canonical commutation relations for details. As Hamiltonian mechanics are generalized by symplectic geometry and canonical transformations are generalized by contact transformations, so the 19th century definition of canonical coordinates in classical mechanics may be generalized to a more abstract 20th century definition of coordinates on the cotangent bundle of a manifold (the mathematical notion of phase space). Definition in classical mechanics In classical mechanics, canonical coordinates are coordinates q^i and p_i in phase space that are used in the Hamiltonian formalism. The canonical coordinates satisfy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uncertainty Principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physical quantities of a particle, such as position, ''x'', and momentum, ''p'', can be predicted from initial conditions. Such variable pairs are known as complementary variables or canonically conjugate variables; and, depending on interpretation, the uncertainty principle limits to what extent such conjugate properties maintain their approximate meaning, as the mathematical framework of quantum physics does not support the notion of simultaneously well-defined conjugate properties expressed by a single value. The uncertainty principle implies that it is in general not possible to predict the value of a quantity with arbitrary certainty, even if all initial conditions are specified. Introduced first in 1927 by the German physicist Werner ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville's Theorem (Hamiltonian)
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that ''the phase-space distribution function is constant along the trajectories of the system''—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time. This time-independent density is in statistical mechanics known as the classical a priori probability. There are related mathematical results in symplectic topology and ergodic theory; systems obeying Liouville's theorem are examples of incompressible dynamical systems. There are extensions of Liouville's theorem to stochastic systems. Liouville equations The Liouville equation describes the time evolution of the ''phase space distribution function''. Although the equation is usually referred to as the "Liouville equation", Josiah Willard Gibbs was the first to recognize the impor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Transformation
In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations. This is sometimes known as form invariance. It need not preserve the form of the Hamiltonian itself. Canonical transformations are useful in their own right, and also form the basis for the Hamilton–Jacobi equations (a useful method for calculating conserved quantities) and Liouville's theorem (itself the basis for classical statistical mechanics). Since Lagrangian mechanics is based on generalized coordinates, transformations of the coordinates do not affect the form of Lagrange's equations and, hence, do not affect the form of Hamilton's equations if we simultaneously change the momentum by a Legendre transformation into P_i=\frac. Therefore, coordinate transformations (also called point transformations) are a ''type'' of canonical transformation. However, the class of canonical transformations is much broader, since the old generalized ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase Space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. It is the outer product of direct space and reciprocal space. The concept of phase space was developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré, and Josiah Willard Gibbs. Introduction In a phase space, every degree of freedom or parameter of the system is represented as an axis of a multidimensional space; a one-dimensional system is called a phase line, while a two-dimensional system is called a phase plane. For every possible state of the system or allowed combination of values of the system's parameters, a point is included in the multidimensional space. The system's evolving state over time traces a path (a phase-space trajectory for the system) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eye Of A Needle
The term "eye of a needle" is used as a metaphor for a very narrow opening. It occurs several times throughout the Talmud. The New Testament quotes Jesus as saying in Luke 18:25 that "it is easier for a camel to go through the eye of a needle than for a rich man to enter the kingdom of God". It also appears in the Qur'an 7:40, "Indeed, those who deny Our verses and are arrogant toward them – the gates of Heaven will not be opened for them, nor will they enter Paradise until a camel enters into the eye of a needle. And thus do We recompense the criminals." Aphorisms Judaism The Babylonian Talmud applies the aphorism to unthinkable thoughts. To explain that dreams reveal the thoughts of a man's heart and are the product of reason rather than the absence of it, some rabbis say: A midrash on the Song of Songs uses the phrase to speak of God's willingness and ability beyond comparison to accomplish the salvation of a sinner: Rav Sheishet of Nehardea applied the same aphorism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Leonidovich Gromov
Mikhael Leonidovich Gromov (also Mikhail Gromov, Michael Gromov or Misha Gromov; russian: link=no, Михаи́л Леони́дович Гро́мов; born 23 December 1943) is a Russian-French mathematician known for his work in geometry, analysis and group theory. He is a permanent member of IHÉS in France and a professor of mathematics at New York University. Gromov has won several prizes, including the Abel Prize in 2009 "for his revolutionary contributions to geometry". Biography Mikhail Gromov was born on 23 December 1943 in Boksitogorsk, Soviet Union. His Russian father Leonid Gromov and his Jewish mother Lea Rabinovitz were pathologists. His mother was the cousin of World Chess Champion Mikhail Botvinnik, as well as of the mathematician Isaak Moiseevich Rabinovich. Gromov was born during World War II, and his mother, who worked as a medical doctor in the Soviet Army, had to leave the front line in order to give birth to him. When Gromov was nine years old, his mother ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ian Stewart (mathematician)
Ian Nicholas Stewart (born 24 September 1945) is a British mathematician and a popular-science and science-fiction writer. He is Emeritus Professor of Mathematics at the University of Warwick, England. Education and early life Stewart was born in 1945 in Folkestone, England. While in the sixth form at Harvey Grammar School in Folkestone he came to the attention of the mathematics teacher. The teacher had Stewart sit mock A-level examinations without any preparation along with the upper-sixth students; Stewart was placed first in the examination. He was awarded a scholarship to study at the University of Cambridge as an undergraduate student of Churchill College, Cambridge, where he studied the Mathematical Tripos and obtained a first-class Bachelor of Arts degree in mathematics in 1966. Stewart then went to the University of Warwick where his PhD on Lie algebras was supervised by Brian Hartley and completed in 1969. Career and research After his PhD, Stewart was offered an a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Form
In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument separately; ; Alternating: holds for all ; and ; Non-degenerate: for all implies that . If the underlying field has characteristic not 2, alternation is equivalent to skew-symmetry. If the characteristic is 2, the skew-symmetry is implied by, but does not imply alternation. In this case every symplectic form is a symmetric form, but not vice versa. Working in a fixed basis, ''ω'' can be represented by a matrix. The conditions above are equivalent to this matrix being skew-symmetric, nonsingular, and hollow (all diagonal entries are zero). This should not be confused with a symplectic matrix, which represents a symplectic transformation of the space. If ''V'' is finite-dimensional, then its dimension must necessarily be even sinc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
