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Topic: Canonical coordinates

Related Topics

Tangent bundle

Cotangent bundle

Symplectic matrix

Hamiltonian mechanics

	Canonical coordinates - Wikipedia, the free encyclopedia
		In mathematics and physics, the canonical coordinates are a special set of coordinates on the cotangent bundle of a manifold.
		A change of coordinates that preserves this form is a canonical transformation; these are a special case of a symplectomorphism, which are essentially a change of coordinates on a symplectic manifold.
		When a Hamiltonian is defined on the cotangent bundle, then the generalized coordinates are related to the canonical coordinates by means of the Hamilton-Jacobi equations.
	en.wikipedia.org /wiki/Canonical_coordinates (364 words)

	Canonical Transformation -- from Eric Weisstein's World of Physics
		Canonical transformations must be area-preserving (and, more generally, Poincaré invariant preserving), with the added restriction that the sign of the area must also remain positive.
		For any problem, there may be a choice of coordinates which are cyclic.
		It is known as the generating function for a canonical transformation.
	scienceworld.wolfram.com /physics/CanonicalTransformation.html (202 words)

	Guiding Centre Equations
		The Boozer coordinate system introduced in the last section may in general form a non-inertial reference frame since it is defined with respect to the magnetic field structure which may be time dependent.
		It is thus natural to select the Boozer coordinates introduced in §3.1 to form the canonical coordinates, enabling the canonically conjugate momentum variables to be read straight from equation (3.17).
		One method of achieving this involves the observation that the equations of motion remain invariant under the addition of exact differentials to the Lagrangian [52].
	www.ipp.mpg.de /~Simon.Pinches/thesis/node44.html (1681 words)

	Systems Seminar (Site not responding. Last check: 2007-11-04)
		In this seminar we show that canonical correlations may be used to decompose the Wiener filter into a whitening transform coder, a canonical filter, and a coloring transform coder.
		The outputs of the whitening transform coder are canonical coordinates, and it is these coordinates that are reduced in dimension and quantized in a reduced-rank, finite-precision Wiener filter.
		For the circulant Gaussian channel, canonical coordinates may be used to illustrate that the capacity solution is a spread spectrum solution in which all canonical correlations, and direction cosines, are equalized.
	eceserv0.ece.wisc.edu /~gubner/seminar/scharf2.html (145 words)

	[No title]
		Canonical Correlation allows us to investigate the relationship between two sets of variables (it is used as either a hypothesis testing or exploratory method).
		Cartesian coordinates (x, y, or x, y, z; also known as rectangular coordinates) are directed distances from two (or three) perpendicular axes.
		One component graph is produced for each level of the grouping variable (or user-defined subset of data) and all the component graphs are arranged in one display to allow for comparisons between the subsets of data (categories).
	www.statsoft.com /textbook/glosc.html (7574 words)

	Canonical commutation relation - Wikipedia, the free encyclopedia
		The uniqueness of the canonical commutation relations between position and momentum is guaranteed by the Stone-von Neumann theorem.
		Canonical quantization is performed, by definition, on canonical coordinates.
		However, in the presence of an electromagnetic field, the canonical momentum p is not gauge invariant.
	en.wikipedia.org /wiki/Canonical_commutation_relation (563 words)

	Observations of highly correlated near-simultaneous magnetic field perturbations at contraposed ground stations (Site not responding. Last check: 2007-11-04)
		The analysis method of canonical correlation is presented in section 3 and applied to several years of data to produce the results presented in section 4.
		It again shows the canonical components for the CANOPUS pair, but the Fort Smith data are from a day in 1997, and the Contwoyto Lake data are from a day in 1998.
		Canonical correlation is quite powerful but not capable of responding to all kinds of global events.
	www.phys.ucalgary.ca /~bjackel/cami4/cancor/cami4_cancor.html (7954 words)

	Canonical transformation - Wikipedia, the free encyclopedia
		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).
		However, the class of canonical transformations is much broader, since the old generalized coordinates, momenta and even time may be combined to form the new generalized coordinates and momenta.
		The superscript conveys the contravariant transformation properties of the generalized coordinates, and does not mean that the coordinate is being raised to a power.
	en.wikipedia.org /wiki/Canonical_transformation (1281 words)

	PlanetMath: Hamilton equations
		The relation with the former definition is that in canonical local coordinates
		Cross-references: state space, product, configuration space, local coordinates, relation, coordinates, vector field, flow, sign, canonical, manifold, cotangent bundle, Hamiltonian vector field, symplectic form, symplectic manifold, represent, function, Hamiltonian, solution, curve, smooth function, interval, open set, equations
		Object id is 6410, canonical name is HamiltonianEquations.
	planetmath.org /encyclopedia/HamiltonianEquations.html (158 words)

	HNBody 1.0.3: Integration Algorithms
		ZWPs are integrated as perturbed two-body orbits, and perturb nothing themselves; their integration coordinates can be either barycentric (specifically, the barycenter of the HWP/LWP subsystem) or canonical bodycentric, and their coordinates (as well as the timestep used) need not agree with those used for the other two groups.
		An optional Modified bodycentric coordinate choice (see IntegCoord) implements Touma and Wisdom's form of the Hamiltonian splitting, which is exact for the two-body problem (unlike canonical bodycentric coordinates).
		To the base coordinates a number of predefined modifiers can be added specifying things such as a 4th order integration or use of a symplectic corrector.
	janus.astro.umd.edu /HNBody/doc/html/integ_algo.html (575 words)

	Symplectic Boundary Form (Site not responding. Last check: 2007-11-04)
		In place of an orthonormal basis we must construct a canonical basis, whose idiosyncracies are familiar to all persons who have worked with Poisson brackets and canonical coordinates in classical mechanics.
		These include use of the Gram-Schmidt process to construct a canonical basis, use of the Gram matrix or the Gram determinant to ascertain linear dependence, or construction of conjugates to aid in the isolation of components of a vector.
		Characteristically, canonical coordinates partition space into two parts, in a fashion very reminiscent of the coordinates and momenta into which phase space is divided.
	delta.cs.cinvestav.mx /~mcintosh/comun/quant/node5.html (1327 words)

	p-Adic TGD (Site not responding. Last check: 2007-11-04)
		As far as applications are considered, the additional crucial elements are the canonical identification mapping p-adics to reals and making possible to test the predictions of the p-adic physics and the p-adic length scale hypothesis, suggesting which p-adic primes are the most interesting ones physically.
		Canonical identification is the key element in defining the p-adic version of definite integral, which in turn is absolutely crucial for generalizing the variational principles of physics to the p-adic context.
		Canonical identification makes possible to induce the ordering of real numbers to the set of p-adic numbers and this ordering makes it also possible to define homology and differential forms and to lift the standard concepts of real topology to p-adic context.
	www.helsinki.fi /~matpitka/padic2.html (4174 words)

	[No title]
		It is proved that their fluctuation operators form a quantum canonical pair, which decouples from the other degrees of freedom of the system.
		In particular the ground state situation is interesting, because it yields a non-trivial quantum mechanical canonical pair of conjugate operators, giving an explicit representation of the field variables of the socalled Goldstone boson.
		We call it the canonical order parameter operator, it is an order parameter constructed directly from the two given quantities, the Hamiltonian and the symmetry generator.
	www.ma.utexas.edu /mp_arc/papers/00-33 (7454 words)

	PlanetMath: Poincaré $1$-form
		Anyone with an account can edit this entry.
		Cross-references: canonical projection, symplectic form, geometry, transformation, coordinates, Poincaré, local coordinates, canonical, cotangent bundle, manifold
		Object id is 6405, canonical name is Poincare1Form.
	planetmath.org /encyclopedia/Poincare1Form.html (61 words)

	Number Theory Seminar (Site not responding. Last check: 2007-11-04)
		We introduce a global version of the Serre-Tate coordinates for a family of ordinary abelian varieties.
		These "canonical coordinates" are used to study the p-adic monodromy and the Hecke orbit problem in characteristic p>0.
		Among the conjectures is an analogue of the Mumford-Tate conjecture for families of ordinary abelian varieties.
	www.math.princeton.edu /~ytschink/seminar/chai01.html (67 words)

	Integration methods based on canonical coordinates of the second kind (Site not responding. Last check: 2007-11-04)
		Canonical coordinates of the second kind is used for representing the Lie group locally by means of its corresponding Lie algebra.
		The coordinate map itself can, in many cases, be computed inexpensively, but the approach also involves the inversion of its differential, a task that can be challenging.
		To succeed, it is necessary to consider carefully how to choose a basis for the Lie algebra, and the ordering of the basis is important as well.
	www.ii.uib.no /~pinar/seminar/arnema.html (207 words)

	[No title] (Site not responding. Last check: 2007-11-04)
		The canonical coordinates are derived from a fundamental characterization of channel propagation dynamics in terms of discrete multipath-delayed and Doppler-shifted copies of the spread-spectrum signaling waveforms.
		Second, the same coordinates provide a canonical subspace-based representation of the desired signal and interference which fully incorporates channel dispersion effects.
		Finally, the maximally parsimonious nature of the coordinates and their simple computation afford a direct handle on transceiver complexity.
	homepages.cae.wisc.edu /~akbar/papers/all98_abs.html (202 words)

	[No title]
		] Any set of generalized coordinates of a system together with their conjugate momenta.
		] The density of members of the canonical ensemble in phase space.
		] A transformation which occurs among the coordinates and momenta describing the state of a classical dynamical system and which leaves the form of Hamilton's equations of motion unchanged.
	www.accessscience.com /Dictionary/C/C5/DictC5.html (1816 words)

	Formulation of Hamilton's equations for the problem
		The suffix 1 is used when referring to a property of Earth and 2 when referring to the Moon.
		The canonically conjugate momentum coordinates are related to, but not the same as, the actual momentum (defined as
		The relation between canonical momentum coordinates and momentum coordinates depends on the particular coordinate system chosen.
	www.physics.uq.edu.au /people/jones/ph362/cphys/node16.html (488 words)

	canoni.html
		to be used for representing the canonical dependent variable
		canoni tries to find a set of transformations from the original coordinates to the canonical coordinates by the knowledge of the coefficients of the symmetry generator (infinitesimals) of a 1-parameter Lie group.
		the change of variables, using the canonical coordinates, reducing eq to a quadrature (see ?dchange)
	lie.uwaterloo.ca /help/canoni1.html (204 words)

	Decentralized Multiuser Detection for Time-Varying Multipath Channels
		The canonical coordinates are defined by a fixed basis derived from a fundamental characterization of propagation effects.
		The framework builds on the notion of active coordinates that carry the desired signal energy, facilitate maximal exploitation of channel diversity, and provide minimum-complexity MAI suppression.
		Signal space partitioning in terms of active/inactive coordinates provides a direct handle on controlling receiver complexity to achieve a desired level of performance.
	www.comsoc.org /comm/private/2000/nov/1840_48comm11-kadous.html (600 words)

	Development of Model
		The specific form of the Hamiltonian, and therefore the equations of motion to which it gives rise, depends upon the canonical coordinates used.
		The crux for finding simple canonical variables lies in the proper choice of the spatial coordinates which should be suited to the geometry of the magnetic field.
		From the point of view of describing the waves present, all that is needed is the adoption of a set of flux coordinates since it is natural to decompose the waves into Fourier harmonics in the toroidal and poloidal directions.
	www.ipp.mpg.de /~Simon.Pinches/thesis/node42.html (427 words)

	Analysis of wave fields by Fourier integral operators and their application for radio occultations
		Geometric optical rays are described by the canonical Hamilton system, which can be written in different canonical coordinates in the phase space.
		The FIO associated with a canonical transform maps the wave field to a different representation.
		Mapping to the representation of ray impact parameter was used in the formulation of the canonical transform (CT) method for processing radio occultation data.
	www.agu.org /pubs/crossref/2004/2003RS002971.shtml (358 words)

	Enhancing Data Interoperability with Ontologies, Canonical Forms, and Include Files
		This is read as: "The canonical form of the Length class are instances that have a 'value' property in canonical decimal form, and a 'units' property in canonical Length-Unit-of-Measure form.
		If the value of "units" is Kilometer then the function converts the data to canonical form by returning an identical XML fragment, except the content of the element has been multiplied by 1000, and is set to have the value #Meter.
		If the value of "units" is Mile then the function converts the input to canonical form by returning an identical XML fragment, except the content of the element has been multiplied by 1609.344, and is set to have the value #Meter.
	www.xfront.com /interoperability/CanonicalForms.html (2833 words)

	Poisson brackets
		For instance, a Cartesian coordinate has an associated linear momentum, an angular coordinate has an associated angular momentum, etc.
		expressed in terms of the classical coordinates and canonical momenta.
		are regarded as functions of the coordinates and momenta
	farside.ph.utexas.edu /teaching/qm/lectures/node21.html (524 words)

	Time in (quantum-)mechanics
		the time coordinate of the point, is a canonical coordinate of this
		tensor with respect to this coordinate patch is g_mn.
		For a general p-brane the coordinate fields are functions of the d base space
	www-stud.uni-essen.de /~sb0264/TimeInQM.html (6954 words)

	Computer Programs
		This program does canonical analysis of principal coordinates (canonical correlation analysis or canonical discriminant analysis) based on any distance measure as described by Anderson and Robinson (2003) and Anderson and Willis (2003).
		The test is done by permutation (using the trace and first canonical root statistics) and canonical axes for ordination are also given in the output.
		PERMDISP2 is a modification of PERMDISP which handles unequal numbers of observations per group and which does tests based on distances to either centroids or spatial medians.
	www.stat.auckland.ac.nz /~mja/Programs.htm (384 words)

	Louis L. Scharf (Site not responding. Last check: 2007-11-04)
		Canonical correlations may be used to decompose the Wiener filter into a whitening transform coder, a canonical filter, and a coloring transform coder.
		The outputs of the whitening transform coder are canonical coordinates, and it is these coordinates that are reduced in dimension and quantized in a finite-precision Wiener filter.
		They bring insightful decompositions of entropy, mutual information, and rate.
	www.eecs.umich.edu /systems/scharfFal98.html (128 words)

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