Showing papers on "Canonical coordinates published in 2005"
TL;DR: In this paper, the authors present a consistent description of Hamiltonian dynamics on the "symplectic extended phase space" that is analogous to that of a time-independent Hamiltonian system on the conventional symplectic phase space.
Abstract: We will present a consistent description of Hamiltonian dynamics on the 'symplectic extended phase space' that is analogous to that of a time-independent Hamiltonian system on the conventional symplectic phase space The extended Hamiltonian H1 and the pertaining extended symplectic structure that establish the proper canonical extension of a conventional Hamiltonian H will be derived from a generalized formulation of Hamilton's variational principle The extended canonical transformation theory then naturally permits transformations that also map the time scales of the original and destination system, while preserving the extended Hamiltonian H1, and hence the form of the canonical equations derived from H1 The Lorentz transformation, as well as time scaling transformations in celestial mechanics, will be shown to represent particular canonical transformations in the symplectic extended phase space Furthermore, the generalized canonical transformation approach allows us to directly map explicitly time-dependent Hamiltonians into time-independent ones An 'extended' generating function that defines transformations of this kind will be presented for the time-dependent damped harmonic oscillator and for a general class of explicitly time-dependent potentials In the appendix, we will re-establish the proper form of the extended Hamiltonian H1 by means of a Legendre transformation of the extended Lagrangian L1
85 citations
TL;DR: In this article, the authors present a Hamiltonian approach to model spacecraft motion rela- tive to a circular reference orbit based on a derivation of canonical coordinates for the relative state-space dynamics.
Abstract: This paper presents a Hamiltonian approach to modelling spacecraft motion rela- tive to a circular reference orbit based on a derivation of canonical coordinates for the relative state-space dynamics. The Hamiltonian formulation facilitates the modelling of high-order terms and orbital perturbations within the context of the Clohessy-Wiltshire solution. First, the Hamiltonian is partitioned into a linear term and a high-order term. The Hamilton-Jacobi equations are solved for the linear part by separation, and new constants for the relative motions are obtained, called epicyclic elements. The influence of higher order terms and perturbations, such as Earth's oblateness, are incorporated into the analysis by a variation of parameters procedure. As an example, closed-form solutions for J2-invariant orbits are obtained.
55 citations
TL;DR: In this paper, the authors considered a real manifold as a real Frobenius manifold, and the number of coordinates is twice as large as the numbers of coordinates on the Hurwitz spaces of Dubrovin.
Abstract: New Frobenius structures on Hurwitz spaces are found. A Hurwitz space is considered as a real manifold; therefore the number of coordinates is twice as large as the number of coordinates on Hurwitz Frobenius manifolds of Dubrovin. Simple branch points of a ramified covering and their complex conjugates play the role of canonical coordinates on the constructed Frobenius manifolds. Corresponding solutions to WDVV equations and G-functions are obtained.
27 citations
TL;DR: A unified framework for reduced-rank Wiener filtering is presented, and an alternating power method is proposed to recursively compute the canonical coordinate and half-canonical coordinate mappings.
Abstract: The problem of two-channel constrained least squares (CLS) filtering under various sets of constraints is considered, and a general set of solutions is derived. For each set of constraints, the solution is determined by a coupled (asymmetric) generalized eigenvalue problem. This eigenvalue problem establishes a connection between two-channel CLS filtering and transform methods for resolving channel measurements into canonical or half-canonical coordinates. Based on this connection, a unified framework for reduced-rank Wiener filtering is presented. Then, various representations of reduced-rank Wiener filters in canonical and half-canonical coordinates are introduced. An alternating power method is proposed to recursively compute the canonical coordinate and half-canonical coordinate mappings. A deflation process is introduced to extract the mappings associated with the dominant coordinates. The correctness of the alternating power method is demonstrated on a synthesized data set, and conclusions are drawn.
25 citations
TL;DR: In this article, it was shown that any almost one-to-one factor map between two dynamical systems may be lifted to a factor map which is injective on the local stable sets (i.e., s-resolving).
Abstract: We consider Smale spaces, that is, homeomorphisms of a compact metric spaces possessing canonical coordinates of contracting (stable) and expanding (unstable) directions. Examples of such dynamical systems include the basic sets for Smale's Axiom A systems. We also assume that each point of the space is non-wandering and that there is a dense orbit. We show that any almost one-to-one factor map between two such systems may be lifted in a certain sense to a factor map which is injective on the local stable sets (i.e., s-resolving). We derive several corollaries. One is a refinement of Bowen's result that every irreducible Smale space is a factor of an irreducible shift of finite type by an almost one-to-one factor map. We are able to show that there exists such a factor which is the composition of an s-resolving map and a u-resolving map.
23 citations
TL;DR: In this paper, the integrable Camassa-Holm (CH) equation on the line with positive initial data rapidly decaying at infinity is considered, and a one-parameter family of integrably hierarchies that preserve the mixed spectrum of the associated string spectral problem is constructed.
Abstract: We consider the integrable Camassa-Holm (CH) equation on the line with positive initial data rapidly decaying at infinity. On such a phase space we construct a one-parameter family of integrable hierarchies that preserves the mixed spectrum of the associated string spectral problem. This family includes the CH hierarchy. We demonstrate that the constructed flows can be interpreted as Hamiltonian flows on the space of Weyl functions of the associated string spectral problem. The corresponding Poisson bracket is the Atiyah-Hitchin bracket. Using an infinite dimensional version of the Jacobi ellipsoidal coordinates, we obtain a one-parameter family of canonical coordinates linearizing the flows. © 2005 Wiley Periodicals, Inc.
20 citations
TL;DR: In this article, the authors propose a canonical transformation for a quadratic Hamiltonian of 2 degrees of freedom with a symmetry that is often present in celestial mechanics, which has the added advantage of having a clear geometric interpretation and of being a generalization of the so-called reducing transformation.
Abstract: It is always possible to find canonical forms for quadratic Hamiltonians. In cases in which the eigenvalues of the associated linear system are simple, either real or pure imaginary, the canonical form introduces action-angle coordinates that are most useful for the application of perturbation theory; this is the case in which the quadratic Hamiltonian is the first term in the expansion of a nonlinear Hamiltonian around an equilibrium. The general theory is rather involved, and it may be worthwhile to find shortcuts in simple situations. We present here such a shortcut for a quadratic Hamiltonian of 2 degrees of freedom with a symmetry that is often present in celestial mechanics. The canonical transformation proposed has the added advantage of having a clear geometric interpretation and of being a generalization of the so-called reducing transformation that has been useful in several problems.
20 citations
TL;DR: In this paper, a non-linear canonical form observer design approach for multi-input multi-output systems with linearizable error dynamics is proposed, which involves a transformation of the original nonlinear system in a reduced generalized observer canonical form (RGOCF) by a nonlinear state transformation.
Abstract: A non-linear canonical form observer design approach for multi-input multi-output systems , y = h ( x , u ), with linearizable error dynamics is proposed in the paper. The method involves a transformation of the original non-linear system in a reduced generalized observer canonical form (RGOCF) by a non-linear state transformation. The name reduced comes from the fact that this non-linear canonical form has a reduced dependency on derivatives of the inputs. Necessary and sufficient conditions for transformability are derived and a transformation algorithm based on those conditions is developed. The non-linear observer is firstly designed in the canonical coordinates of the RGOCF and then it is presented in the original and observability coordinates. A final example including simulation is given for illustration.
19 citations
TL;DR: In this article, a supersymmetric quantum mechanics in terms of two real supercharges on non-commutative space in arbitrary dimensions was constructed, and the exact eigenspectra of the two-and three-dimensional superoscillators were obtained.
Abstract: We construct a supersymmetric quantum mechanics in terms of two real supercharges on non-commutative space in arbitrary dimensions. We obtain the exact eigenspectra of the two- and three-dimensional non-commutative superoscillators. We further show that a reduction in the phase space occurs for a critical surface in the space of parameters. At this critical surface, the energy spectrum of the bosonic sector is infinitely degenerate, while the degeneracy in the spectrum of the fermionic sector gets enhanced by a factor of two for each pair of reduced canonical coordinates. For the two-dimensional non-commutative “inverted superoscillator”, we find exact eigenspectra with a well-defined ground state for certain regions in the parameter space, which have no smooth limit to the ordinary commutative space.
19 citations
TL;DR: In this paper, the authors consider a homogeneous fluid membrane described by the Helfrich-Canham energy, quadratic in the mean curvature of the membrane surface and introduce a Hamiltonian formulation of this equation which dismantles it into a set of coupled first-order equations.
Abstract: Consider a homogeneous fluid membrane described by the Helfrich–Canham energy, quadratic in the mean curvature of the membrane surface. The shape equation that determines equilibrium configurations is fourth order in derivatives and cubic in the mean curvature. We introduce a Hamiltonian formulation of this equation which dismantles it into a set of coupled first-order equations. This involves interpreting the Helfrich–Canham energy as an action; equilibrium surfaces are generated by the evolution of space curves. Two features complicate the implementation of a Hamiltonian framework. (i) The action involves second derivatives. This requires treating the velocity as a phase-space variable and the introduction of its conjugate momentum. The canonical Hamiltonian is constructed on this phase space. (ii) The action possesses a local symmetry—reparametrization invariance. The two labels we use to parametrize points on the surface are themselves physically irrelevant. This symmetry implies primary constraints, one for each label, that need to be implemented within the Hamiltonian. The two Lagrange multipliers associated with these constraints are identified as the components of the acceleration tangential to the surface. The conservation of the primary constraints implies two secondary constraints, fixing the tangential components of the momentum conjugate to the position. Hamilton's equations are derived and the appropriate initial conditions on the phase-space variables are identified. Finally, it is shown how the shape equation can be reconstructed from these equations.
15 citations
TL;DR: In this article, a class of functions used to fill the disks is derived imposing conditions on the first and second derivatives to generate physically acceptable disks, and the analysis of the function's curvature further restricts the ranges of the free parameters that allow physically acceptable disk.
Abstract: New families of exact general relativistic thick disks are constructed using the 'displace, cut, fill, and reflect' method. A class of functions used to fill the disks is derived imposing conditions on the first and second derivatives to generate physically acceptable disks. The analysis of the function's curvature further restricts the ranges of the free parameters that allow physically acceptable disks. Then this class of functions together with the Schwarzschild metric is employed to construct thick disks in isotropic, Weyl and Schwarzschild canonical coordinates. In these last coordinates an additional function must be added to one of the metric coefficients to generate exact disks. Disks in isotropic and Weyl coordinates satisfy all energy conditions, but those in Schwarzschild canonical coordinates do not satisfy the dominant energy condition.
TL;DR: In this article, a Hamiltonian formulation of nonlinear, parallel propagating, travelling whistler waves is developed, where the complete system of equations reduces to two coupled differential equations for the transverse electron speed and a phase variable representing the difference in the phases of transverse complex velocities of the protons and the electrons.
Abstract: . A Hamiltonian formulation of nonlinear, parallel propagating, travelling whistler waves is developed. The complete system of equations reduces to two coupled differential equations for the transverse electron speed and a phase variable representing the difference in the phases of the transverse complex velocities of the protons and the electrons. Two integrals of the equations are obtained. The Hamiltonian integral H, is used to classify the trajectories in the phase plane, where and w=u2 are the canonical coordinates. Periodic, oscilliton solitary wave and compacton solutions are obtained, depending on the value of the Hamiltonian integral H and the Alfven Mach number M of the travelling wave. The second integral of the equations of motion gives the position x in the travelling wave frame as an elliptic integral. The dependence of the spatial period, L, of the compacton and periodic solutions on the Hamiltonian integral H and the Alfven Mach number M is given in terms of complete elliptic integrals of the first and second kind. A solitary wave solution, with an embedded rotational discontinuity is obtained in which the transverse Reynolds stresses of the electrons are balanced by equal and opposite transverse stresses due to the protons. The individual electron and proton phase variables and are determined in terms of and . An alternative Hamiltonian formulation in which is the new independent variable replacing x is used to write the travelling wave solutions parametrically in terms of .
TL;DR: In this article, the theoretical foundations of neutral two-body systems exposed to an inhomogeneous magnetic field are investigated and various representations for the Hamiltonian describing the coupled centre-of-mass and internal motion are derived.
Abstract: We investigate the theoretical foundations of neutral two-body systems exposed to an inhomogeneous magnetic field. Various representations for the Hamiltonian describing the coupled centre-of-mass and internal motion are derived. For the specific case of a magnetic quadrupole field we establish the continuous and discrete symmetries and show that the energy levels of the interacting system are two-fold degenerate. We exploit the symmetries in two alternative ways and derive corresponding effective equations of motion. The first approach eliminates two of the six spatial degrees of freedom and leads to an (infinite) set of coupled channel equations for the spin and spatial degrees of freedom. The second approach introduces the projection of the total angular momentum onto the symmetry axis of the quadrupole field as a canonical momentum, thereby eliminating the corresponding cyclic angle.
TL;DR: In this article, a variational approach for the canonical momentum equation in the Eulerian description is presented, which is based on an extremum principle for the total potential energy functional defined in terms of the inverse deformation function.
Abstract: This paper aims at the exploitation of material forces to find an optimum mesh in the finite element method (FEM). The classical variational formulation provides the linear momentum equation in a Lagrangian description. A variational setting for the derivation of the canonical momentum equation in the Eulerian description is presented. The latter is based on an extremum principle for the total potential energy functional defined in terms of the inverse deformation function. This constitutes a theoretical framework which allows the formulation of the finite element method for the canonical momentum equation as well as the computation of the material forces arising from the discretization. Thus, apart from the finite element solution for the standard boundary value problem of elastostatics, a second one for the canonical momentum equation can be formulated and solved numerically. The former provides an optimum deformation by minimizing the standard total potential energy, namely solving the physical forces ...
TL;DR: In this paper, the quadratic field theory on seven dimensional spacetime constructed by a direct product of Calabi-Yau three-fold by a real time axis is studied.
Abstract: Studying the quadratic field theory on seven dimensional spacetime constructed by a direct product of Calabi-Yau three-fold by a real time axis, with phase space being the third cohomology of the Calabi-Yau three-fold, the generators of translation along moduli directions of Calabi-Yau three-fold are constructed. The algebra of these generators is derived which take a simple form in canonical coordinates. Applying the Dirac method of quantization of second class constraint systems, we show that the Schrodinger equations corresponding to these generators are equivalent to the holomorphic anomaly equations if one defines the action functional of the quadratic field theory with a proper factor one-half.
TL;DR: In this paper, the Hamiltonian and Lagrangian structures of the Maxwell-Bloch equations are described. Butler et al. describe two families of solutions of the equations which are expressed in terms of the C Neumann system and the 2π-pulse soliton.
Abstract: The Maxwell–Bloch equations are represented as the equation of motion for a continuous chain of coupled C Neumann oscillators on the three-dimensional sphere. This description enables us to find new Hamiltonian and Lagrangian structures of the Maxwell–Bloch equations. The symplectic structure contains a topologically non-trivial magnetic term which is responsible for the coupling. The coupling forces are geometrized by means of an analogue of Kaluza–Klein theory. The conjugate momentum of the additional degree of freedom is precisely the speed of light in the medium. It can also be thought of as the strength of the coupling. The Lagrangian description has a structure similar to that of the Wess–Zumino–Witten–Novikov action. We describe two families of solutions of the Maxwell–Bloch equations which are expressed in terms of the C Neumann system. One family describes travelling non-linear waves whose constituent oscillators are the C Neumann oscillators in the same way as the harmonic oscillators are the constituent oscillators of the harmonic waves. The 2π-pulse soliton is a member of this family.
TL;DR: In this article, the quadratic field theory on seven dimensional spacetime constructed by a direct product of Calabi-Yau three-fold by a real time axis is studied.
Abstract: Studying the quadratic field theory on seven dimensional spacetime constructed by a direct product of Calabi-Yau three-fold by a real time axis, with phase space being the third cohomology of the Calabi-Yau three-fold, the generators of translation along moduli directions of
Calabi-Yau three-fold are constructed. The algebra of these generators is derived which take a simple form in canonical coordinates. Applying the Dirac method of quantization of second class constraint systems, we show that the Schr\"{o}dinger equations corresponding to these generators are equivalent to the holomorphic anomaly equations if one defines the action functional of the quadratic field theory with a proper factor one-half.
TL;DR: In this paper, the conservative dynamics of two point masses given in harmonic coordinates up to the third post-Newtonian order are treated within the framework of constrained canonical dynamics, and a representation of the approximate Poincar\'e algebra is constructed with the aid of Dirac brackets.
Abstract: The conservative dynamics of two point masses given in harmonic coordinates up to the third post-Newtonian order is treated within the framework of constrained canonical dynamics. A representation of the approximate Poincar\'e algebra is constructed with the aid of Dirac brackets. Uniqueness of the generators of the Poincar\'e group or the integrals of motion is achieved by imposing their action on the point mass coordinates to be identical with that of the usual infinitesimal Poincar\'e transformations. The second post-Coulombian approximation to the dynamics of two point charges as predicted by Feynman-Wheeler electrodynamics in Lorentz gauge is treated similarly.
Journal Article•
TL;DR: In this article, the authors show that when the methods of (2) are combined with the explicit stratification and orbital parameters of (9) and (10), the result is a construction of explicit analytic canonical coordinates for any coadjoint orbit of a completely solvable Lie group.
Abstract: We show that when the methods of (2) are combined with the explicit stratification and orbital parameters of (9) and (10), the result is a construction of explicit analytic canonical coordinates for any coadjoint orbit O of a completely solvable Lie group. For each layer in the stratification, the canonical coordinates and the orbital cross-section together constitute an analytic parametrization for the layer. Finally, we quantize the minimal open layer with the Moyal star product and prove that the coordinate functions are in a convenient comple- tion of spaces of polynomial functions on g , for a metric topology naturally related to the star product. Mathematics Subject Index: 22E25, 22E27, 53D55
TL;DR: In this paper, a general problem of visual representation of objects, individuals or populations, generally called profiles, characterized by a set of measurements as points in a low-dimensional Euclidean space is considered.
Abstract: Publisher Summary This chapter discusses canonical coordinates (variates) methods for reduction of dimensionality and graphical representation problems Canonical coordinates is the first attempt to give a general theory for reduction of high-dimensional data for graphical representation, which included the principal component analysis as a special case The general problem considered in the chapter is that of visual representation of objects, individuals or populations, generally called profiles, characterized by a set of measurements as points in a low-dimensional Euclidean space for an exploratory data analysis of the affinities between profiles The first step in such a problem is the specification of the basic metric space S, in which the profiles with the given set of measurements can be considered as points The second is the development of a methodology for transforming the points in S to Euclidean space Ek of k dimensions, preferably for k = 2 or 3 The choice of the basic space S and the metric (distance between points) in S has to be made on practical considerations relevant to the problem under investigation The chapter reviews methodology known by different names in statistical literature
TL;DR: In this paper, the Poisson structure for vortex strings is analyzed in detail and a Fock-like space of quantum states for the simplest case of bosonic vortex loops, with natural, nonlocal creation and annihilation operators for the quantized vortex filaments.
Abstract: In the framework of geometric quantization, filaments of vorticity in a two-dimensional, ideal incompressible superfluid belong to certain coadjoint orbits of the group of area-preserving diffeomorphisms. The Poisson structure for such vortex strings is analyzed in detail. While the Lie algebra associated with area-preserving diffeomorphisms is noncanonical, we can nevertheless find canonical coordinates and their conjugate momenta that describe these systems. We then introduce a Fock-like space of quantum states for the simplest case of bosonic vortex loops, with natural, nonlocal creation and annihilation operators for the quantized vortex filaments.
Posted Content•
TL;DR: This paper shows how the problem of feedback equivalence of generic germs of affine systems with two-dimensional input in state space of dimensions 4 and 5 can be reduced to the same problem for affine system with scalar input.
Abstract: The paper is devoted to the local classification of generic control-affine systems on an n-dimensional manifold with scalar input for any n>3 or with two inputs for n=4 and n=5, up to state-feedback transformations, preserving the affine structure. First using the Poincare series of moduli numbers we introduce the intrinsic numbers of functional moduli of each prescribed number of variables on which a classification problem depends. In order to classify affine systems with scalar input we associate with such a system the canonical frame by normalizing some structural functions in a commutative relation of the vector fields, which define our control system. Then, using this canonical frame, we introduce the canonical coordinates and find a complete system of state-feedback invariants of the system. It also gives automatically the micro-local (i.e. local in state-input space) classification of the generic non-affine n-dimensional control system with scalar input for n>2. Further we show how the problem of feedback-equivalence of affine systems with two-dimensional input in state space of dimensions 4 and 5 can be reduced to the same problem for affine systems with scalar input. In order to make this reduction we distinguish the subsystem of our control system, consisting of the directions of all extremals in dimension 4 and all abnormal extremals in dimension 5 of the time optimal problem, defined by the original control system. In each classification problem under consideration we find the intrinsic numbers of functional moduli of each prescribed number of variables according to its Poincare series.
TL;DR: In the canonical formulation of a classical field theory, symmetry properties are encoded in the Poisson bracket algebra, which may have a central term as mentioned in this paper, and the related Lagrangian form of the central term is derived from this well-understood canonical structure.
Abstract: In the canonical formulation of a classical field theory, symmetry properties are encoded in the Poisson bracket algebra, which may have a central term. Starting from this well-understood canonical structure, we derive the related Lagrangian form of the central term.
TL;DR: In this article, the authors present a Hamiltonian approach to modeling stellar motion by the derivation of canonical coordinates for the dynamics of a star relative to a star cluster, and a numerical optimization technique is developed based on the new orbital elements, and quasiperiodic stellar orbits are found.
TL;DR: In this article, the reaction path Hamiltonian is formulated using the canonical transformation theory of classical mechanics and a method for checking the accuracy of reaction path approach is presented, being based on the definition of a functional depending on the system Lagrangian.
Abstract: The reaction path Hamiltonian is formulated using the canonical transformation theory of classical mechanics. A method for checking the accuracy of the reaction path approach is presented, being based on the definition of a functional depending on the system Lagrangian. It is shown that the difference between the classical exact and the reaction path Hamiltonian dynamics can be expressed in terms of the potential energy sampled during the corresponding trajectories.
01 Jan 2005
TL;DR: In this article, it was shown that Eq. (4) is not gauge-invariant, and hence the law of conservation of the canonical momentum has a restricted meaning.
Abstract: μπe×=GGG (6) (μG is the magnetic moment). Eq. (4) has been also derived in ref. [4] for a non-relativistic system of mechanically free charged particles, using the Darwin gauge. However, one can see that in a general case Eq. (4) is not gauge-invariant, and hence the law of conservation of the canonical momentum has a restricted meaning. Moreover, Eq. (4) is not Lorentz-invariant. Indeed, its generalization to four-space gives
TL;DR: In this paper, the Dirac analogy between the dynamics of a single mode of the electromagnetic field and a simple harmonic oscillator is used to derive a generalized electromagnetic momentum from the average canonical momentum of harmonic oscillators.
TL;DR: In this article, the supersymmetric positivity induced by the Hilbert-Krein structure of the superspace is used to canonically quantize massive and massless chiral,antichiral and vector fields.
Abstract: We present unitarily represented supersymmetric canonical commutation relations which are subsequently used to canonically quantize massive and massless chiral,antichiral and vector fields. The massless fields, especially the vector one, show new facets which do not appear in the non superymmetric case. Our tool is the supersymmetric positivity induced by the Hilbert-Krein structure of the superspace.
Posted Content•
TL;DR: The representation of a Schrodinger equation as a classic Hamiltonian system allows to construct a unified perturbation theory both in classic, and in a quantum mechanics grounded on the theory of canonical transformations as discussed by the authors.
Abstract: The representation of a Schrodinger equations as a classic Hamiltonian system allows to construct a unified perturbation theory both in classic, and in a quantum mechanics grounded on the theory of canonical transformations, and also to receive asymptotic estimations of affinity of the precisian approximated solutions of Schrodinger equations
Posted Content•
TL;DR: In this paper, the question of whether the transformation that leaves the Euler-Lagrange equation of motion invariant is also a canonical transformation is addressed, and it is shown that it is not.
Abstract: In classical mechanics, we can describe the dynamics of a given system using either the Lagrangian formalism or the Hamiltonian formalism, the choice of either one being determined by whether one wants to deal with a second degree differential equation or a pair of first degree ones. For the former approach, we know that the Euler-Lagrange equation of motion remains invariant under additive total derivative with respect to time of any function of coordinates and time in the Lagrangian function, whereas the latter one is invariant under canonical transformations. In this short paper we address the question whether the transformation that leaves the Euler-Lagrange equation of motion invariant is also a canonical transformation and show that it is not.