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138
A Poisson integrator for Gaussian wavepacket dynamics
 Comput. Visual. Sci
, 2005
"... Dedicated to Peter Deuflhard on the occasion of his sixtieth birthday. Abstract We consider the variational approximation of the timedependent Schr"odinger equation by Gaussian wavepackets. The corresponding finitedimensional dynamical system inherits a Poisson (or noncanonically symplec ..."
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Cited by 15 (4 self)
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Dedicated to Peter Deuflhard on the occasion of his sixtieth birthday. Abstract We consider the variational approximation of the timedependent Schr&quot;odinger equation by Gaussian wavepackets. The corresponding finitedimensional dynamical system inherits a Poisson (or noncanonically symplectic) structure from the Schr&quot;odinger equation by its construction via the DiracFrenkelMcLachlan variational principle. The variational splitting between kinetic and potential energy turns out to yield an explicit, easily implemented numerical scheme. This method is a timereversible Poisson integrator, which also preserves the L 2 norm and linear and angular momentum. Using backward error analysis, we show longtime energy conservation for this splitting scheme. In the semiclassical limit the numerical approximations to position and momentum converge to those obtained by applying the St&quot;ormerVerlet method to the classical limit system. 1 Introduction Gaussian wavepacket dynamics is widely used in quantum molecular dynamics as an approximation to the timedependent Schr&quot;odinger equation, which we write as
The GuilleminSternberg conjecture for noncompact groups and spaces
, 2008
"... The Guillemin–Sternberg conjecture states that “quantisation commutes with reduction” in a specific technical setting. So far, this conjecture has almost exclusively been stated and proved for compact Lie groups G acting on compact symplectic manifolds, and, largely due to the use of Spin c Dirac op ..."
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Cited by 14 (4 self)
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The Guillemin–Sternberg conjecture states that “quantisation commutes with reduction” in a specific technical setting. So far, this conjecture has almost exclusively been stated and proved for compact Lie groups G acting on compact symplectic manifolds, and, largely due to the use of Spin c Dirac operator techniques, has reached a high degree of perfection under these compactness assumptions. In this paper we formulate an appropriate Guillemin– Sternberg conjecture in the general case, under the main assumptions that the Lie group action is proper and cocompact. This formulation is motivated by our interpretation of the “quantisation commuates with reduction ” phenomenon as a special case of the functoriality of quantisation, and uses equivariant Khomology and the Ktheory of the group C ∗algebra C ∗ (G) in a crucial way. For example, the equivariant index which in the compact case takes values in the representation ring R(G) is replaced by the analytic assembly map which takes values in K0(C ∗ (G)) familiar from the Baum–Connes conjecture in noncommutative geometry. Under the usual freeness assumption on the action, we prove our conjecture for all Lie groups G having a discrete normal subgroup Γ with compact quotient G/Γ, but we believe
A Hamiltonian Particle Method for Diffeomorphic Image Registration
, 2007
"... Diffeomorphic image registration, where images are aligned using diffeomorphic warps, is a popular subject for research in medical image analysis. We introduce a novel algorithm for computing diffeomorphic warps that solves the Euler equations on the diffeomorphism group explicitly, based on a disc ..."
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Cited by 12 (0 self)
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Diffeomorphic image registration, where images are aligned using diffeomorphic warps, is a popular subject for research in medical image analysis. We introduce a novel algorithm for computing diffeomorphic warps that solves the Euler equations on the diffeomorphism group explicitly, based on a discretisation of the Hamiltonian, rather than using an optimiser. The result is an algorithm that is many times faster than those considered previously.
Multiple hamiltonian structure of BogoyavlenskyToda lattices
 Rev. Math. Phys
"... This paper is mainly a review of the multi–Hamiltonian nature of Toda and generalized Toda lattices corresponding to the classical simple Lie groups but it includes also some new results. The areas investigated include master symmetries, recursion operators, higher Poisson brackets, invariants and g ..."
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Cited by 12 (5 self)
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This paper is mainly a review of the multi–Hamiltonian nature of Toda and generalized Toda lattices corresponding to the classical simple Lie groups but it includes also some new results. The areas investigated include master symmetries, recursion operators, higher Poisson brackets, invariants and group symmetries for the systems. In addition to the positive hierarchy we also consider the negative hierarchy which is crucial in establishing the bi–Hamiltonian structure for each particular simple Lie group. Finally, we include some results on point and Noether symmetries and an interesting connection with the exponents of simple Lie groups.
The geometry of the twocomponent CamassaHolm and DegasperisProcesi equations
 J. Geom. Phys
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Magnetic fields in noncommutative quantum mechanics
"... We discuss various descriptions of a quantum particle on noncommutative space in a (possibly nonconstant) magnetic field. We have tried to present the basic facts in a unified and synthetic manner, and to clarify the relationship between various approaches and results that are scattered in the lite ..."
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We discuss various descriptions of a quantum particle on noncommutative space in a (possibly nonconstant) magnetic field. We have tried to present the basic facts in a unified and synthetic manner, and to clarify the relationship between various approaches and results that are scattered in the literature. 1 We dedicate these notes to the memory of Julius Wess whose scientific work was largely devoted to the study of gauge fields and whose latest interests concerned physical theories on noncommutative space. The fundamental and inspiring contributions of Julius to Theoretical Physics will always bear with us, but his great kindness, his clear and enthusiastic presentations, and his precious advice will be missed by all those who had the chance to meet him. 2
Variational integrators for constrained dynamical systems
, 2007
"... Key words Variational time integration, constrained dynamical systems, differential algebraic equations, flexible multibody dynamics. A variational formulation of constrained dynamics is presented in the continuous and in the discrete setting. The existing theory on variational integration of constr ..."
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Key words Variational time integration, constrained dynamical systems, differential algebraic equations, flexible multibody dynamics. A variational formulation of constrained dynamics is presented in the continuous and in the discrete setting. The existing theory on variational integration of constrained problems is extended by aspects on the initialization of simulations, the discrete Legendre transform and certain postprocessing steps. Furthermore, the discrete null space method which has been introduced in the framework of energymomentum conserving integration of constrained systems is adapted to the framework of variational integrators. It eliminates the constraint forces (including the Lagrange multipliers) from the timestepping scheme and subsequently reduces its dimension to the minimal possible number. While retaining the structure preserving properties of the specific integrator, the solution of the smaller dimensional system saves computational costs and does not suffer from conditioning problems. The performance of the variational discrete null space method is illustrated by numerical examples dealing with mass point systems, a closed kinematic chain of rigid bodies and flexible multibody dynamics and the solutions are compared to those obtained by an energymomentum scheme. © 2008 WILEYVCH Verlag GmbH & Co. KGaA, Weinheim 1
Nparticle dynamics of the Euler equations for planar diffeomorphisms
 DYNAM. SYSTEMS
"... The Euler equations associated with diffeomorphism groups have received much recent study because of their links with fluid dynamics, computer vision, and mechanics. In this paper, we consider the dynamics of N point particles or ‘blobs’ moving under the action of the Euler equations associated with ..."
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Cited by 9 (3 self)
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The Euler equations associated with diffeomorphism groups have received much recent study because of their links with fluid dynamics, computer vision, and mechanics. In this paper, we consider the dynamics of N point particles or ‘blobs’ moving under the action of the Euler equations associated with the group of diffeomorphisms of the plane in a variety of different metrics. This dynamical system is already in widespread use in the field of image registration, where the point particles correspond to image landmarks, but its dynamical behaviour has not previously been studied. The 2 body problem is always integrable, and we analyze its phase portrait under different metrics. In particular, we show that 2body capturing orbits (in which the distances between the particles tend to 0 as t → ∞) can occur when the kernel is sufficiently smooth and the relative initial velocity of the particles is sufficiently large. We compute the dynamics of these ‘dipoles ’ with respect to other test particles, and supplement the calculations with simulations for larger N that illustrate the different regimes.
Internal variables and dynamic degrees of freedom
 J. NonEquilib. Thermodynamics
"... Abstract. Dynamic degrees of freedom and internal variables are treated in a uniform way. The unification is achieved by means of the introduction of a dual internal variable, which provides the corresponding evolution equations under assumption that the OnsagerCasimir reciprocity relations are not ..."
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Cited by 9 (4 self)
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Abstract. Dynamic degrees of freedom and internal variables are treated in a uniform way. The unification is achieved by means of the introduction of a dual internal variable, which provides the corresponding evolution equations under assumption that the OnsagerCasimir reciprocity relations are not required in general. 1.
On algebraic structures of numerical integration on vector spaces and manifolds
 IRMA Lectures in Mathematics and Theoretical Physics
, 2012
"... Abstract. Numerical analysis of timeintegration algorithms has been applying advanced algebraic techniques for more than fourty years. An explicit description of the group of characters in the Butcher–Connes–Kreimer Hopf algebra first appeared in Butcher’s work on composition of integration method ..."
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Cited by 8 (1 self)
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Abstract. Numerical analysis of timeintegration algorithms has been applying advanced algebraic techniques for more than fourty years. An explicit description of the group of characters in the Butcher–Connes–Kreimer Hopf algebra first appeared in Butcher’s work on composition of integration methods in 1972. In more recent years, the analysis of structure preserving algorithms, geometric integration techniques and integration algorithms on manifolds have motivated the incorporation of other algebraic structures in numerical analysis. In this paper we will survey algebraic structures that have found applications within these areas. This includes preLie structures for the geometry of flat and torsion free connections appearing in the analysis of numerical flows on vector spaces. The much more recent postLie and Dalgebras appear in the analysis of flows on manifolds with flat connections with constant torsion. Dynkin and Eulerian idempotents appear in the analysis of nonautonomous flows and in backward error analysis. Noncommutative Bell polynomials and a noncommutative Faa ̀ di Bruno Hopf algebra are other examples of structures appearing naturally in the numerical analysis of integration on manifolds.