45 results on '"Euler-Poincaré equations"'
Search Results
2. A natural 4th-order generalization of the geodesic problem.
- Author
-
Camarinha, Margarida
- Subjects
- *
EULER-Lagrange equations , *LIE algebras , *LIE groups , *RIEMANNIAN manifolds , *PROBLEM solving - Abstract
We propose a fourth-order extension of the geodesic problem arising in the continuity of the study of Riemannian cubics. We consider the variational problem in a Riemannian manifold and derive the Euler-Lagrange equation. For the special situation of Lie groups, we use Euler-Poincaré reduction to obtain the reduced equation on the Lie algebra. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
3. On the Continuity of the Solution Map of the Euler–Poincaré Equations in Besov Spaces.
- Author
-
Li, Min and Liu, Huan
- Abstract
By constructing a series of perturbation functions through localization in the Fourier domain and using a symmetric form of the system, we show that the data-to-solution map for the Euler–Poincaré equations is nowhere uniformly continuous in B p , r s (R d) with s > max { 1 + d 2 , 3 2 } and (p , r) ∈ (1 , ∞) × [ 1 , ∞) . This improves our previous result (Li et al. in Nonlinear Anal RWA 63:103420, 2022) which shows the data-to-solution map for the Euler–Poincaré equations is non-uniformly continuous on a bounded subset of B p , r s (R d) near the origin. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
4. Reduction in optimal control with broken symmetry for collision and obstacle avoidance of multi-agent system on Lie groups
- Author
-
Efstratios Stratoglou, Alexandre Anahory Simoes, and Leonardo J. Colombo
- Subjects
lagrangian systems ,symmetry reduction ,euler-poincaré equations ,multi-agent control systems ,lie-poisson integrators ,Analytic mechanics ,QA801-939 - Abstract
We study the reduction by symmetry for optimality conditions in optimal control problems of left-invariant affine multi-agent control systems, with partial symmetry breaking cost functions for continuous-time and discrete-time systems. We recast the optimal control problem as a constrained variational problem with a partial symmetry breaking Lagrangian and obtain the reduced optimality conditions from a reduced variational principle via symmetry reduction techniques in both settings continuous-time, and discrete-time. We apply the results to a collision and obstacle avoidance problem for multiple vehicles evolving on $ SE(2) $ in the presence of a static obstacle.
- Published
- 2023
- Full Text
- View/download PDF
5. Reduction in optimal control with broken symmetry for collision and obstacle avoidance of multi-agent system on Lie groups.
- Author
-
Stratoglou, Efstratios, Simoes, Alexandre Anahory, and Colombo, Leonardo J.
- Subjects
- *
MULTIAGENT systems , *SYMMETRY breaking , *DISCRETE-time systems , *VARIATIONAL principles , *COST functions , *LIE groups - Abstract
We study the reduction by symmetry for optimality conditions in optimal control problems of left-invariant affine multi-agent control systems, with partial symmetry breaking cost functions for continuous-time and discrete-time systems. We recast the optimal control problem as a constrained variational problem with a partial symmetry breaking Lagrangian and obtain the reduced optimality conditions from a reduced variational principle via symmetry reduction techniques in both settings continuous-time, and discrete-time. We apply the results to a collision and obstacle avoidance problem for multiple vehicles evolving on S E (2) in the presence of a static obstacle. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
6. Decomposing Euler–Poincaré Flow on the Space of Hamiltonian Vector Fields.
- Author
-
Esen, Oğul, De Lucas, Javier, Muñoz, Cristina Sardon, and Zając, Marcin
- Subjects
- *
VECTOR fields , *VECTOR spaces , *SYMMETRIC spaces , *FLUID flow , *LIE algebras - Abstract
The main result of this paper is a matched-pair decomposition of the space of symmetric contravariant tensors T Q . From this procedure two complementary Lie subalgebras of T Q under mutual interaction arise. Introducing a lift operator, the matched pair decomposition of the space of Hamiltonian vector fields is determined. According to this realization, the Euler–Poincaré flows on such spaces are decomposed into two subdynamics: one is the Euler–Poincaré formulation of isentropic fluid flows, and the other one corresponds with Euler–Poincaré equations on contravariant tensors of order n ⩾ 2 . [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
7. Variational Discretization Framework for Geophysical Flow Models
- Author
-
Bauer, Werner, Gay-Balmaz, François, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Woeginger, Gerhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Nielsen, Frank, editor, and Barbaresco, Frédéric, editor
- Published
- 2019
- Full Text
- View/download PDF
8. Tulczyjew’s triplet for lie groups III: Higher order dynamics and reductions for iterated bundles
- Author
-
Esen Oğul, Gümral Hasan, and Sütlü Serkan
- Subjects
euler-poincaré equations ,lie-poisson equations ,higher order dynamics on lie groups ,Mechanics of engineering. Applied mechanics ,TA349-359 - Abstract
Given a Lie group 𝐺, we elaborate the dynamics on 𝑇*𝑇*𝐺 and 𝑇*𝑇𝐺, which is given by a Hamiltonian, as well as the dynamics on the Tulczyjew symplectic space 𝑇𝑇*𝐺, which may be defined by a Lagrangian or a Hamiltonian function. As the trivializations we adapted respect the group structures of the iterated bundles, we exploit all possible subgroup reductions (Poisson, symplectic or both) of higher order dynamics.
- Published
- 2021
- Full Text
- View/download PDF
9. Decomposing Euler–Poincaré Flow on the Space of Hamiltonian Vector Fields
- Author
-
Oğul Esen, Javier De Lucas, Cristina Sardon Muñoz, and Marcin Zając
- Subjects
matched pair lie algebras ,symmetric contravariant tensors ,Hamiltonian vector fields ,Euler–Poincaré equations ,Mathematics ,QA1-939 - Abstract
The main result of this paper is a matched-pair decomposition of the space of symmetric contravariant tensors TQ. From this procedure two complementary Lie subalgebras of TQ under mutual interaction arise. Introducing a lift operator, the matched pair decomposition of the space of Hamiltonian vector fields is determined. According to this realization, the Euler–Poincaré flows on such spaces are decomposed into two subdynamics: one is the Euler–Poincaré formulation of isentropic fluid flows, and the other one corresponds with Euler–Poincaré equations on contravariant tensors of order n⩾2.
- Published
- 2022
- Full Text
- View/download PDF
10. Ill-posedness for the Euler–Poincaré equations in Besov spaces.
- Author
-
Li, Min and Guo, Yingying
- Subjects
- *
BESOV spaces , *INITIAL value problems , *EQUATIONS , *TRIANGULAR norms - Abstract
We prove that the initial value problem for the Euler–Poincaré equations is not locally well-posed for initial data in the Besov space B p , ∞ σ whenever σ > 2 + max { 1 + d p , 3 2 }. By presenting a new construction of initial data u 0 , we prove the corresponding solution of Euler–Poincaré equations starting from u 0 is discontinuous at t = 0 in the norm of B p , ∞ σ , which implies the ill-posedness. Since this problem is locally well-posed in the Besov space B p , r σ for r < ∞ and the same σ , our result suggests that well-posedness does not hold at the endpoint r = ∞. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
11. On the Hamel Coefficients and the Boltzmann–Hamel Equations for the Rigid Body.
- Author
-
Müller, Andreas
- Abstract
The Boltzmann–Hamel (BH) equations are central in the dynamics and control of nonholonomic systems described in terms of quasi-velocities. The rigid body is a classical example of such systems, and it is well-known that the BH-equations are the Newton–Euler (NE) equations when described in terms of rigid body twists as quasi-velocities. It is further known that the NE-equations are the Euler–Poincaré, respectively, the reduced Euler–Lagrange equations on SE(3) when using body-fixed or spatial representation of rigid body twists. The connection between these equations are the Hamel coefficients, which are immediately identified as the structure constants of SE(3). However, an explicit coordinate-free derivation has not been presented in the literature. In this paper the Hamel coefficients for the rigid body are derived in a coordinate-free way without resorting to local coordinates describing the rigid body motion. The three most relevant choices of quasi-velocities (body-fixed, spatial, and hybrid representation of rigid body twists) are considered. The corresponding BH-equations are derived explicitly for the rotating and free floating body. Further, the Hamel equations for nonholonomically constrained rigid bodies are discussed, and demonstrated for the inhomogenous ball rolling on a plane. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
12. Euler-Poincare reduction in principal fibre bundles and the problem of Lagrange
- Author
-
Castrillón López, Marco, Rodrigo, César, Garcia, Pedro L., Castrillón López, Marco, Rodrigo, César, and Garcia, Pedro L.
- Abstract
We compare Euler–Poincaré reduction in principal fibre bundles, as a constrained variational problem on the connections of this fibre bundle and constraint defined by the vanishing of the curvature of the connection, with the corresponding problem of Lagrange. Under certain cohomological condition we prove the equality of the sets of critical sections of both problems with the one obtained by application of the Lagrange multiplier rule. We compute the corresponding Cartan form and characterise critical sections as the set of holonomic solutions of the Cartan equation and, in particular, under a certain regularity condition for the problem, we prove the holonomy of any solution of this equation., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub
- Published
- 2023
13. Reduction of Forward Difference Operators in Principal G-bundles.
- Author
-
Casimiro, Ana and Rodrigo, César
- Subjects
LIE groups ,DIFFERENCE operators ,NUMERICAL integration ,ORDINARY differential equations ,DISCRETIZATION methods ,ANALYSIS of covariance - Abstract
Retraction maps on Lie groups can be successfully used in mechanics and control theory to generate numerical integration schemes, for ordinary differential equations with a variational origin, recovering at the same time a discrete version of the energy and symplectic structure conservation properties, that are characteristic of smooth variational mechanics. The present work fixes the specific tool that plays in gauge field theories the same role as retraction maps on geometric mechanics. This tool, the covariant reduced projectable forward difference operator, can be used for a covariant discretization of the main elements of a variational theory: the jet bundle, the Lagrangian density and the associated action functional. Particular interest is dedicated to the trivialized formulation of a gauge field theory, and its reduction into a theory where fields are given as principal connections and H-structures. Main characteristics of the presented method are its covariance by gauge transformations and the commutation of the discretization and the reduction processes. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
14. Intrinsic dynamics and total energy-shaping control of the ballbot system.
- Author
-
Satici, A. C., Donaire, A., and Siciliano, B.
- Subjects
- *
BIPEDALISM , *MOBILE robots , *CONSTRAINT satisfaction , *EULER characteristic , *EQUATIONS of motion - Abstract
Research on bipedal locomotion has shown that a dynamic walking gait is energetically more efficient than a statically stable one. Analogously, even though statically stable multi-wheeled robots are easier to control, they are energetically less efficient and have low accelerations to avoid tipping over. In contrast, the ballbot is an underactuated, nonholonomically constrained mobile robot, whose upward equilibrium point has to be stabilised by active control. In this work, we derive coordinate-invariant, reduced, Euler–Poincaré equations of motion for the ballbot. By means of partial feedback linearisation, we obtain two independent passive outputs with corresponding storage functions and utilise these to come up with energy-shaping control laws which move the system along the trajectories of a new Lagrangian system whose desired equilibrium point is asymptotically stable by construction. The basin of attraction of this controller is shown to be almost global under certain conditions on the design of the mechanism which are reflected directly in the mass matrix of the unforced equations of motion. [ABSTRACT FROM PUBLISHER]
- Published
- 2017
- Full Text
- View/download PDF
15. On extensions, Lie-Poisson systems, and dissipation
- Author
-
Esen, Oğul, Özcan, Gökhan, Sütlü, Serkan, Işık Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Işık University, Faculty of Arts and Sciences, Department of Mathematics, and Sütlü, Serkan
- Subjects
Moments ,Fluids ,Metriplectic system ,Semidirect products ,Geometry ,Unified product ,Bracket formulation ,Noether's theorem ,Lie-Poisson equation ,Dynamics ,Magnetohydrodynamics ,Algebra ,Euler-poincare equations ,Lagrangian ,Reduction - Abstract
Acknowledgments. This paper is a part of the project “Matched pairs of Lagrangian and Hamiltonian Systems” supported by TÜBİTAK (the Scientific and Technological Research Council of Turkey) with the project number 117F426, the support of which is acknowledged by the authors. Lie-Poisson systems on the dual spaces of unified products are studied. Having been equipped with a twisted 2-cocycle term, the extending structure framework allows not only to study the dynamics on 2-cocycle extensions, but also to (de)couple mutually interacting Lie-Poisson systems. On the other hand, symmetric brackets; such as the double bracket, the Cartan-Killing bracket, the Casimir dissipation bracket, and the Hamilton dissipation bracket are worked out in detail. Accordingly, the collective motion of two mutually interacting irreversible dynamics, as well as the mutually interacting metriplectic flows, are obtained. The theoretical results are illustrated in three examples. As an infinite-dimensional physical model, decompositions of the BBGKY hierarchy are presented. As for the finite-dimensional examples, the coupling of two Heisenberg algebras, and the coupling of two copies of 3D dynamics are studied. Q4 WOS:000817805300003
- Published
- 2022
16. Model Formulation Over Lie Groups and Numerical Methods to Simulate the Motion of Gyrostats and Quadrotors
- Author
-
Simone Fiori
- Subjects
lagrange–d’alembert principle ,non-conservative dynamical system ,euler–poincaré equations ,gyrostat satellite ,quadcopter drone ,forward euler method ,explicit runge–kutta method ,lie group ,Mathematics ,QA1-939 - Abstract
The present paper recalls a formulation of non-conservative system dynamics through the Lagrange−d’Alembert principle expressed through a generalized Euler−Poincaré form of the system equation on a Lie group. The paper illustrates applications of the generalized Euler−Poincaré equations on the rotation groups to a gyrostat satellite and a quadcopter drone. The numerical solution of the dynamical equations on the rotation groups is tackled via a generalized forward Euler method and an explicit Runge−Kutta integration method tailored to Lie groups.
- Published
- 2019
- Full Text
- View/download PDF
17. Tulczyjew’s triplet for lie groups III: Higher order dynamics and reductions for iterated bundles
- Author
-
Serkan Sütlü, Oğul Esen, Hasan Gümral, Işık Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Işık University, Faculty of Arts and Sciences, Department of Mathematics, and Sütlü, Serkan
- Subjects
Pure mathematics ,Nonholonomic systems ,Computational Mechanics ,Lie—Poisson equations ,FOS: Physical sciences ,lie-poisson equations ,Space (mathematics) ,euler-poincaré equations ,symbols.namesake ,FOS: Mathematics ,Order (group theory) ,higher order dynamics on lie groups ,Mathematics::Symplectic Geometry ,Lagrangian ,Mathematical Physics ,Mathematics ,Higher order dynamics on Lie groups ,Hamiltonian mechanics ,Group (mathematics) ,Applied Mathematics ,Mechanical Engineering ,Mechanics of engineering. Applied mechanics ,Lie group ,Mathematical Physics (math-ph) ,TA349-359 ,Iterated function ,Mathematics - Symplectic Geometry ,Euler—Poincaré equations ,symbols ,Symplectic Geometry (math.SG) ,Multi-symplectic ,Hamiltonian (control theory) ,Symplectic geometry - Abstract
Given a Lie group $G$, we elaborate the dynamics on $T^*T^*G$ and $T^*TG$, which is given by a Hamiltonian, as well as the dynamics on the Tulczyjew symplectic space $TT^*G$, which may be defined by a Lagrangian or a Hamiltonian function. As the trivializations we adapted respect the group structures of the iterated bundles, we exploit all possible subgroup reductions (Poisson, symplectic or both) of higher order dynamics., arXiv admin note: substantial text overlap with arXiv:1503.06568
- Published
- 2021
18. Dinamik sistemlerin eşlenmesi.
- Author
-
Esen, Oğul
- Abstract
The equations (matched Lie-Poisson and matched Euler-Poincaré) are written for a couple of mutually interacting physical systems. It is shown that the matched dynamics is a generalization of the well-developed semi-direct product theory. Two examples are provided. The first one is to write the matched equations for the matched pair of upper and lower triangular matrix groups whose diagonal entries are 1. The second example is to write the matched equations for the Lie group obtained by matching a nilpotent group of class two by itself. Two new open problems are presented. One of these is to write pure geometric relation between the plasma and fluid in the framework of the matched dynamics. The other is to match two discrete systems under mutual interaction. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
19. Lagrangian dynamics on matched pairs.
- Author
-
Esen, Oğul and Sütlü, Serkan
- Subjects
- *
LAGRANGIAN functions , *LIE groups , *EULER-Lagrange system , *GROUP theory , *DYNAMICS - Abstract
Given a matched pair of Lie groups, we show that the tangent bundle of the matched pair group is isomorphic to the matched pair of the tangent groups. We thus obtain the Euler–Lagrange equations on the trivialized matched pair of tangent groups, as well as the Euler–Poincaré equations on the matched pair of Lie algebras. We show explicitly how these equations cover those of the semi-direct product theory. In particular, we study the trivialized, and the reduced Lagrangian dynamics on the group S L ( 2 , C ) . [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
20. Euler-Poincaré equations on semi-direct products.
- Author
-
Cismas, Emanuel-Ciprian
- Abstract
We study the Euler-Poincaré equations on the semi-direct products of the group of orientation preserving diffeomorphisms of the circle with itself. We establish a reduction result to the direct product structure which will allow us to investigate the well-posedness, in the smooth category, using geodesic flows on infinite dimensional Lie groups. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
21. Variational principles for fluid dynamics on rough paths.
- Author
-
Crisan, Dan, Holm, Darryl D., Leahy, James-Michael, and Nilssen, Torstein
- Subjects
- *
FLUID dynamics , *VARIATIONAL principles , *GEOPHYSICAL fluid dynamics , *BROWNIAN motion , *STOCHASTIC integrals , *VECTOR fields , *EULERIAN graphs , *LIE groups - Abstract
In recent works, beginning with [76] , several stochastic geophysical fluid dynamics (SGFD) models have been derived from variational principles. In this paper, we introduce a new framework for parametrization schemes (PS) in GFD. We derive a class of rough geophysical fluid dynamics (RGFD) models as critical points of rough action functionals using the theory of controlled rough paths. These RGFD models characterize Lagrangian trajectories in fluid dynamics as geometric rough paths (GRP) on the manifold of diffeomorphic maps. We formulate three constrained variational approaches for the derivation of these models. The first is the Clebsch formulation, in which the constraints are imposed as rough advection laws. The second is the Hamilton-Pontryagin formulation, in which the constraints are imposed as right-invariant rough vector fields. And the third is the Euler–Poincaré formulation, in which the variations are constrained. These constrained rough variational principles lead directly to the Lie–Poisson Hamiltonian formulation of fluid dynamics on GRP. The GRP framework preserves the geometric structure of fluid dynamics obtained by using Lie group reduction to pass from Lagrangian to Eulerian variational principles, yielding a rough formulation of the Kelvin circulation theorem. The rough formulation enhances its stochastic counterpart developed in [76] , and extended to semimartingales in [109]. For example, the rough-path variational approach includes non-Markovian perturbations of the Lagrangian fluid trajectories. In particular, memory effects can be introduced through a judicious choice of the rough path (e.g. a realization of a fractional Brownian motion). In the particular case when the rough path is a realization of a semimartingale, we recover the SGFD models in [76,109]. However, by eliminating the need for stochastic variational tools, we retain a pathwise interpretation of the Lagrangian trajectories. In contrast, the Lagrangian trajectories in the stochastic framework are described by stochastic integrals, which do not have a pathwise interpretation. Thus, the rough path formulation restores this property. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
22. Generalized Hunter-Saxton Equations, Optimal Information Transport, and Factorization of Diffeomorphisms.
- Author
-
Modin, Klas
- Abstract
We study geodesic equations for a family of right-invariant Riemannian metrics on the group of diffeomorphisms of a compact manifold. The metrics descend to Fisher's information metric on the space of smooth probability densities. The right reduced geodesic equations are higher-dimensional generalizations of the μ-Hunter-Saxton equation, used to model liquid crystals under the influence of magnetic fields. Local existence and uniqueness results are established by proving smoothness of the geodesic spray. The descending property of the metrics is used to obtain a novel factorization of diffeomorphisms. Analogous to the polar factorization in optimal mass transport, this factorization solves an optimal information transport problem. It can be seen as an infinite-dimensional version of QR factorization of matrices. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
23. Euler–Poincaré reduction in principal bundles by a subgroup of the structure group.
- Author
-
Castrillón López, M., García, P.L., and Rodrigo, C.
- Subjects
- *
EULER characteristic , *JET bundles (Mathematics) , *GROUP theory , *CONSERVATION laws (Physics) , *GAUGE invariance , *HOLONOMIC constraints - Abstract
Abstract: Given a Lagrangian density defined on the 1-jet bundle of a principal -bundle invariant with respect to a subgroup of , the reduction of the variational problem defined by to , where is the bundle of connections in , is studied. It is shown that the reduced Lagrangian density defines a zero order variational problem on connections and -structures of with non-holonomic constraints and and set of admissible variations those induced by the infinitesimal gauge transformations in and . The Euler–Poincaré equations for critical reduced sections are obtained as well as the reconstruction process to the unreduced problem. The corresponding conservation laws and their relationship with the Noether theory are also analyzed. Finally, some instances are studied: the heavy top and affine principal bundles, the main application of which is used for molecular strands. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
- View/download PDF
24. Constraints in Euler-Poincaré Reduction of Field Theories.
- Author
-
Castrillón López, M.
- Subjects
- *
EULER characteristic , *LAGRANGE multiplier , *FIELD theory (Physics) , *MATHEMATICAL symmetry , *CALCULUS , *MATHEMATICAL models , *MATHEMATICAL optimization - Abstract
The goal of this short note is to show the geometric structure of the Euler-Poincaré reduction procedure in Field Theories with special emphasis on the nature of the set of variations and the set of admissible sections. The method of Lagrange multipliers is also applied for a deeper study of these constraints. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
25. G-Strands.
- Author
-
Holm, Darryl, Ivanov, Rossen, and Percival, James
- Subjects
- *
LIE groups , *INVARIANTS (Mathematics) , *LAGRANGIAN functions , *RIGID bodies , *HAMILTONIAN systems , *SOBOLEV spaces , *EULER characteristic - Abstract
A G-strand is a map g( t, s):ℝ×ℝ→ G for a Lie group G that follows from Hamilton's principle for a certain class of G-invariant Lagrangians. The SO(3)-strand is the G-strand version of the rigid body equation and it may be regarded physically as a continuous spin chain. Here, SO(3)-strand dynamics for ellipsoidal rotations is derived as an Euler-Poincaré system for a certain class of variations and recast as a Lie-Poisson system for coadjoint flow with the same Hamiltonian structure as for a perfect complex fluid. For a special Hamiltonian, the SO(3)-strand is mapped into a completely integrable generalization of the classical chiral model for the SO(3)-strand. Analogous results are obtained for the Sp(2)-strand. The Sp(2)-strand is the G-strand version of the Sp(2) Bloch-Iserles ordinary differential equation, whose solutions exhibit dynamical sorting. Numerical solutions show nonlinear interactions of coherent wave-like solutions in both cases. Diff(ℝ)-strand equations on the diffeomorphism group G=Diff(ℝ) are also introduced and shown to admit solutions with singular support (e.g., peakons). [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
26. DISCRETE SECOND-ORDER EULER-POINCARÉ EQUATIONS:: APPLICATIONS TO OPTIMAL CONTROL.
- Author
-
COLOMBO, LEONARDO, JIMÉNEZ, FERNANDO, and DE DIEGO, DAVID MARTÍN
- Subjects
- *
CONTROL theory (Engineering) , *NUMERICAL analysis , *LAGRANGE equations , *LIE groups , *CALCULUS , *MECHANICS (Physics) , *EULER characteristic - Abstract
In this paper we will discuss some new developments in the design of numerical methods for optimal control problems of Lagrangian systems on Lie groups. We will construct these geometric integrators using discrete variational calculus on Lie groups, deriving a discrete version of the second-order Euler-Lagrange equations. Interesting applications as, for instance, a discrete derivation of the Euler-Poincaré equations for second-order Lagrangians and its application to optimal control of a rigid body, and of a Cosserat rod are shown at the end of the paper. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
27. Exact geometric theory of dendronized polymer dynamics
- Author
-
Gay-Balmaz, François, Holm, Darryl D., Putkaradze, Vakhtang, and Ratiu, Tudor S.
- Subjects
- *
GEOMETRIC analysis , *DENDRIMERS , *POLYMERS , *LAGRANGIAN functions , *ITERATIVE methods (Mathematics) , *MATHEMATICAL symmetry - Abstract
Abstract: Dendronized polymers consist of an elastic backbone with a set of iterated branch structures (dendrimers) attached at every base point of the backbone. The conformations of such molecules depend on the elastic deformation of the backbone and the branches, as well as on nonlocal (e.g., electrostatic, or Lennard–Jones) interactions between the elementary molecular units comprising the dendrimers and/or backbone. We develop a geometrically exact theory for the dynamics of such polymers, taking into account both local (elastic) and nonlocal interactions. The theory is based on applying symmetry reduction of Hamiltonʼs principle for a Lagrangian defined on the tangent bundle of iterated semidirect products of the rotation groups that represent the relative orientations of the dendritic branches of the polymer. The resulting symmetry-reduced equations of motion are written in conservative form. [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
- View/download PDF
28. Higher order Lagrange-Poincaré and Hamilton-Poincaré reductions.
- Author
-
Gay-Balmaz, François, Holm, Darryl, and Ratiu, Tudor
- Subjects
- *
TANGENT bundles , *LAGRANGE equations , *LIE groups , *POISSON brackets , *VARIATIONAL principles , *MATHEMATICAL analysis - Abstract
Motivated by the problem of longitudinal data assimilation, e.g., in the registration of a sequence of images, we develop the higher-order framework for Lagrangian and Hamiltonian reduction by symmetry in geometric mechanics. In particular, we obtain the reduced variational principles and the associated Poisson brackets. The special case of higher order Euler-Poincaré and Lie-Poisson reduction is also studied in detail. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
29. Continuous and Discrete Clebsch Variational Principles.
- Author
-
Cotter, C. and Holm, D.
- Subjects
- *
VECTOR spaces , *LIE groups , *ALGEBRA , *EQUATIONS , *DYNAMICS - Abstract
The Clebsch method provides a unifying approach for deriving variational principles for continuous and discrete dynamical systems where elements of a vector space are used to control dynamics on the cotangent bundle of a Lie group via a velocity map. This paper proves a reduction theorem which states that the canonical variables on the Lie group can be eliminated, if and only if the velocity map is a Lie algebra action, thereby producing the Euler–Poincaré (EP) equation for the vector space variables. In this case, the map from the canonical variables on the Lie group to the vector space is the standard momentum map defined using the diamond operator. We apply the Clebsch method in examples of the rotating rigid body and the incompressible Euler equations. Along the way, we explain how singular solutions of the EP equation for the diffeomorphism group (EPDiff) arise as momentum maps in the Clebsch approach. In the case of finite-dimensional Lie groups, the Clebsch variational principle is discretized to produce a variational integrator for the dynamical system. We obtain a discrete map from which the variables on the cotangent bundle of a Lie group may be eliminated to produce a discrete EP equation for elements of the vector space. We give an integrator for the rotating rigid body as an example. We also briefly discuss how to discretize infinite-dimensional Clebsch systems, so as to produce conservative numerical methods for fluid dynamics. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
30. VARIATIONAL CALCULUS ON LIE ALGEBROIDS.
- Author
-
Martínez, Eduardo
- Subjects
- *
LIE algebroids , *ALGEBROIDS , *CALCULUS of variations , *MATHEMATICAL optimization , *MAXIMA & minima - Abstract
It is shown that the Lagrange's equations for a Lagrangian system on a Lie algebroid are obtained as the equations for the critical points of the action functional defined on a Banach manifold of curves. The theory of Lagrangian reduction and the relation with the method of Lagrange multipliers are also studied. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
31. Euler–Poincaré reduction in principal fibre bundles and the problem of Lagrange
- Author
-
Castrillón, Marco, García, Pedro L., and Rodrigo, César
- Subjects
- *
EULER characteristic , *FIBER bundles (Mathematics) , *LAGRANGE problem , *HOLONOMY groups - Abstract
Abstract: We compare Euler–Poincaré reduction in principal fibre bundles, as a constrained variational problem on the connections of this fibre bundle and constraint defined by the vanishing of the curvature of the connection, with the corresponding problem of Lagrange. Under certain cohomological condition we prove the equality of the sets of critical sections of both problems with the one obtained by application of the Lagrange multiplier rule. We compute the corresponding Cartan form and characterise critical sections as the set of holonomic solutions of the Cartan equation and, in particular, under a certain regularity condition for the problem, we prove the holonomy of any solution of this equation. [Copyright &y& Elsevier]
- Published
- 2007
- Full Text
- View/download PDF
32. Noise and Dissipation on Coadjoint Orbits
- Author
-
Alex Castro, Darryl D. Holm, and Alexis Arnaudon
- Subjects
Technology ,EULER-POINCARE EQUATIONS ,Mathematics, Applied ,Dynamical Systems (math.DS) ,SELECTIVE DECAY ,37H10 ,01 natural sciences ,010305 fluids & plasmas ,0102 Applied Mathematics ,Lie algebra ,Attractor ,Mathematics - Dynamical Systems ,Mathematical Physics ,Physics ,Semidirect product ,Applied Mathematics ,Random attractors ,Mathematical analysis ,General Engineering ,Mathematical Physics (math-ph) ,RIGID-BODY ,Physics, Mathematical ,Modeling and Simulation ,Physical Sciences ,symbols ,DYNAMICAL-SYSTEMS ,math.DS ,Dynamical systems theory ,Integrable system ,Fluids & Plasmas ,math-ph ,FOS: Physical sciences ,Lyapunov exponent ,Mechanics ,Article ,Poisson bracket ,symbols.namesake ,math.MP ,FLUIDS ,POISSON BRACKETS ,0103 physical sciences ,FOS: Mathematics ,Stochastic geometric mechanics ,0101 mathematics ,ATTRACTORS ,Science & Technology ,STABILITY ,nlin.CD ,010102 general mathematics ,Lyapunov exponents ,Nonlinear Sciences - Chaotic Dynamics ,Symmetry (physics) ,Euler-Poincaré theory ,Nonlinear Sciences::Chaotic Dynamics ,Euler-Poincare theory ,37J15 ,60H10 ,Chaotic Dynamics (nlin.CD) ,Coadjoint orbits ,Invariant measures ,Mathematics - Abstract
We derive and study stochastic dissipative dynamics on coadjoint orbits by incorporating noise and dissipation into mechanical systems arising from the theory of reduction by symmetry, including a semidirect product extension. Random attractors are found for this general class of systems when the Lie algebra is semi-simple, provided the top Lyapunov exponent is positive. We study in details two canonical examples, the free rigid body and the heavy top, whose stochastic integrable reductions are found and numerical simulations of their random attractors are shown.
- Published
- 2017
33. Euler–Poincaré Reduction on Principal Bundles.
- Author
-
Castrillón López, M., García Pérez, P., and Ratiu, T.
- Abstract
Let π: P → M be an arbitrary principal G-bundle. We give a full proof of the Euler–Poincaré reduction for a G-invariant Lagrangian L: J
1 P → R as well as the study of the second variation formula, the conservations laws, and study some of their properties. [ABSTRACT FROM AUTHOR]- Published
- 2001
- Full Text
- View/download PDF
34. G-Strands and Peakon Collisions on Diff(R)
- Author
-
Darryl D. Holm and Rossen I. Ivanov
- Subjects
Hamilton's principle ,continuum spin chains ,Euler-Poincaré equations ,Sobolev norms ,singular momentum maps ,diffeomorphisms ,harmonic maps ,Mathematics ,QA1-939 - Abstract
A G-strand is a map g: R×R→G for a Lie group G that follows from Hamilton's principle for a certain class of G-invariant Lagrangians. Some G-strands on finite-dimensional groups satisfy 1+1 space-time evolutionary equations that admit soliton solutions as completely integrable Hamiltonian systems. For example, the SO(3)-strand equations may be regarded physically as integrable dynamics for solitons on a continuous spin chain. Previous work has shown that G-strands for diffeomorphisms on the real line possess solutions with singular support (e.g. peakons). This paper studies collisions of such singular solutions of G-strands when G=Diff(R) is the group of diffeomorphisms of the real line R, for which the group product is composition of smooth invertible functions. In the case of peakon-antipeakon collisions, the solution reduces to solving either Laplace's equation or the wave equation (depending on a sign in the Lagrangian) and is written in terms of their solutions. We also consider the complexified systems of G-strand equations for G=Diff(R) corresponding to a harmonic map g: C→Diff(R) and find explicit expressions for its peakon-antipeakon solutions, as well.
- Published
- 2013
- Full Text
- View/download PDF
35. Intrinsic dynamics and total energy-shaping control of the ballbot system
- Author
-
Satici, Aykut Cihan, Donaire, Alejandro, Siciliano, Bruno, Satici, Aykut Cihan, Donaire, Alejandro, and Siciliano, Bruno
- Abstract
Research on bipedal locomotion has shown that a dynamic walking gait is energetically more efficient than a statically stable one. Analogously, even though statically stable multi-wheeled robots are easier to control, they are energetically less efficient and have low accelerations to avoid tipping over. In contrast, the ballbot is an underactuated, nonholonomically constrained mobile robot, whose upward equilibrium point has to be stabilised by active control. In this work, we derive coordinate-invariant, reduced, Euler–Poincaré equations of motion for the ballbot. By means of partial feedback linearisation, we obtain two independent passive outputs with corresponding storage functions and utilise these to come up with energy-shaping control laws which move the system along the trajectories of a new Lagrangian system whose desired equilibrium point is asymptotically stable by construction. The basin of attraction of this controller is shown to be almost global under certain conditions on the design of the mechanism which are reflected directly in the mass matrix of the unforced equations of motion. © 2016 Informa UK Limited, trading as Taylor & Francis Group
- Published
- 2017
36. Lagrangian Dynamics on Matched Pairs
- Author
-
Oğul Esen, Serkan Sütlü, Işık Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Işık University, Faculty of Arts and Sciences, Department of Mathematics, and Sütlü, Serkan Selçuk
- Subjects
Tangent bundle ,Mathematics - Differential Geometry ,Pure mathematics ,70H03, 76M60, 37J15 ,Lie-groups ,General Physics and Astronomy ,Matched pair of Lie groups ,Mechanics ,01 natural sciences ,0103 physical sciences ,Lie algebra ,0101 mathematics ,Lagrangian ,Mathematical Physics ,Mathematics ,Reduction ,010308 nuclear & particles physics ,Group (mathematics) ,Semidirect products ,010102 general mathematics ,Systems ,Lie group ,Tangent ,Euler–Poincaré equations ,Hopf algebra ,Poisson ,Optimal control ,Lagrangian function ,Product (mathematics) ,Cover (algebra) ,Hopf-algebras ,Geometry and Topology ,Matched pair of Lie algebras ,Lagrangian reduction ,Trivialized Euler–Lagrange equations - Abstract
Given a matched pair of Lie groups, we show that the tangent bundle of the matched pair group is isomorphic to the matched pair of the tangent groups. We thus obtain the Euler–Lagrange equations on the trivialized matched pair of tangent groups, as well as the Euler–Poincare equations on the matched pair of Lie algebras. We show explicitly how these equations cover those of the semi-direct product theory. In particular, we study the trivialized, and the reduced Lagrangian dynamics on the group S L ( 2 , C ) .
- Published
- 2015
37. Euler–Poincaré reduction in principal fibre bundles and the problem of Lagrange
- Author
-
César Rodrigo, Marco Castrillón, and Pedro L. García
- Subjects
Problem of Lagrange ,Lagrange multipliers ,Holonomic ,Mathematical analysis ,Holonomy ,Variational problems ,Euler–Poincaré equations ,Curvature ,Connection (mathematics) ,symbols.namesake ,Constraint algorithm ,Computational Theory and Mathematics ,Lagrange multiplier ,Euler's formula ,symbols ,Fiber bundle ,Geometry and Topology ,Analysis ,Mathematics - Abstract
We compare Euler–Poincare reduction in principal fibre bundles, as a constrained variational problem on the connections of this fibre bundle and constraint defined by the vanishing of the curvature of the connection, with the corresponding problem of Lagrange. Under certain cohomological condition we prove the equality of the sets of critical sections of both problems with the one obtained by application of the Lagrange multiplier rule. We compute the corresponding Cartan form and characterise critical sections as the set of holonomic solutions of the Cartan equation and, in particular, under a certain regularity condition for the problem, we prove the holonomy of any solution of this equation.
- Published
- 2007
- Full Text
- View/download PDF
38. Model Formulation Over Lie Groups and Numerical Methods to Simulate the Motion of Gyrostats and Quadrotors.
- Author
-
Fiori, Simone
- Subjects
- *
LIE groups , *GYROSCOPES , *EULER method , *RUNGE-Kutta formulas , *NUMERICAL solutions to equations , *EULER equations (Rigid dynamics) , *DYNAMICAL systems , *EULER equations - Abstract
The present paper recalls a formulation of non-conservative system dynamics through the Lagrange–d'Alembert principle expressed through a generalized Euler–Poincaré form of the system equation on a Lie group. The paper illustrates applications of the generalized Euler–Poincaré equations on the rotation groups to a gyrostat satellite and a quadcopter drone. The numerical solution of the dynamical equations on the rotation groups is tackled via a generalized forward Euler method and an explicit Runge–Kutta integration method tailored to Lie groups. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
39. Covariant and dynamical reduction for principal bundle field theories
- Author
-
Castrillón López, Marco and Marsden, Jerrold E.
- Published
- 2008
- Full Text
- View/download PDF
40. Exact geometric theory of dendronized polymer dynamics
- Author
-
Darryl D. Holm, Tudor S. Ratiu, François Gay-Balmaz, Vakhtang Putkaradze, Laboratoire de Météorologie Dynamique (UMR 8539) (LMD), Département des Géosciences - ENS Paris, École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École des Ponts ParisTech (ENPC)-École polytechnique (X)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC), Department of Mathematics [Imperial College London], Imperial College London, Department of Mathematics [Fort Collins], College of Natural Sciences [Fort Collins], Colorado State University [Fort Collins] (CSU)-Colorado State University [Fort Collins] (CSU), Section de Mathématiques and Bernoulli Center, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland, Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-École polytechnique (X)-École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)-Département des Géosciences - ENS Paris, École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École normale supérieure - Paris (ENS-PSL), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)
- Subjects
Tangent bundle ,Poisson bracket ,Dendrimers ,Nonlocal potential ,Momentum map ,[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] ,FOS: Physical sciences ,Euler–Lagrange equations ,Euler-Poincare equations ,02 engineering and technology ,010402 general chemistry ,01 natural sciences ,Symmetry reduction ,Variational principle ,Cocycle ,Mathematical Physics ,Mathematics ,Semidirect product ,Quantitative Biology::Biomolecules ,Applied Mathematics ,Mathematical analysis ,Modeling ,Equations of motion ,Euler–Poincaré equations ,Mathematical Physics (math-ph) ,021001 nanoscience & nanotechnology ,Dendronized polymer ,Nonlinear Sciences - Chaotic Dynamics ,0104 chemical sciences ,Condensed Matter::Soft Condensed Matter ,[CHIM.POLY]Chemical Sciences/Polymers ,Geometric group theory ,Iterated function ,Euler-Lagrange equations ,Polymer dynamics ,Chaotic Dynamics (nlin.CD) ,0210 nano-technology ,Rotation (mathematics) - Abstract
Dendronized polymers consist of an elastic backbone with a set of iterated branch structures (dendrimers)attached at every base point of the backbone. The conformations of such molecules depend on the elastic deformation of the backbone and the branches, as well as on nonlocal (e.g., electrostatic, or Lennard-Jones) interactions between the elementary molecular units comprising the dendrimers and/or backbone. We develop a geometrically exact theory for the dynamics of such polymers, taking into account both local (elastic) and nonlocal interactions. The theory is based on applying symmetry reduction of Hamilton's principle for a Lagrangian defined on the tangent bundle of iterated semidirect products of the rotation groups that represent the relative orientations of the dendritic branches of the polymer. The resulting symmetry-reduced equations of motion are written in conservative form., Comment: 33 pages, 2 figures, first version, please send comments
- Published
- 2012
41. Singular Solutions of Coss-coupled EPDiff Equations: Waltzing Peakons and Compacton Pairs
- Author
-
Cotter, Colin, Holm, Darryl, Ivanov, Rossen, Percival, James, SFI, and Basic Research Grant
- Subjects
Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Non-linear Dynamics ,Dynamic Systems ,Ordinary Differential Equations and Applied Dynamics ,Partial Differential Equations ,Numerical Analysis and Computation ,Peakons ,Camassa-Holm equation ,Euler-Poincare Equations ,Nonlinear Sciences::Pattern Formation and Solitons - Abstract
We introduce EPDiff equations as Euler-Poincare´ equations related to Lagrangian provided by a metric, invariant under the Lie Group Diff(Rn). Then we proceed with a particular form of EPDiff equations, a cross coupled two-component system of Camassa-Holm type. The system has a new type of peakon solutions, 'waltzing' peakons and compacton pairs.
- Published
- 2012
- Full Text
- View/download PDF
42. Symmetry Reduced Dynamics of Charged Molecular Strands
- Author
-
Vakhtang Putkaradze, David C. P. Ellis, Darryl D. Holm, Tudor S. Ratiu, and François Gay-Balmaz
- Subjects
Physics ,Hamiltonian mechanics ,Mechanical Engineering ,Equations of motion ,Symmetry group ,Elastic Rods ,Poisson bracket ,symbols.namesake ,Molecular dynamics ,Mathematics (miscellaneous) ,Classical mechanics ,Phase space ,symbols ,Semidirect Products ,Calculus of variations ,Euler-Poincare Equations ,Hamiltonian (quantum mechanics) ,Analysis ,Reduction ,Mathematical physics - Abstract
The equations of motion are derived for the dynamical folding of charged molecular strands (such as DNA) modeled as flexible continuous filamentary distributions of interacting rigid charge conformations. The new feature is that these equations are nonlocal when the screened Coulomb interactions, or Lennard-Jones potentials between pairs of charges, are included. The nonlocal dynamics is derived in the convective representation of continuum motion by using modified Euler-Poincar, and Hamilton-Pontryagin variational formulations that illuminate the various approaches within the framework of symmetry reduction of Hamilton's principle for exact geometric rods. In the absence of nonlocal interactions, the equations recover the classical Kirchhoff theory of elastic rods. The motion equations in the convective representation are shown to arise by a classical Lagrangian reduction associated to the symmetry group of the system. This approach uses the process of affine Euler-Poincar, reduction initially developed for complex fluids. On the Hamiltonian side, the Poisson bracket of the molecular strand is obtained by reduction of the canonical symplectic structure on phase space. A change of variables allows a direct passage from this classical point of view to the covariant formulation in terms of Lagrange-Poincar, equations of field theory. In another revealing perspective, the convective representation of the nonlocal equations of molecular strand motion is transformed into quaternionic form.
- Published
- 2010
43. Geometrija mnogostrukosti u lievom endomorfskom prostoru i njihovi parovi u frakcijskom aktivnom varijacijskom pristupu
- Author
-
El-Nabulsi, Rami Ahmad
- Subjects
fractional action-like variational approach ,symmetry ,constants of motion ,Euler-Poincaré equations ,Lie Algebra and their duals ,Kelvin-Noether theorem - Abstract
Some interesting fractional features of the geometry of manifolds on Lie endomorphism space and their duals are discussed within the framework of fractional action-like variational approach (fractionally differentiated Lagrangian function) formulated recently by the author., Raspravljaju se neke zanimljive odlike geometrije mnogostrukosti u Lievom endomorfskom prostoru u frakcijskom aktivnom varijacijskom pristupu (frakcijski diferenciranoj Lagrangeovoj funkciji) kako je to autor nedavno objavio.
- Published
- 2007
44. Inexact trajectory planning and inverse problems in the Hamilton-Pontryagin framework.
- Author
-
Burnett CL, Holm DD, and Meier DM
- Abstract
We study a trajectory-planning problem whose solution path evolves by means of a Lie group action and passes near a designated set of target positions at particular times. This is a higher-order variational problem in optimal control, motivated by potential applications in computational anatomy and quantum control. Reduction by symmetry in such problems naturally summons methods from Lie group theory and Riemannian geometry. A geometrically illuminating form of the Euler-Lagrange equations is obtained from a higher-order Hamilton-Pontryagin variational formulation. In this context, the previously known node equations are recovered with a new interpretation as Legendre-Ostrogradsky momenta possessing certain conservation properties. Three example applications are discussed as well as a numerical integration scheme that follows naturally from the Hamilton-Pontryagin principle and preserves the geometric properties of the continuous-time solution.
- Published
- 2013
- Full Text
- View/download PDF
45. The geometric structure of complex fluids
- Author
-
Tudor S. Ratiu and François Gay-Balmaz
- Subjects
Hall magnetohydrodynamics ,Diffeomorphism group ,Microfluids ,FOS: Physical sciences ,Affine Lie–Poisson equations ,Yang-Mills magnetohydrodynamics ,Poisson bracket ,symbols.namesake ,Yang–Mills magnetohydrodynamics ,Spin glasses ,Semidirect Products ,Superfluid dynamics ,Affine Euler–Poincaré equations ,Euler-Poincare Equations ,Affine Euler-Poincare equations ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Mathematical physics ,Mathematics ,Complex fluid ,Affine Lie-Poisson equations ,Complex fluids ,Applied Mathematics ,Liquid crystals ,Mathematical Physics (math-ph) ,Nonlinear Sciences - Chaotic Dynamics ,Euler equations ,Poisson brackets ,symbols ,Cotangent bundle ,Affine transformation ,Yang-Mills Fluids ,Magnetohydrodynamics ,Poisson's equation ,Chaotic Dynamics (nlin.CD) ,Hamiltonian (quantum mechanics) - Abstract
This paper develops the theory of affine Euler-Poincare and affine Lie-Poisson reductions and applies these processes to various examples of complex fluids, including Yang-Mills and Hall magnetohydrodynamics for fluids and superfluids, spin glasses, microfluids, and liquid crystals. As a consequence of the Lagrangian approach, the variational formulation of the equations is determined. On the Hamiltonian side, the associated Poisson brackets are obtained by reduction of a canonical cotangent bundle. A Kelvin-Noether circulation theorem is presented and is applied to these examples. (C) 2008 Elsevier Inc. All rights reserved.
- Full Text
- View/download PDF
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.