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Journal articles on the topic 'Hamiltonian and Lagrangian dynamics'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Cheng, Xu-Hui, and Guo-Qing Huang. "A Comparison between Second-Order Post-Newtonian Hamiltonian and Coherent Post-Newtonian Lagrangian in Spinning Compact Binaries." Symmetry 13, no. 4 (April 1, 2021): 584. http://dx.doi.org/10.3390/sym13040584.

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In relativistic celestial mechanics, post-Newtonian (PN) Lagrangian and PN Hamiltonian formulations are not equivalent to the same PN order as our previous work in PRD (2015). Usually, an approximate Lagrangian is used to discuss the difference between a PN Hamiltonian and a PN Lagrangian. In this paper, we investigate the dynamics of compact binary systems for Hamiltonians and Lagrangians, including Newtonian, post-Newtonian (1PN and 2PN), and spin–orbit coupling and spin–spin coupling parts. Additionally, coherent equations of motion for 2PN Lagrangian are adopted here to make the comparison with Hamiltonian approaches and approximate Lagrangian approaches at the same condition and same PN order. The completely opposite nature of the dynamics shows that using an approximate PN Lagrangian is not convincing. Hence, using the coherent PN Lagrangian is necessary for obtaining an exact result in the research of dynamics of compact binary at certain PN order. Meanwhile, numerical investigations from the spinning compact binaries show that the 2PN term plays an important role in causing chaos in the PN Hamiltonian system.

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2

Rosas-Ortiz, Oscar. "Lagrangian and Hamiltonian dynamics." Contemporary Physics 60, no. 1 (January 2, 2019): 85–86. http://dx.doi.org/10.1080/00107514.2019.1580314.

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3

MALIK, R. P. "HAMILTONIAN AND LAGRANGIAN DYNAMICS IN A NONCOMMUTATIVE SPACE." Modern Physics Letters A 18, no. 39 (December 21, 2003): 2795–806. http://dx.doi.org/10.1142/s0217732303012350.

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We discuss the dynamics of a particular two-dimensional (2D) physical system in the four-dimensional (4D) (non-)commutative phase space by exploiting the consistent Hamiltonian and Lagrangian formalisms based on the symplectic structures defined on the 4D (non-)commutative cotangent manifolds. The noncommutativity exists equivalently in the coordinate or the momentum planes embedded in the 4D cotangent manifolds. The signature of this noncommutativity is reflected in the derivation of the first-order Lagrangians where we exploit the most general form of the Legendre transformation defined on the (non-)commutative (co-)tangent manifolds. The second-order Lagrangian, defined on the 4D tangent manifold, turns out to be the same irrespective of the noncommutativity present in the 4D cotangent manifolds for the discussion of the Hamiltonian formulation. A connection with the noncommutativity of the dynamics, associated with the quantum groups on the q-deformed 4D cotangent manifolds, is also pointed out.

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4

Wheeler, James T. "Not-so-classical mechanics: unexpected symmetries of classical motion." Canadian Journal of Physics 83, no. 2 (February 1, 2005): 91–138. http://dx.doi.org/10.1139/p05-003.

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A survey of topics of recent interest in Hamiltonian and Lagrangian dynamical systems, including accessible discussions of regularization of the central-force problem; inequivalent Lagrangians and Hamiltonians; constants of central-force motion; a general discussion of higher order Lagrangians and Hamiltonians, with examples from Bohmian quantum mechanics, the Kortewegde Vries equation, and the logistic equation; gauge theories of Newtonian mechanics; and classical spin, Grassmann numbers, and pseudomechanics. PACS No.: 45.25.Jj

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5

KIJOWSKI, J., G. MAGLI, and D. MALAFARINA. "LAGRANGIAN AND HAMILTONIAN FORMULATION OF SPHERICAL SHELL DYNAMICS." International Journal of Geometric Methods in Modern Physics 02, no. 05 (October 2005): 887–94. http://dx.doi.org/10.1142/s021988780500082x.

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Lagrangian and Hamiltonian descriptions of the dynamics of a self-gravitating matter shell in General Relativity are discussed in general. The case of a spherical shell composed of an elastic fluid is then considered, its Lagrangian function is derived from first principles and the Hamiltonian is calculated. Known results for dust shells are recovered as particular cases.

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6

Entov, Michael, and Leonid Polterovich. "Lagrangian tetragons and instabilities in Hamiltonian dynamics." Nonlinearity 30, no. 1 (November 17, 2016): 13–34. http://dx.doi.org/10.1088/0951-7715/30/1/13.

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7

Bernard, Patrick. "The Lax–Oleinik semi-group: a Hamiltonian point of view." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 142, no. 6 (November 27, 2012): 1131–77. http://dx.doi.org/10.1017/s0308210511000059.

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The weak KAM theory was developed by Fathi in order to study the dynamics of convex Hamiltonian systems. It somehow makes a bridge between viscosity solutions of the Hamilton–Jacobi equation and Mather invariant sets of Hamiltonian systems, although this was fully understood only a posteriori. These theories converge under the hypothesis of convexity, and the richness of applications mostly comes from this remarkable convergence. In this paper, we provide an elementary exposition of some of the basic concepts of weak KAM theory. In a companion paper, Albert Fathi exposed the aspects of his theory which are more directly related to viscosity solutions. Here, on the contrary, we focus on dynamical applications, even if we also discuss some viscosity aspects to underline the connections with Fathi's lecture. The fundamental reference on weak KAM theory is the still unpublished book Weak KAM theorem in Lagrangian dynamics by Albert Fathi. Although we do not offer new results, our exposition is original in several aspects. We only work with the Hamiltonian and do not rely on the Lagrangian, even if some proofs are directly inspired by the classical Lagrangian proofs. This approach is made easier by the choice of a somewhat specific setting. We work on ℝd and make uniform hypotheses on the Hamiltonian. This allows us to replace some compactness arguments by explicit estimates. For the most interesting dynamical applications, however, the compactness of the configuration space remains a useful hypothesis and we retrieve it by considering periodic (in space) Hamiltonians. Our exposition is centred on the Cauchy problem for the Hamilton–Jacobi equation and the Lax–Oleinik evolution operators associated to it. Dynamical applications are reached by considering fixed points of these evolution operators, the weak KAM solutions. The evolution operators can also be used for their regularizing properties; this opens an alternative route to dynamical applications.

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8

Bokhove, Onno, and Marcel Oliver. "Parcel Eulerian–Lagrangian fluid dynamics of rotating geophysical flows." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462, no. 2073 (March 30, 2006): 2575–92. http://dx.doi.org/10.1098/rspa.2006.1656.

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Parcel Eulerian–Lagrangian Hamiltonian formulations have recently been used in structure-preserving numerical schemes, asymptotic calculations and in alternative explanations of fluid parcel (in)stabilities. A parcel formulation describes the dynamics of one fluid parcel with a Lagrangian kinetic energy but an Eulerian potential evaluated at the parcel's position. In this paper, we derive the geometric link between the parcel Eulerian–Lagrangian formulation and well-known variational and Hamiltonian formulations for three models of ideal and geophysical fluid flow: generalized two-dimensional vorticity–stream function dynamics, the rotating two-dimensional shallow-water equations and the rotating three-dimensional compressible Euler equations.

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9

Zivieri, Roberto, and Giancarlo Consolo. "Hamiltonian and Lagrangian Dynamical Matrix Approaches Applied to Magnetic Nanostructures." Advances in Condensed Matter Physics 2012 (2012): 1–16. http://dx.doi.org/10.1155/2012/765709.

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Two micromagnetic tools to study the spin dynamics are reviewed. Both approaches are based upon the so-called dynamical matrix method, a hybrid micromagnetic framework used to investigate the spin-wave normal modes of confined magnetic systems. The approach which was formulated first is the Hamiltonian-based dynamical matrix method. This method, used to investigate dynamic magnetic properties of conservative systems, was originally developed for studying spin excitations in isolated magnetic nanoparticles and it has been recently generalized to study the dynamics of periodic magnetic nanoparticles. The other one, the Lagrangian-based dynamical matrix method, was formulated as an extension of the previous one in order to include also dissipative effects. Such dissipative phenomena are associated not only to intrinsic but also to extrinsic damping caused by injection of a spin current in the form of spin-transfer torque. This method is very accurate in identifying spin modes that become unstable under the action of a spin current. The analytical development of the system of the linearized equations of motion leads to a complex generalized Hermitian eigenvalue problem in the Hamiltonian dynamical matrix method and to a non-Hermitian one in the Lagrangian approach. In both cases, such systems have to be solved numerically.

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10

MUKHANOV, V., and A. WIPF. "ON THE SYMMETRIES OF HAMILTONIAN SYSTEMS." International Journal of Modern Physics A 10, no. 04 (February 10, 1995): 579–610. http://dx.doi.org/10.1142/s0217751x95000267.

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In this paper we show how the well-known local symmetries of Lagrangian systems, and in particular the diffeomorphism invariance, emerge in the Hamiltonian formulation. We show that only the constraints which are linear in the momenta generate transformations which correspond to symmetries of the corresponding Lagrangian system. The non-linear constraints (which we have, for instance, in gravity, supergravity and string theory) generate the dynamics of the corresponding Lagrangian system. Only in a very special combination with "trivial" transformations proportional to the equations of motion do they lead to symmetry transformations. We show the importance of these special "trivial" transformations for the interconnection theorems which relate the symmetries of a system with its dynamics. We prove these theorems for general Hamiltonian systems. We apply the developed formalism to concrete physically relevant systems, in particular those which are diffeomorphism-invariant. The connection between the parameters of the symmetry transformations in the Hamiltonian and Lagrangian formalisms is found. The possible applications of our results are discussed.

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11

Ilinskii, Yurii A., Mark F. Hamilton, and Evgenia A. Zabolotskaya. "Bubble interaction dynamics in Lagrangian and Hamiltonian mechanics." Journal of the Acoustical Society of America 121, no. 2 (February 2007): 786–95. http://dx.doi.org/10.1121/1.2404798.

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12

Lalonde, Fran�ois, and Daniel Gatien. "Holomorphic cylinders with Lagrangian boundaries and Hamiltonian dynamics." Duke Mathematical Journal 102, no. 3 (May 2000): 485–511. http://dx.doi.org/10.1215/s0012-7094-00-10236-0.

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13

Wheeler, James T. "Gauging Newton's law." Canadian Journal of Physics 85, no. 4 (April 1, 2007): 307–44. http://dx.doi.org/10.1139/p07-052.

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We derive both Lagrangian and Hamiltonian mechanics as gauge theories of Newtonian mechanics. Systematic development of the distinct symmetries of dynamics and measurement suggest that gauge theory may be motivated as a reconciliation of dynamics with measurement. Applying this principle to Newton's law with the simplest measurement theory leads to Lagrangian mechanics, while use of conformal measurement theory leads to Hamiltonian mechanics. PACS Nos.: 45.20.Jj, 11.25.Hf, 45.10.–b

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14

BIZDADEA, CONSTANTIN, MARIA-MAGDALENA BÂRCAN, MIHAELA TINCA MIAUTĂ, and SOLANGE-ODILE SALIU. "SECOND-ORDER LAGRANGIAN DYNAMICS IN THE PHASE-SPACE: SOME EXAMPLES." Modern Physics Letters A 27, no. 10 (March 28, 2012): 1250062. http://dx.doi.org/10.1142/s0217732312500629.

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By means of a class of nondegenerate models with a finite number of degrees of freedom, it is proved that given a Hamiltonian formulation of dynamics, there exists an equivalent second-order Lagrangian formulation whose configuration space coincides with the Hamiltonian phase-space. The above result is extended to scalar field theories in a Lorentz-covariant manner.

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15

Hajihashemi, Mahdi, and Ahmad Shirzad. "A generalized model for the classical relativistic spinning particle." International Journal of Modern Physics A 31, no. 07 (March 2, 2016): 1650027. http://dx.doi.org/10.1142/s0217751x16500275.

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Following the Poincaré algebra, in the Hamiltonian approach, for a free spinning particle and using the Casimirs of the algebra, we construct systematically a set of Lagrangians for the relativistic spinning particle which includes the Lagrangian given in the literature. We analyze the dynamics of this generalized system in the Lagrangian formulation and show that the equations of motion support an oscillatory solution corresponding to the spinning nature of the system. Then we analyze the canonical structure of the system and present the correct gauge suitable for the spinning motion of the system.

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16

Kaparulin, Dmitry S., Simon L. Lyakhovich, and Oleg D. Nosyrev. "Extended Chern–Simons Model for a Vector Multiplet." Symmetry 13, no. 6 (June 3, 2021): 1004. http://dx.doi.org/10.3390/sym13061004.

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We consider a gauge theory of vector fields in 3D Minkowski space. At the free level, the dynamical variables are subjected to the extended Chern–Simons (ECS) equations with higher derivatives. If the color index takes n values, the third-order model admits a 2n-parameter series of second-rank conserved tensors, which includes the canonical energy–momentum. Even though the canonical energy is unbounded, the other representatives in the series have a bounded from below the 00-component. The theory admits consistent self-interactions with the Yang–Mills gauge symmetry. The Lagrangian couplings preserve the energy–momentum tensor that is unbounded from below, and they do not lead to a stable non-linear theory. The non-Lagrangian couplings are consistent with the existence of a conserved tensor with a 00-component bounded from below. These models are stable at the non-linear level. The dynamics of interacting theory admit a constraint Hamiltonian form. The Hamiltonian density is given by the 00-component of the conserved tensor. In the case of stable interactions, the Poisson bracket and Hamiltonian do not follow from the canonical Ostrogradski construction. Particular attention is paid to the “triply massless” ECS theory, which demonstrates instability even at the free level. It is shown that the introduction of extra scalar field, serving as Higgs, can stabilize the dynamics in the vicinity of the local minimum of energy. The equations of motion of the stable model are non-Lagrangian, but they admit the Hamiltonian form of dynamics with a Hamiltonian that is bounded from below.

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17

SENGUPTA, AMBAR N. "SYMPLECTIC REDUCTION FOR YANG–MILLS ON A CYLINDER." International Journal of Geometric Methods in Modern Physics 01, no. 04 (August 2004): 289–98. http://dx.doi.org/10.1142/s0219887804000204.

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An account of the Lagrangian and Hamiltonian dynamics of the pure Yang–Mills system is presented. This framework is applied to the case of (1+1)-dimensional cylindrical spacetime. Hamiltonian dynamics on the space of connections over a circle is often identified with dynamics on the cotangent bundle of the gauge group by means of the holonomy. In support of this procedure we show that the symplectic structure for Hamiltonian dynamics for connections on a circle is identifiable with the natural symplectic structure on the cotangent bundle of the gauge group.

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18

Han, Yu, Yongge Ma, and Xiangdong Zhang. "Connection dynamics of higher-dimensional scalar–tensor theories of gravity." Modern Physics Letters A 29, no. 28 (September 14, 2014): 1450134. http://dx.doi.org/10.1142/s021773231450134x.

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The scalar–tensor theories (STTs) of gravity in spacetime dimensions (D+1)>2 are studied. By performing Hamiltonian analysis, we obtain the geometrical dynamics of the theories from their Lagrangian. The Hamiltonian formalism indicates that the theories are naturally divided into two sectors by the coupling parameter ω. The Hamiltonian structures in both sectors are similar to the corresponding structures of four-dimensional cases. It turns out that, similar to the case of general relativity (GR), there is also a symplectic reduction from the canonical structure of so (D+1) Yang–Mills theories coupled to the scalar field to the canonical structure of the geometrical STTs. Therefore, the non-perturbative loop quantum (LQG) gravity techniques can also be applied to the STTs in D+1 dimensions based on their connection-dynamical formalism.

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19

GRABOWSKA, KATARZYNA, PAWEŁ URBAŃSKI, and JANUSZ GRABOWSKI. "GEOMETRICAL MECHANICS ON ALGEBROIDS." International Journal of Geometric Methods in Modern Physics 03, no. 03 (May 2006): 559–75. http://dx.doi.org/10.1142/s0219887806001259.

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A natural geometric framework is proposed, based on ideas of W. M. Tulczyjew, for constructions of dynamics on general algebroids. One obtains formalisms similar to the Lagrangian and the Hamiltonian ones. In contrast with recently studied concepts of Analytical Mechanics on Lie algebroids, this approach requires much less than the presence of a Lie algebroid structure on a vector bundle, but it still reproduces the main features of the Analytical Mechanics, like the Euler–Lagrange-type equations, the correspondence between the Lagrangian and Hamiltonian functions (Legendre transform) in the hyperregular cases, and a version of the Noether Theorem.

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20

Bolivar, A. O. "Dynamical quantization and classical limit." Canadian Journal of Physics 81, no. 4 (April 1, 2003): 663–73. http://dx.doi.org/10.1139/p02-121.

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We have worked out a quantization method directly from classical dynamics without using Hamiltonian and Lagrangian functions; we call it dynamical quantization. The present article compares such a method with the Dirac and Feynman quantization procedures and also verifies the logical consistence of the dynamical quantization calculating the classical limit of a Brownian particle, for example. PACS Nos.: 03.65.w, 05.30.d, 05.40.+j, 52.65.Ff

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21

IVANCEVIC, VLADIMIR G., and TIJANA T. IVANCEVIC. "HUMAN VERSUS HUMANOID ROBOT BIODYNAMICS." International Journal of Humanoid Robotics 05, no. 04 (December 2008): 699–713. http://dx.doi.org/10.1142/s0219843608001595.

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In this paper we compare and contrast modern dynamical methodologies common to both humanoid robotics and human biomechanics. While the humanoid robot's motion is defined on the system of constrained rotational Lie groups SO(3) acting in all major robot joints, human motion is defined on the corresponding system of constrained Euclidean groups SE(3) of the full (rotational + translational) rigid motions acting in all synovial human joints. In both cases the smooth configuration manifolds, Q rob and Q hum , respectively, can be constructed. The autonomous Lagrangian dynamics are developed on the corresponding tangent bundles, TQ rob and TQ hum , respectively, which are themselves smooth Riemannian manifolds. Similarly, the autonomous Hamiltonian dynamics are developed on the corresponding cotangent bundles, T*Q rob and T*Q hum , respectively, which are themselves smooth symplectic manifolds. In this way a full rotational + translational biodynamics simulator has been created with 270 DOFs in total, called the Human Biodynamics Engine, which is currently in its validation stage. Finally, in both the human and the humanoid case, the time-dependent biodynamics generalizing the autonomous Lagrangian (of Hamiltonian) dynamics is naturally formulated in terms of jet manifolds.

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22

Koon, Wang Sang, and Jerrold E. Marsden. "The Hamiltonian and Lagrangian approaches to the dynamics of nonholonomic systems." Reports on Mathematical Physics 40, no. 1 (August 1997): 21–62. http://dx.doi.org/10.1016/s0034-4877(97)85617-0.

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23

Grabowski, Janusz, Katarzyna Grabowska, and Paweł Urbański. "Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings." Journal of Geometric Mechanics 6, no. 4 (2014): 503–26. http://dx.doi.org/10.3934/jgm.2014.6.503.

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24

MALIK, R. P., A. K. MISHRA, and G. RAJASEKARAN. "DYNAMICS IN A NONCOMMUTATIVE PHASE SPACE." International Journal of Modern Physics A 13, no. 27 (October 30, 1998): 4759–75. http://dx.doi.org/10.1142/s0217751x98002249.

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Dynamics in a noncommutative phase space is considered. The noncommuting phase space is taken to be invariant under the quantum group GLq,p(2). The q-deformed differential calculus on the phase space is formulated and using this, both the Hamiltonian and Lagrangian forms of dynamics have been constructed. In contrast to earlier forms of q-dynamics, our formalism has the advantage of preserving the conventional symmetries such as rotational or Lorentz invariance.

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25

GÜREL, BAŞAK Z. "TOTALLY NON-COISOTROPIC DISPLACEMENT AND ITS APPLICATIONS TO HAMILTONIAN DYNAMICS." Communications in Contemporary Mathematics 10, no. 06 (December 2008): 1103–28. http://dx.doi.org/10.1142/s0219199708003198.

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In this paper, we prove the Conley conjecture and the almost existence theorem in a neighborhood of a closed nowhere coisotropic submanifold under certain natural assumptions on the ambient symplectic manifold. Essential to the proofs is a displacement principle for such submanifolds. Namely, we show that a topologically displaceable nowhere coisotropic submanifold is also displaceable by a Hamiltonian diffeomorphism, partially extending the well-known non-Lagrangian displacement property.

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26

WEBB, G. M., M. BRIO, and G. P. ZANK. "Lagrangian and Hamiltonian aspects of wave mixing in non-uniform media: waves on strings and waves in gas dynamics." Journal of Plasma Physics 60, no. 2 (September 1998): 341–82. http://dx.doi.org/10.1017/s002237789800693x.

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Hamiltonian and Lagrangian perturbation theory is used to describe linear wave propagation in inhomogeneous media. In particular, the problems of wave propagation on an inhomogeneous string, and the propagation of sound waves and entropy waves in gas dynamics in one Cartesian space dimension are investigated. For the case of wave propagation on an inhomogeneous heavy string, coupled evolution equations are obtained describing the interaction of the backward and forward waves via wave reflection off gradients in the string density. Similarly, in the case of gas dynamics the backward and forward sound waves and the entropy wave interact with each other via gradients in the background flow. The wave coupling coefficients in the gas-dynamical case depend on the gradients of the Riemann invariants R± and entropy S of the background flow. Coupled evolution equations describing the interaction of the different wave modes are obtained by exploiting the Hamiltonian and Poisson-bracket structure of the governing equations. Both Lagrangian and Clebsch-variable formulations are used. The similarity of the equations to equations obtained by Heinemann and Olbert describing the propagation of bidirectional Alfvén waves in the solar wind is pointed out.

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27

ECHEVERRÍA ENRÍQUEZ, A., M. C. MUÑOZ LECANDA, and N. ROMÁN ROY. "GEOMETRICAL SETTING OF TIME-DEPENDENT REGULAR SYSTEMS: ALTERNATIVE MODELS." Reviews in Mathematical Physics 03, no. 03 (September 1991): 301–30. http://dx.doi.org/10.1142/s0129055x91000114.

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We analyse exhaustively the geometric formulations of the time-dependent regular dynamical systems, both the Hamiltonian and the Lagrangian formalisms. We study the equivalence between the different models and, in each case, between the Lagrangian and the Hamiltonian formulations, giving the suitable definitions of the Legendre transformation. In addition, we include the variational formalisms as well as the Klein formalism.

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28

Román-Roy, Narciso. "An Overview of the Hamilton–Jacobi Theory: the Classical and Geometrical Approaches and Some Extensions and Applications." Mathematics 9, no. 1 (January 3, 2021): 85. http://dx.doi.org/10.3390/math9010085.

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This work is devoted to review the modern geometric description of the Lagrangian and Hamiltonian formalisms of the Hamilton–Jacobi theory. The relation with the “classical” Hamiltonian approach using canonical transformations is also analyzed. Furthermore, a more general framework for the theory is also briefly explained. It is also shown how, from this generic framework, the Lagrangian and Hamiltonian cases of the theory for dynamical systems are recovered, as well as how the model can be extended to other types of physical systems, such as higher-order dynamical systems and (first-order) classical field theories in their multisymplectic formulation.

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Tar, J. K., I. J. Rudas, and J. F. Bitó. "Group theoretical approach in using canonical transformations and symplectic geometry in the control of approximately modelled mechanical systems interacting with an unmodelled environment." Robotica 15, no. 2 (March 1997): 163–79. http://dx.doi.org/10.1017/s0263574797000192.

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In spite of its simpler structure than that of the Euler-Lagrange equations-based model, the Hamiltonian formulation of Classical Mechanics (CM) gained only limited application in the Computed Torque Control (CTC) of the rather conventional robots. A possible reason for this situation may be, that while the independent variables of the Lagrangian model are directly measurable by common industrial sensors and encoders, the Hamiltonian canonical coordinates are not measurable and can also not be computed in the lack of detailed information on the dynamics of the system under control. As a consequence, transparent and lucid mathematical methods bound to the Hamiltonian model utilizing the special properties of such concepts as Canonical Transformations, Symplectic Geometry, Symplectic Group, Symplectizing Algorithm, etc. remain out of the reach of Dynamic Control approaches based on the Lagrangian model. In this paper the preliminary results of certain recent investigations aiming at the introduction of these methods in dynamic control are summarized and illustrated by simulation results. The proposed application of the Hamiltonian model makes it possible to achieve a rigorous deductive analytical treatment up to a well defined point exactly valid for a quite wide range of many different mechanical systems. From this point on it reveals such an ample assortment of possible non-deductive, intuitive developments and approaches even within the investigations aiming at a particular paradigm that publication of these very preliminary and early results seems to have definite reason, too.

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Patra, Pinaki, Md Raju, Gargi Manna, and Jyoti Prasad Saha. "Modified Hamiltonian Formalism for Regge-Teitelboim Cosmology." Physics Research International 2014 (December 28, 2014): 1–6. http://dx.doi.org/10.1155/2014/606727.

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The Ostrogradski approach for the Hamiltonian formalism of higher derivative theory is not satisfactory because the Lagrangian cannot be viewed as a function on the tangent bundle to coordinate manifold. In this paper, we have used an alternative approach which leads directly to the Lagrangian which, being a function on the tangent manifold, gives correct equation of motion; no new coordinate variables need to be added. This approach can be used directly to the singular (in Ostrogradski sense) Lagrangian. We have used this method for the Regge-Teitelboim (RT) minisuperspace cosmological model. We have obtained the Hamiltonian of the dynamical equation of the scale factor of RT model.

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Katsanikas, M., Víctor J. García-Garrido, and S. Wiggins. "Detection of Dynamical Matching in a Caldera Hamiltonian System Using Lagrangian Descriptors." International Journal of Bifurcation and Chaos 30, no. 09 (July 2020): 2030026. http://dx.doi.org/10.1142/s0218127420300268.

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The goal of this paper is to apply the method of Lagrangian descriptors to reveal the phase space mechanism by which a Caldera-type potential energy surface (PES) exhibits the dynamical matching phenomenon. Using this technique, we can easily establish that the nonexistence of dynamical matching is a consequence of heteroclinic connections between the unstable manifolds of the unstable periodic orbits (UPOs) of the upper index-1 saddles (entrance channels to the Caldera) and the stable manifolds of the family of UPOs of the central minimum of the Caldera, resulting in the temporary trapping of trajectories. Moreover, dynamical matching will occur when there is no heteroclinic connection, which allows trajectories to enter and exit the Caldera without interacting with the shallow region of the central minimum. Knowledge of this phase space mechanism is relevant because it allows us to effectively predict the existence, and nonexistence, of dynamical matching. In this work, we explore a stretched Caldera potential by means of Lagrangian descriptors, allowing us to accurately compute the critical value for the stretching parameter for which dynamical matching behavior occurs in the system. This approach is shown to provide a tremendous advantage for exploring this mechanism in comparison to other methods from nonlinear dynamics that use phase space dividing surfaces.

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NESTERENKO, V. V., and NGUYEN SUAN HAN. "THE HAMILTONIAN FORMALISM IN THE MODEL OF THE RELATIVISTIC STRING WITH RIGIDITY." International Journal of Modern Physics A 03, no. 10 (October 1988): 2315–29. http://dx.doi.org/10.1142/s0217751x88000977.

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The model of the bosonic string with the Nambu-Goto action extended by the term proportional to the external curvature of the world sheet is explored. The external curvature of the world sheet is determined by the second order derivatives of the string coordinates. Thus, a modified Lagrangian is the singular Lagrangian of second order. The canonical variables for this theory are introduced, constraints are found and the Hamiltonian formulation of the classical dynamics in this theory is constructed. The transition to the quantum theory is briefly discussed.

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Koenigstein, Adrian, Johannes Kirsch, Horst Stoecker, Juergen Struckmeier, David Vasak, and Matthias Hanauske. "Gauge theory by canonical transformations." International Journal of Modern Physics E 25, no. 07 (July 2016): 1642005. http://dx.doi.org/10.1142/s0218301316420052.

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Electromagnetism, the strong and the weak interactions are commonly formulated as gauge theories in a Lagrangian description. In this paper, we present an alternative formal derivation of [Formula: see text]-gauge theory in a manifestly covariant Hamilton formalism. We make use of canonical transformations as our guiding tool to formalize the gauging procedure. The introduction of the gauge field, its transformation behavior and a dynamical gauge field Lagrangian/Hamiltonian are unavoidable consequences of this formalism, whereas the form of the free gauge Lagrangian/Hamiltonian depends on the selection of the gauge dependence of the canonically conjugate gauge fields.

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WEBB, G. M., R. RATKIEWICZ, M. BRIO, and G. P. ZANK. "Multidimensional simple waves in gas dynamics." Journal of Plasma Physics 59, no. 3 (April 1998): 417–60. http://dx.doi.org/10.1017/s0022377897006375.

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A formalism for multidimensional simple waves in gas dynamics using ideas developed by Boillat is investigated. For simple-wave solutions, the physical variables depend on a single function (r, t). The wave phase (r, t) is implicitly determined by an equation of the form f( )=r·n( )−λ( )t, where n( ) denotes the normal to the wave front, λ is the characteristic speed of the wave mode of interest, r is the position vector, t is the time, and the function f( ) determines whether the wave is a centred (f( )=0) or a non-centred (f( )≠0) wave. Examples are given of time-dependent vortex waves, shear waves and sound waves in one or two space dimensions. The streamlines for the wave reduce to two coupled ordinary differential equations in which the wave phase plays the role of a parameter along the streamlines. The streamline equations are expressed in Hamiltonian form. The roles of Clebsch variables, Lagrangian variables, Hamiltonian formulations and characteristic surfaces are briefly discussed.

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35

Holm, Darryl D. "Elliptical vortices and integrable Hamiltonian dynamics of the rotating shallow-water equations." Journal of Fluid Mechanics 227 (June 1991): 393–406. http://dx.doi.org/10.1017/s0022112091000162.

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The problem of the dynamics of elliptical-vortex solutions of the rotating shallow-water equations is solved in Lagrangian coordinates using methods of Hamiltonian mechanics. All such solutions are shown to be quasi-periodic by reducing the problem to quadratures in terms of physically meaningful variables. All of the relative equilibria - including the well-known rodon solution - are shown to be orbitally Lyapunov stable to perturbations in the class of elliptical-vortex solutions.

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36

Luan, Pi-Gang. "Effective Electrodynamics Theory for the Hyperbolic Metamaterial Consisting of Metal–Dielectric Layers." Crystals 10, no. 10 (September 24, 2020): 863. http://dx.doi.org/10.3390/cryst10100863.

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In this work, we study the dynamical behaviors of the electromagnetic fields and material responses in the hyperbolic metamaterial consisting of periodically arranged metallic and dielectric layers. The thickness of each unit cell is assumed to be much smaller than the wavelength of the electromagnetic waves, so the effective medium concept can be applied. When electromagnetic (EM) fields are present, the responses of the medium in the directions parallel to and perpendicular to the layers are similar to those of Drude and Lorentz media, respectively. We derive the time-dependent energy density of the EM fields and the power loss in the effective medium based on Poynting theorem and the dynamical equations of the polarization field. The time-averaged energy density for harmonic fields was obtained by averaging the energy density in one period, and it reduces to the standard result for the lossless dispersive medium when we turn off the loss. A numerical example is given to reveal the general characteristics of the direction-dependent energy storage capacity of the medium. We also show that the Lagrangian density of the system can be constructed. The Euler–Lagrange equations yield the correct dynamical equations of the electromagnetic fields and the polarization field in the medium. The canonical momentum conjugates to every dynamical field can be derived from the Lagrangian density via differentiation or variation with respect to that field. We apply Legendre transformation to this system and find that the resultant Hamiltonian density is identical to the energy density up to an irrelevant divergence term. This coincidence implies the correctness of the energy density formula we obtained before. We also give a brief discussion about the Hamiltonian dynamics description of the system. The Lagrangian description and Hamiltonian formulation presented in this paper can be further developed for studying the elementary excitations or quasiparticles in other hyperbolic metamaterials.

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37

EL-NABULSI, AHMAD RAMI. "FRACTIONAL QUANTUM EULER–CAUCHY EQUATION IN THE SCHRÖDINGER PICTURE, COMPLEXIFIED HARMONIC OSCILLATORS AND EMERGENCE OF COMPLEXIFIED LAGRANGIAN AND HAMILTONIAN DYNAMICS." Modern Physics Letters B 23, no. 28 (November 10, 2009): 3369–86. http://dx.doi.org/10.1142/s0217984909021387.

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Fractional quantum Euler–Cauchy equation in the Schrödinger picture is derived from the fractional action-like variational approach recently introduced by the author. Many interesting consequences are revealed and explored, in particular the emergence of complexified harmonic oscillators, complexified Lagrangian and Hamiltonian and complexified fractional action integral.

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IVANCEVIC, VLADIMIR. "LIE–LAGRANGIAN MODEL FOR REALISTIC HUMAN BIODYNAMICS." International Journal of Humanoid Robotics 03, no. 02 (June 2006): 205–18. http://dx.doi.org/10.1142/s0219843606000680.

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We present a sophisticated Lagrangian model for anatomically and physiologically realistic human biodynamics (RHB), to accompany the recently reported Hamiltonian formulation.1 The present RHB formulation is designed around three main modules: (i) A Riemannian configuration manifold, composed of gauge Lie groups of constrained 3D rotations and translations, which includes more than 300 degrees of freedom (DOF); (ii) exterior Lagrangian dynamics of the human musculo-skeletal system, including all natural conservative, dissipative and driving forces, powered by 600 equivalent muscles; and (iii) hierarchical nonlinear control, based on an iterative Lie derivative formalism, resembling both spinal reflexes and coordination-control of the human cerebellum. RHB is driven by individual, user supplied musculo-skeletal data. It is modeled in the computer algebra system Mathematica™, simulated in Delphi™ and animated in the 3DS Max™ graphical environment. As an applied example of RHB, we present the full spine simulator, with 150 DOF (25 movable joints each with three constrained rotations and translations), muscular excitation and contraction dynamics, spring-and-damper ligament-like dynamics, spinal-like and cerebellar-like control, and external torques and forces (including inertial, gravitational, viscous, elastic and various types of impacts).

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39

Hasan-Zadeh, Atefeh. "DYNAMIC OPTIMAL CONTROL PROBLEMS IN HAMILTONIAN AND LAGRANGIAN SYSTEMS." Advances in Differential Equations and Control Processes 24, no. 2 (April 20, 2021): 175–85. http://dx.doi.org/10.17654/de024020175.

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40

Caselles, Antonio, Joan C. Micó, and Salvador Amigó. "Energy and Personality: A Bridge between Physics and Psychology." Mathematics 9, no. 12 (June 9, 2021): 1339. http://dx.doi.org/10.3390/math9121339.

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The objective of this paper is to present a mathematical formalism that states a bridge between physics and psychology, concretely between analytical dynamics and personality theory, in order to open new insights in this theory. In this formalism, energy plays a central role. First, the short-term personality dynamics can be measured by the General Factor of Personality (GFP) response to an arbitrary stimulus. This GFP dynamical response is modeled by a stimulus–response model: an integro-differential equation. The bridge between physics and psychology appears when the stimulus–response model can be formulated as a linear second order differential equation and, subsequently, reformulated as a Newtonian equation. This bridge is strengthened when the Newtonian equation is derived from a minimum action principle, obtaining the current Lagrangian and Hamiltonian functions. However, the Hamiltonian function is non-conserved energy. Then, some changes lead to a conserved Hamiltonian function: Ermakov–Lewis energy. This energy is presented, as well as the GFP dynamical response that can be derived from it. An application case is also presented: an experimental design in which 28 individuals consumed 26.51 g of alcohol. This experiment provides an ordinal scale for the Ermakov–Lewis energy that predicts the effect of a single dose of alcohol.

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Colombo, Leonardo, Manuel De Léon, Pedro Daniel Prieto-Martínez, and Narciso Román-Roy. "Unified formalism for the generalized kth-order Hamilton–Jacobi problem." International Journal of Geometric Methods in Modern Physics 11, no. 09 (October 2014): 1460037. http://dx.doi.org/10.1142/s0219887814600378.

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The geometric formulation of the Hamilton–Jacobi theory enables us to generalize it to systems of higher-order ordinary differential equations. In this work we introduce the unified Lagrangian–Hamiltonian formalism for the geometric Hamilton–Jacobi theory on higher-order autonomous dynamical systems described by regular Lagrangian functions.

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Khanin, Konstantin, and Andrei Sobolevski. "Particle dynamics inside shocks in Hamilton–Jacobi equations." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 368, no. 1916 (April 13, 2010): 1579–93. http://dx.doi.org/10.1098/rsta.2009.0283.

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The characteristic curves of a Hamilton–Jacobi equation can be seen as action-minimizing trajectories of fluid particles. For non-smooth ‘viscosity’ solutions, which give rise to discontinuous velocity fields, this description is usually pursued only up to the moment when trajectories hit a shock and cease to minimize the Lagrangian action. In this paper we show that, for any convex Hamiltonian, there exists a uniquely defined canonical global non-smooth coalescing flow that extends particle trajectories and determines the dynamics inside shocks. We also provide a variational description of the corresponding effective velocity field inside shocks, and discuss the relation to the ‘dissipative anomaly’ in the limit of vanishing viscosity.

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Smilga, Andrei. "Classical and quantum dynamics of higher-derivative systems." International Journal of Modern Physics A 32, no. 33 (November 30, 2017): 1730025. http://dx.doi.org/10.1142/s0217751x17300253.

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A brief review of the physics of systems including higher derivatives in the Lagrangian is given. All such systems involve ghosts, i.e. the spectrum of the Hamiltonian is not bounded from below and the vacuum ground state is absent. Usually, this leads to collapse and loss of unitarity. In certain special cases, this does not happen, however, ghosts are benign. We speculate that the Theory of Everything is a higher-derivative field theory, characterized by the presence of such benign ghosts and defined in a higher-dimensional bulk. Our Universe then represents a classical solution in this theory, having the form of a 3-brane embedded in the bulk.

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García-Garrido, Víctor J., Shibabrat Naik, and Stephen Wiggins. "Tilting and Squeezing: Phase Space Geometry of Hamiltonian Saddle-Node Bifurcation and its Influence on Chemical Reaction Dynamics." International Journal of Bifurcation and Chaos 30, no. 04 (March 30, 2020): 2030008. http://dx.doi.org/10.1142/s0218127420300086.

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In this article, we present the influence of a Hamiltonian saddle-node bifurcation on the high-dimensional phase space structures that mediate reaction dynamics. To achieve this goal, we identify the phase space invariant manifolds using Lagrangian descriptors, which is a trajectory-based diagnostic suitable for the construction of a complete “phase space tomography” by means of analyzing dynamics on low-dimensional slices. First, we build a Hamiltonian system with one degree-of-freedom (DoF) that models reaction, and study the effect of adding a parameter to the potential energy function that controls the depth of the well. Then, we extend this framework to a saddle-node bifurcation for a two DoF Hamiltonian, constructed by coupling a harmonic oscillator, i.e. a bath mode, to the other reactive DoF in the system. For this problem, we describe the phase space structures associated with the rank-1 saddle equilibrium point in the bottleneck region, which is a Normally Hyperbolic Invariant Manifold (NHIM) and its stable and unstable manifolds. Finally, we address the qualitative changes in the reaction dynamics of the Hamiltonian system due to changes in the well depth of the potential energy surface that gives rise to the saddle-node bifurcation.

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Kang, Junjie, and Zheng H. Zhu. "Hamiltonian formulation and energy-based control for space tethered system deployment and retrieval." Transactions of the Canadian Society for Mechanical Engineering 43, no. 4 (December 1, 2019): 463–70. http://dx.doi.org/10.1139/tcsme-2018-0215.

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Usually, the dynamic equations of tethered systems are derived using Lagrangian formulation. However, Hamiltonian formulation is also widely used for mechanical systems because of its well-known symplectic structure property. In this paper, the dynamic equations of the tethered system are deduced using Hamiltonian formulation. The goodness of the Hamiltonian formulation is intuitive to reveal the energy balance property that corresponds to the passive property. Furthermore, the Hamiltonian function of the tethered system is employed to facilitate the design of the controller. The energy-based control is to achieve the tether deployment/retrieval with suppressing the tether liberation. Numerical simulations are used to demonstrate the effectiveness of the designed controller.

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IVANCEVIC, V. G., and D. J. REID. "GEOMETRICAL AND TOPOLOGICAL DUALITY IN CROWD DYNAMICS." International Journal of Biomathematics 03, no. 04 (December 2010): 493–507. http://dx.doi.org/10.1142/s1793524510001082.

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The purpose of this paper is to establish strong theoretical basis for solving practical problems in modeling the behavior of crowds. Based on the previously developed (entropic) geometrical model of crowd behavior dynamics, in this paper we formulate two duality theorems related to the crowd manifold. Firstly, we formulate the geometrical crowd-duality theorem and prove it using Lie-functorial and Riemannian proofs. Secondly, we formulate the topological crowd-duality theorem and prove it using cohomological and homological proofs. After that we discuss the related question of the connection between Lagrangian and Hamiltonian crowd-duality, and finally establish the globally dual structure of crowd dynamics. All used terms from algebraic topology are defined in Appendix A.

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47

DUVIRYAK, A. "FOKKER-TYPE CONFINEMENT MODELS FROM EFFECTIVE LAGRANGIAN IN CLASSICAL YANG–MILLS THEORY." International Journal of Modern Physics A 14, no. 28 (November 10, 1999): 4519–47. http://dx.doi.org/10.1142/s0217751x99002128.

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Abelian potentials of pointlike moving sources are obtained from the nonstandard theory of Yang–Mills field. They are used for the construction of the time-symmetric and time-asymmetric Fokker-type action integrals describing the dynamics of two-particle system with confinement interaction. The time-asymmetric model is reformulated in the framework of the Hamiltonian formalism. The corresponding two-body problem is reduced to quadratures. The behavior of Regge trajectories is estimated within the semiclassical consideration.

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48

Dubos, T., S. Dubey, M. Tort, R. Mittal, Y. Meurdesoif, and F. Hourdin. "DYNAMICO-1.0, an icosahedral hydrostatic dynamical core designed for consistency and versatility." Geoscientific Model Development 8, no. 10 (October 7, 2015): 3131–50. http://dx.doi.org/10.5194/gmd-8-3131-2015.

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Abstract. The design of the icosahedral dynamical core DYNAMICO is presented. DYNAMICO solves the multi-layer rotating shallow-water equations, a compressible variant of the same equivalent to a discretization of the hydrostatic primitive equations in a Lagrangian vertical coordinate, and the primitive equations in a hybrid mass-based vertical coordinate. The common Hamiltonian structure of these sets of equations is exploited to formulate energy-conserving spatial discretizations in a unified way. The horizontal mesh is a quasi-uniform icosahedral C-grid obtained by subdivision of a regular icosahedron. Control volumes for mass, tracers and entropy/potential temperature are the hexagonal cells of the Voronoi mesh to avoid the fast numerical modes of the triangular C-grid. The horizontal discretization is that of Ringler et al. (2010), whose discrete quasi-Hamiltonian structure is identified. The prognostic variables are arranged vertically on a Lorenz grid with all thermodynamical variables collocated with mass. The vertical discretization is obtained from the three-dimensional Hamiltonian formulation. Tracers are transported using a second-order finite-volume scheme with slope limiting for positivity. Explicit Runge–Kutta time integration is used for dynamics, and forward-in-time integration with horizontal/vertical splitting is used for tracers. Most of the model code is common to the three sets of equations solved, making it easier to develop and validate each piece of the model separately. Representative three-dimensional test cases are run and analyzed, showing correctness of the model. The design permits to consider several extensions in the near future, from higher-order transport to more general dynamics, especially deep-atmosphere and non-hydrostatic equations.

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49

Dubos, T., S. Dubey, M. Tort, R. Mittal, Y. Meurdesoif, and F. Hourdin. "DYNAMICO, an icosahedral hydrostatic dynamical core designed for consistency and versatility." Geoscientific Model Development Discussions 8, no. 2 (February 19, 2015): 1749–800. http://dx.doi.org/10.5194/gmdd-8-1749-2015.

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Abstract. The design of the icosahedral dynamical core DYNAMICO is presented. DYNAMICO solves the multi-layer rotating shallow-water equations, a compressible variant of the same equivalent to a discretization of the hydrostatic primitive equations in a Lagrangian vertical coordinate, and the primitive equations in a hybrid mass-based vertical coordinate. The common Hamiltonian structure of these sets of equations is exploited to formulate energy-conserving spatial discretizations in a unified way. The horizontal mesh is a quasi-uniform icosahedral C-grid obtained by subdivision of a regular icosahedron. Control volumes for mass, tracers and entropy/potential temperature are the hexagonal cells of the Voronoi mesh to avoid the fast numerical modes of the triangular C-grid. The horizontal discretization is that of Ringler et al. (2010), whose discrete quasi-Hamiltonian structure is identified. The prognostic variables are arranged vertically on a Lorenz grid with all thermodynamical variables collocated with mass. The vertical discretization is obtained from the three-dimensional Hamiltonian formulation. Tracers are transported using a second-order finite volume scheme with slope limiting for positivity. Explicit Runge–Kutta time integration is used for dynamics and forward-in-time integration with horizontal/vertical splitting is used for tracers. Most of the model code is common to the three sets of equations solved, making it easier to develop and validate each piece of the model separately. Representative three-dimensional test cases are run and analyzed, showing correctness of the model. The design permits to consider several extensions in the near future, from higher-order transport to more general dynamics, especially deep-atmosphere and non-hydrostatic equations.

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50

Peletminskii, A., and S. Peletminskii. "Lagrangian and Hamiltonian formalisms for relativistic dynamics of a charged particle with dipole moment." European Physical Journal C 42, no. 4 (August 2005): 505–17. http://dx.doi.org/10.1140/epjc/s2005-02336-4.

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