Статті в журналах: "Principe de Hamilton"

1

Flament, Dominique. "W. R. Hamilton." Revista Brasileira de História da Ciência 1, no. 1 (June 3, 2008): 71–93. http://dx.doi.org/10.53727/rbhc.v1i1.389.

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Hamilton prend le risque d’une proposition en apparence ouvertement métaphysique et en rupture avec celles de travaux réformateurs déjà en cours de développement depuis le début des années 1820. Il surprend quand, opposé par principe aux autres réformateurs qui prônent la nécessité d’une algèbre symbolique, il substitue l’”ordre en progression” à la grandeur, après avoir signalé l’évidence de la relation entre le temps et les progrès de l’algèbre. Il construit une science mathématique du temps pur: elle “existe”, est naturellement comparable à l’algèbre (comme Science), elle coïncide avec elle et, pour finir, est l’algèbre elle-même. L’algèbre est ainsi tirée d’affaire en devenant une branche de la philosophie de l’esprit. La “découverte” du quaternion sera considérée, dès les “opposants” de l’École algébrique anglaise, comme l’ouverture sur la liberté de “créer” en mathématiques.

2

Boyle, Deborah. "Elizabeth Hamilton on Sympathy and the Selfish Principle." Journal of Scottish Philosophy 19, no. 3 (September 2021): 219–41. http://dx.doi.org/10.3366/jsp.2021.0309.

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In A Series of Popular Essays (1813 ) , Scottish philosopher Elizabeth Hamilton (1758–1816) identifies two ‘principles’ in the human mind: sympathy and the selfish principle. While sharing Adam Smith's understanding of sympathy as a capacity for fellow-feeling, Hamilton also criticizes Smith's account of sympathy as involving the imagination. Even more important for Hamilton is the selfish principle, a ‘propensity to expand or enlarge the idea of self’ that she distinguishes from both selfishness and self-love. Counteracting the selfish principle requires cultivating sympathy and benevolent affections from birth. Since no one can do this alone, Hamilton's prescription appeals ineliminably to the caregivers of the very young; and Hamilton was ahead of her time in claiming that these caregivers need not be female.

3

Junker, Philipp, and Daniel Balzani. "An extended Hamilton principle as unifying theory for coupled problems and dissipative microstructure evolution." Continuum Mechanics and Thermodynamics 33, no. 4 (June 7, 2021): 1931–56. http://dx.doi.org/10.1007/s00161-021-01017-z.

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AbstractAn established strategy for material modeling is provided by energy-based principles such that evolution equations in terms of ordinary differential equations can be derived. However, there exist a variety of material models that also need to take into account non-local effects to capture microstructure evolution. In this case, the evolution of microstructure is described by a partial differential equation. In this contribution, we present how Hamilton’s principle provides a physically sound strategy for the derivation of transient field equations for all state variables. Therefore, we begin with a demonstration how Hamilton’s principle generalizes the principle of stationary action for rigid bodies. Furthermore, we show that the basic idea behind Hamilton’s principle is not restricted to isothermal mechanical processes. In contrast, we propose an extended Hamilton principle which is applicable to coupled problems and dissipative microstructure evolution. As example, we demonstrate how the field equations for all state variables for thermo-mechanically coupled problems, i.e., displacements, temperature, and internal variables, result from the stationarity of the extended Hamilton functional. The relation to other principles, as the principle of virtual work and Onsager’s principle, is given. Finally, exemplary material models demonstrate how to use the extended Hamilton principle for thermo-mechanically coupled elastic, gradient-enhanced, rate-dependent, and rate-independent materials.

4

Marrocco, Michele. "“A call to action”: Schrödinger's representation of quantum mechanics via Hamilton's principle." American Journal of Physics 91, no. 2 (February 2023): 110–15. http://dx.doi.org/10.1119/5.0083015.

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A few years ago, one of the former Editors of this journal launched “a call to action” (E. F. Taylor, Am. J. Phys. 71, 423–425 (2003)) for a revision of teaching methods in physics in order to emphasize the importance of the principle of least action. In response, we suggest the use of Hamilton's principle of stationary action to introduce the Schrödinger equation. When considering the geometric interpretation of the Hamilton–Jacobi theory, the real part of the action [Formula: see text] defines the phase of the wave function [Formula: see text], and requiring the Hamilton–Jacobi wave function to obey wave-front propagation (i.e., [Formula: see text] is a constant of the motion) yields the Schrödinger equation.

5

Fusco Girard, Mario. "Evaluation of the Feynman Propagator by Means of the Quantum Hamilton-Jacobi Equation." Quanta 12, no. 1 (April 24, 2023): 22–26. http://dx.doi.org/10.12743/quanta.v12i1.223.

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It is shown that the complex phase of the Feynman propagator is a solution of the quantum Hamilton–Jacobi equation, namely, it is the quantum Hamilton's principal function (or quantum action). Therefore, the Feynman propagator can be computed either by means of the path integration, or by the way of the Hamilton–Jacobi equation. This is analogous to what happens in classical mechanics, where the Hamilton's principal function can be computed either by integrating the Lagrangian along the extremal paths, or as a solution of partial differential equation, namely the classical Hamilton–Jacobi equation. If the path is decomposed in the classical one and quantum fluctuations, the contribution of these quantum fluctuations satisfies a non-linear partial differential equation, whose coefficients depend on the classical action. When the contribution of the quantum fluctuations depend only on the time, it can be computed by means of a simple integration. The final results for the propagators in this case are equal to the Van Vleck–Pauli–Morette expressions, even though the two derivations are quite different.Quanta 2023; 12: 22–26.

6

Fusco Girard, Mario. "The Quantum Hamilton–Jacobi Equation and the Link Between Classical and Quantum Mechanics." Quanta 11, no. 1 (November 3, 2022): 42–52. http://dx.doi.org/10.12743/quanta.v11i1.202.

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We study how the classical Hamilton's principal and characteristic functions are generated from the solutions of the quantum Hamilton–Jacobi equation. While in the classically forbidden regions these quantum quantities directly tend to the classical ones, this is not the case in the allowed regions. There, the limit is reached only if the quantum fluctuations are eliminated by means of coarse-graining averages. Analogously, the classical Hamilton–Jacobi scheme bringing to the motion's equations arises from a similar formal quantum procedure.Quanta 2022; 11: 42–52.

7

Tabarrok, B., and C. M. Leech. "Hamiltonian Mechanics for Functionals Involving Second-Order Derivatives." Journal of Applied Mechanics 69, no. 6 (October 31, 2002): 749–54. http://dx.doi.org/10.1115/1.1505626.

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Hamilton’s principle was developed for the modeling of dynamic systems in which time is the principal independent variable and the resulting equations of motion are second-order differential equations. This principle uses kinetic energy which is functionally dependent on first-order time derivatives, and potential energy, and has been extended to include virtual work. In this paper, a variant of Hamiltonian mechanics for systems whose motion is governed by fourth-order differential equations is developed and is illustrated by an example invoking the flexural analysis of beams. The variational formulations previously associated with Newton’s second-order equations of motion have been generalized to encompass problems governed by energy functionals involving second-order derivatives. The canonical equations associated with functionals with second order derivatives emerge as four first-order equations in each variable. The transformations of these equations to a new system wherein the generalized variables and momenta appear as constants, can be obtained through several different forms of generating functions. The generating functions are obtained as solutions of the Hamilton-Jacobi equation. This theory is illustrated by application to an example from beam theory the solution recovered using a technique for solving nonseparable forms of the Hamilton-Jacobi equation. Finally whereas classical variational mechanics uses time as the primary independent variable, here the theory is extended to include static mechanics problems in which the primary independent variable is spatial.

8

Miller, Karol, and Boris S. Stevens. "Modeling of Dynamics and Model-Based Control of DELTA Direct-Drive Parallel Robot." Journal of Robotics and Mechatronics 7, no. 4 (August 20, 1995): 344–52. http://dx.doi.org/10.20965/jrm.1995.p0344.

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The term ""Extended Space"" used in this article is hereby defined as a union of the operational and articulation spaces of a manipulator. The advantages in the use of such coordinates (extended space) in the description of DELTA robot is presented here and discussed in some detail. The emerging importance of parallel robots has necessitated an increased sophistication to achieve improved control. A method based on the direct application of the Hamilton's Principle in extended space, has been applied efficiently to solving the inverse problem of dynamics and implemented for real time application in the control law of the direct-drive version of DELTA parallel robot.1-3) The full dynamic model of this robot has been developed herein. The numerical efficiency and other benefits of this approach over the more classical Lagrange and Newton-Euler methods for the inverse dynamics problem solving are also briefly discussed. For similar models, the version obtained by the direct application of Hamilton's principle is found to possess 23% less mathematical operations than for the Lagrangebased model. Frictional effects. being very small in the direct-drive manipulator, are not included in the present Hamilton development but can be handled with a slight modification. Furthermore the acceleration information of the robot are not required as input states to the Hamilton model. The measurement of trajectory tracking performances for different controllers is conducted. The repeatability of the robot trajectory tracking is determined. The improvement obtained in the control algorithm's performance after the Hamilton implementation is proven to be conclusive.

9

SHEEHAN, COLLEEN A. "Madison v. Hamilton: The Battle Over Republicanism and the Role of Public Opinion." American Political Science Review 98, no. 3 (August 2004): 405–24. http://dx.doi.org/10.1017/s0003055404001248.

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This article examines the causes of the dispute between James Madison and Alexander Hamilton in the early 1790s. Though Hamilton initially believed that Madison's opposition to the Federalist administration was probably motivated by personal animosity and political advantage, in later years he concluded what Madison had long argued: the controversy between Republicans and Federalists stemmed from a difference of principle. For Madison, republicanism meant the recognition of the sovereignty of public opinion and the commitment to participatory politics. Hamilton advocated a more submissive role for the citizenry and a more independent status for the political elite. While Madison did not deny to political leaders and enlightened men a critical place in the formation of public opinion, he fought against Hamilton's thin version of public opinion as “confidence” in government. In 1791–92 Madison took the Republican lead in providing a philosophic defense for a tangible, active, and responsible role for the citizens of republican government.

10

Gong, Sheng-nan, and Jing-li Fu. "Noether’s theorems for the relative motion systems on time scales." Applied Mathematics and Nonlinear Sciences 3, no. 2 (December 1, 2018): 513–26. http://dx.doi.org/10.2478/amns.2018.2.00040.

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AbstractThis paper propose Noether symmetries and the conserved quantities of the relative motion systems on time scales. The Lagrange equations with delta derivatives on time scales are presented for the system. Based upon the invariance of Hamilton action on time scales, under the infinitesimal transformations with respect to the time and generalized coordinates, the Hamilton’s principle, the Noether theorems and conservation quantities are given for the systems on time scales. Lastly, an example is given to show the application the conclusion.

11

Ván, P., and B. Nyíri. "Hamilton formalism and variational principle construction." Annalen der Physik 511, no. 4 (April 1999): 331–54. http://dx.doi.org/10.1002/andp.19995110404.

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12

Ván, P., and B. Nyíri. "Hamilton formalism and variational principle construction." Annalen der Physik 8, no. 4 (April 1999): 331–54. http://dx.doi.org/10.1002/(sici)1521-3889(199904)8:4<331::aid-andp331>3.0.co;2-r.

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13

Guangzhou, Ge. "Discussions on the space‐time structure, Hamilton’s field, and Breakthrough Starshot project." Physics Essays 33, no. 3 (September 17, 2020): 243–55. http://dx.doi.org/10.4006/0836-1398-33.3.243.

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This article can be regarded as the author's exploration of the space‐time structure and of field theory. The author first puts forward the equivalence of time and space based on Hamilton’s principle and then applies Newton's laws of motion to the interpretation of an object’s motion in time, thus deducing that Newton's first law of motion and the principle of constancy of light speed were to be unified. And then the author summarizes the three states of motion existing in the physical world, specially explores the state of motion that is not to be limited by force or the speed of light, and further comes up with a new interpretation of the falling apple. Next, pursuing the understanding of Hamilton's principle and Hamilton’s Tension Equation (THE), the author explores the space‐time structure corresponding to the state of super-light-speed and super-force, and puts forward the Hamilton’s field and its full description and main characteristics. The author also indicates that the Hamilton’s field could realize the unification of fields with physical geometrization. Finally, the author applies the principles of Hamilton’s field to the research of the Breakthrough Starshot project and explores the three breakthroughs as needed. A new photoelectric effect is meanwhile presented.

14

Liu, Zong Min, Hai Yan Song, and Ji Ze Mao. "Quasi-Hamilton Principle of Quasi-Crystals Beam." Advanced Materials Research 197-198 (February 2011): 1540–44. http://dx.doi.org/10.4028/www.scientific.net/amr.197-198.1540.

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Quasi-crystals is not only a new structure of solids but also a new class of functional and structural materials. With the research and development of quasi-crystals, the mechanical properties of quasi-crystals get more and more attention. In the paper, quasi-Hamilton principle of quasi-crystals beam is established in non-conservative systems. And applying the quasi-Hamilton principle, all the equations of non-conservative quasi-crystals beam problem are obtained in the phonon field and the phason field respectively.

15

МОРОЗ, Іван, Володимир ІВАНІЙ, Євгеній ДЄМЄНТЬЄВ, and Аніта ЩУПАЧИНСЬКА. "METHODOLOGICAL FOUNDATION OF HAMILTON-OSTROGRADSKYI VARIATION PRINCIPLE." Scientific papers of Berdiansk State Pedagogical University Series Pedagogical sciences 3 (December 27, 2019): 310–19. http://dx.doi.org/10.31494/2412-9208-2019-1-3-310-319.

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16

LARSSON, JONAS. "A practical form of Lagrange–Hamilton theory for ideal fluids and plasmas." Journal of Plasma Physics 69, no. 3 (April 2003): 211–52. http://dx.doi.org/10.1017/s0022377803002290.

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Lagrangian and Hamiltonian formalisms for ideal fluids and plasmas have, during the last few decades, developed much in theory and applications. The recent formulation of ideal fluid/plasma dynamics in terms of the Euler–Poincaré equations makes a self-contained, but mathematically elementary, form of Lagrange–Hamilton theory possible. The starting point is Hamilton's principle. The main goal is to present Lagrange–Hamilton theory in a way that simplifies its applications within usual fluid and plasma theory so that we can use standard vector analysis and standard Eulerian fluid variables. The formalisms of differential geometry, Lie group theory and dual spaces are avoided and so is the use of Lagrangian fluid variables or Clebsch potentials. In the formal ‘axiomatic’ setting of Lagrange–Hamilton theory the concepts of Lie algebra and Hilbert space are used, but only in an elementary way. The formalism is manifestly non-canonical, but the analogy with usual classical mechanics is striking. The Lie derivative is a most convenient tool when the abstract Lagrange–Hamilton formalism is applied to concrete fluid/plasma models. This directional/dynamical derivative is usually defined within differential geometry. However, following the goals of this paper, we choose to define Lie derivatives within standard vector analysis instead (in terms of the directional field and the div, grad and curl operators). Basic identities for the Lie derivatives, necessary for using them effectively in vector calculus and Lagrange–Hamilton theory, are included. Various dynamical invariants, valid for classes of fluid and plasma models (including both compressible and incompressible ideal magnetohydrodynamics), are given simple and straightforward derivations thanks to the Lie derivative calculus. We also consider non-canonical Poisson brackets and derive, in particular, an explicit result for incompressible and inhomogeneous flows.

17

Brun, J. L. "Hamilton's principle for beginners." European Journal of Physics 28, no. 3 (March 23, 2007): 487–91. http://dx.doi.org/10.1088/0143-0807/28/3/009.

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18

Zhou, Yinqiu, and Xiuming Wang. "A methodology for formulating dynamical equations in analytical mechanics based on the principle of energy conservation." Journal of Physics Communications 6, no. 3 (March 1, 2022): 035006. http://dx.doi.org/10.1088/2399-6528/ac57f8.

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Abstract In this work, a methodology is proposed for formulating general dynamical equations in mechanics under the umbrella of the principle of energy conservation. It is shown that Lagrange’s equation, Hamilton’s canonical equations, and Hamilton-Jacobi’s equation are all formulated based on the principle of energy conservation with a simple energy conservation equation, i.e., the rate of kinetic and potential energy with time is equal to the rate of work with time done by external forces; while D’Alembert’s principle is a special case of the law of the conservation of energy, with either the virtual displacements (‘frozen’ time) or the virtual displacement (‘frozen’ generalized coordinates). It is argued that all of the formulations for characterizing the dynamical behaviors of a system can be derived from the principle of energy conservation, and the principle of energy conservation is an underlying guide for constructing mechanics in a broad sense. The proposed methodology provides an efficient way to tackle the dynamical problems in general mechanics, including dissipation continuum systems, especially for those with multi-physical field interactions and couplings. It is pointed out that, on the contrary to the classical analytical mechanics, especially to existing Hamiltonian mechanics, the physics essences of Hamilton’s variational principle, Lagrange’s equation, and the Newtonian second law of motion, including their derivatives such as momentum and angular momentum conservations, are the consequences of the law of conservation of energy. In addition, our proposed methodology is easier to understand with clear physical meanings and can be used for explaining the existing mechanical principles or theorems. Finally, as an application example, the methodology is applied in fluid mechanics to derive Cauchy’s first law of motion.

19

Liu, Yong-Jin, Kai Tang, and Ajay Joneja. "Modeling dynamic developable meshes by the Hamilton principle." Computer-Aided Design 39, no. 9 (September 2007): 719–31. http://dx.doi.org/10.1016/j.cad.2007.02.013.

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20

Cen, Song, Tao Zhang, Chen-Feng Li, Xiang-Rong Fu, and Yu-Qiu Long. "A hybrid-stress element based on Hamilton principle." Acta Mechanica Sinica 26, no. 4 (June 29, 2010): 625–34. http://dx.doi.org/10.1007/s10409-010-0352-5.

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21

Wu, Xiangyao, Benshan Wu, Hong Li, and Qiming Wu. "From Generalized Hamilton Principle to Generalized Schrodinger Equation." Journal of Modern Physics 14, no. 05 (2023): 676–91. http://dx.doi.org/10.4236/jmp.2023.145039.

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22

Meirovitch, L. "Derivation of Equations for Flexible Multibody Systems in Terms of Quasi-Coordinates from the Extended Hamilton’s Principle." Shock and Vibration 1, no. 2 (1993): 107–19. http://dx.doi.org/10.1155/1993/915264.

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Early derivations of the equations of motion for single rigid bodies, single flexible bodies, and flexible multibody systems in terms of quasi-coordinates have been carried out in two stages. The first consists of the use of the extended Hamilton’s principle to derive standard Lagrange’s equations in terms of generalized coordinates and the second represents a transformation of the Lagrange’s equations to equations in terms of quasi-coordinates. In this article, hybrid (ordinary and partial) differential equations for flexible multibody systems are derived in terms of quasi-coordinates directly from the extended Hamilton's principle. The approach has beneficial implications in an eventual spatial discretization of the problem.

23

Hong-Xia, Zhao, and Ma Shan-Jun. "High-Order Hamilton's Principle and the Hamilton's Principle of High-Order Lagrangian Function." Communications in Theoretical Physics 49, no. 2 (February 2008): 297–302. http://dx.doi.org/10.1088/0253-6102/49/2/08.

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24

He, Ji-Huan. "Hamilton Principle and Generalized Variational Principles of Linear Thermopiezoelectricity." Journal of Applied Mechanics 68, no. 4 (October 19, 2000): 666–67. http://dx.doi.org/10.1115/1.1352067.

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25

Stastna, J. "Hamilton's principle for perfect fluids." International Journal of Mathematical Education in Science and Technology 17, no. 3 (May 1986): 311–14. http://dx.doi.org/10.1080/0020739860170306.

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26

Howarth, J. A., and A. Bedford. "Hamilton's Principle in Continuum Mechanics." Mathematical Gazette 70, no. 454 (December 1986): 329. http://dx.doi.org/10.2307/3616226.

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27

Bedford, A., and S. L. Passman. "Hamilton’s Principle in Continuum Mechanics." Journal of Applied Mechanics 53, no. 3 (September 1, 1986): 731. http://dx.doi.org/10.1115/1.3171846.

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28

Holm, Darryl D., and Vladimir Zeitlin. "Hamilton’s principle for quasigeostrophic motion." Physics of Fluids 10, no. 4 (April 1998): 800–806. http://dx.doi.org/10.1063/1.869623.

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29

Kapsa, V., and L. Skála. "From probabilities to Hamilton's principle." Journal of Physics A: Mathematical and Theoretical 42, no. 31 (July 13, 2009): 315202. http://dx.doi.org/10.1088/1751-8113/42/31/315202.

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30

Jalnapurkar, Sameer M., and Jerrold E. Marsden. "Reduction of Hamilton's variational principle." Dynamics and Stability of Systems 15, no. 3 (September 2000): 287–318. http://dx.doi.org/10.1080/713603744.

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31

Pavon, Michele. "Hamilton’s principle in stochastic mechanics." Journal of Mathematical Physics 36, no. 12 (December 1995): 6774–800. http://dx.doi.org/10.1063/1.531187.

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32

He, Ji-Huan. "Hamilton’s principle for dynamical elasticity." Applied Mathematics Letters 72 (October 2017): 65–69. http://dx.doi.org/10.1016/j.aml.2017.04.008.

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33

Ghori, Q. K., and N. Ahmed. "Hamilton's Principle for Nonholonomic Systems." ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 74, no. 2 (1994): 137–40. http://dx.doi.org/10.1002/zamm.19940740219.

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34

Rotondo, Marcello. "A Wheeler–DeWitt Equation with Time." Universe 8, no. 11 (November 3, 2022): 580. http://dx.doi.org/10.3390/universe8110580.

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The equation for canonical gravity produced by Wheeler and DeWitt in the late 1960s still presents difficulties both in terms of its mathematical solution and its physical interpretation. One of these issues is, notoriously, the absence of an explicit time. In this short note, we suggest one simple and straightforward way to avoid this occurrence. We go back to the classical equation that inspired Wheeler and DeWitt (namely, the Hamilton–Jacobi–Einstein equation) and make explicit, before quantization, the presence of a known, classically meaningful notion of time. We do this by allowing Hamilton’s principal function to be explicitly dependent on this time locally. This choice results in a Wheeler–DeWitt equation with time. A working solution for the de Sitter minisuperspace is shown.

35

MORITA, Susumu, and Toshiyuki OHTSUKA. "Natural Motion Trajectory Generation Based on Hamilton's Principle." Transactions of the Society of Instrument and Control Engineers 42, no. 1 (2006): 1–10. http://dx.doi.org/10.9746/sicetr1965.42.1.

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36

Faraggi, Alon E., and Marco Matone. "Equivalence principle, Planck length and quantum Hamilton–Jacobi equation." Physics Letters B 445, no. 1-2 (December 1998): 77–81. http://dx.doi.org/10.1016/s0370-2693(98)01484-1.

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37

Yoshimura, Hiroaki, and Jerrold E. Marsden. "Reduction of Dirac structures and the Hamilton-Pontryagin principle." Reports on Mathematical Physics 60, no. 3 (December 2007): 381–426. http://dx.doi.org/10.1016/s0034-4877(08)00004-9.

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38

Rochet, J. C. "The taxation principle and multi-time Hamilton-Jacobi equations." Journal of Mathematical Economics 14, no. 2 (January 1985): 113–28. http://dx.doi.org/10.1016/0304-4068(85)90015-1.

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39

Hien, T. D., and M. Kleiber. "Finite element analysis based on stochastic Hamilton variational principle." Computers & Structures 37, no. 6 (January 1990): 893–902. http://dx.doi.org/10.1016/0045-7949(90)90002-j.

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40

Kimball, J. C., and Harold Story. "Fermat's principle, Huygens' principle, Hamilton's optics and sailing strategy." European Journal of Physics 19, no. 1 (January 1, 1998): 15–24. http://dx.doi.org/10.1088/0143-0807/19/1/004.

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41

Zou, Guiping, and Gang Liang. "The hamilton system and hamilton type generalized variational principle for the laminated composite plates and shells." Journal of Shanghai University (English Edition) 1, no. 2 (September 1997): 123–29. http://dx.doi.org/10.1007/s11741-997-0008-2.

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42

Mu, Benrong, Peng Wang, and Haitang Yang. "Covariant GUP Deformed Hamilton-Jacobi Method." Advances in High Energy Physics 2017 (2017): 1–8. http://dx.doi.org/10.1155/2017/3191839.

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We first briefly revisit the original Hamilton-Jacobi method and show that the Hamilton-Jacobi equation for the action I of tunneling of a fermionic particle from a charged black hole can be written in the same form of that for a scalar particle. On the other hand, various theories of quantum gravity suggest the existence of a minimal length scale, incorporating of which into quantum mechanics implies a modification of the uncertainty principle. In the scenario incorporating the generalized uncertainty principle (GUP) into a quantum field theory (QFT) in a covariant way, we derive the deformed model-independent KG/Dirac and Hamilton-Jacobi equations using the methods of effective field theory. For this Lorentz invariant GUP modified QFT, we find that the effect of GUP on the Hamilton-Jacobi equations is simply to “renormalize” the mass of the emitted particles, from m to meff. Therefore, in this scenario, the Hawking temperature of a black hole does not receive any corrections from the GUP effect.

43

Zhang, Yi. "Mei Symmetry and Conservation Laws for Time-Scale Nonshifted Hamilton Equations." Advances in Mathematical Physics 2021 (November 17, 2021): 1–8. http://dx.doi.org/10.1155/2021/7329399.

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The Mei symmetry and conservation laws for time-scale nonshifted Hamilton equations are explored, and the Mei symmetry theorem is presented and proved. Firstly, the time-scale Hamilton principle is established and extended to the nonconservative case. Based on the Hamilton principles, the dynamic equations of time-scale nonshifted constrained mechanical systems are derived. Secondly, for the time-scale nonshifted Hamilton equations, the definitions of Mei symmetry and their criterion equations are given. Thirdly, Mei symmetry theorems are proved, and the Mei-type conservation laws in time-scale phase space are driven. Two examples show the validity of the results.

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Zhdanov, Dmitry V., and Denys I. Bondar. "Joint quantum–classical Hamilton variational principle in the phase space." Journal of Physics A: Mathematical and Theoretical 55, no. 10 (February 17, 2022): 104001. http://dx.doi.org/10.1088/1751-8121/ac4ce7.

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Abstract We show that the dynamics of a closed quantum system obeys the Hamilton variational principle. Even though quantum particles lack well-defined trajectories, their evolution in the Husimi representation can be treated as a flow of multidimensional probability fluid in the phase space. By introducing the classical counterpart of the Husimi representation in a close analogy to the Koopman–von Neumann theory, one can largely unify the formulations of classical and quantum dynamics. We prove that the motions of elementary parcels of both classical and quantum Husimi fluid obey the Hamilton variational principle, and the differences between associated action functionals stem from the differences between classical and quantum pure states. The Husimi action functionals are not unique and defined up to the Skodje flux gauge fixing (Skodje et al 1989 Phys. Rev. A 40 2894). We demonstrate that the gauge choice can dramatically alter flux trajectories. Applications of the presented theory for constructing semiclassical approximations and hybrid classical–quantum theories are discussed.

45

Zhou, Yue Fa, Fang Lue Huang, Zhi Yong Zhang, and Tian Shu Song. "Dynamics Analysis of Multi-Degree-of-Freedom Motion Simulator Based on Hamilton Method." Applied Mechanics and Materials 138-139 (November 2011): 434–41. http://dx.doi.org/10.4028/www.scientific.net/amm.138-139.434.

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In this paper, a new method is given about using Hamilton principle to establish multi-degree-of-freedom electro-hydraulic mix-drive motion simulator model. And dynamic analysis is performed based on kinetics and kinematics. The simulation is done on multi-degree-of-freedom electro-hydraulic mix-drive motion simulator and Hamilton principle. As cases, some calculating results on dynamic simulation are plotted with the help of Matlab Lagrange. The work in the paper could be seen as a theoretical basis to research motion simulator in depth.

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Leech, C. M. "The Hamilton-Jacobi Equation Applied to Continuum." Journal of Applied Mechanics 64, no. 3 (September 1, 1997): 658–63. http://dx.doi.org/10.1115/1.2788943.

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The Hamilton-Jacobi partial differential equation is established for continuum systems; to do this a new concept in material distributions is introduced. The Lagrangian and Hamiltonian are developed, so that the Hamilton-Jacobi equation can be formulated and the principal function defined. Finally the principal function is constructed for the dynamics of a one-dimensional linear elastic bar; the solution for its’ vibrations is then established following the differentiation of the principal function.

47

Song Bai, Wu Jing, and Guo Zeng-Yuan. "Hamilton’s principle based on thermomass theory." Acta Physica Sinica 59, no. 10 (2010): 7129. http://dx.doi.org/10.7498/aps.59.7129.

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48

Hurtado, John E. "Hamilton’s Principle for Variable-Mass Systems." Journal of Guidance, Control, and Dynamics 41, no. 12 (December 2018): 2647–50. http://dx.doi.org/10.2514/1.g003340.

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49

Atanacković, T. M., S. Konjik, Lj Oparnica, and S. Pilipović. "Generalized Hamilton's principle with fractional derivatives." Journal of Physics A: Mathematical and Theoretical 43, no. 25 (May 27, 2010): 255203. http://dx.doi.org/10.1088/1751-8113/43/25/255203.

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50

Yang, Q., F. H. Guan, and Y. R. Liu. "Hamilton’s principle for Green-inelastic bodies." Mechanics Research Communications 37, no. 8 (December 2010): 696–99. http://dx.doi.org/10.1016/j.mechrescom.2010.10.002.

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