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Journal articles on the topic 'Lagrangian functions'

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Author: Grafiati

Published: 4 June 2021

Last updated: 20 February 2023

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1

Musielak, Zdzislaw E., Niyousha Davachi, and Marialis Rosario-Franco. "Special Functions of Mathematical Physics: A Unified Lagrangian Formalism." Mathematics 8, no. 3 (March 9, 2020): 379. http://dx.doi.org/10.3390/math8030379.

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Lagrangian formalism is established for differential equations with special functions of mathematical physics as solutions. Formalism is based on either standard or non-standard Lagrangians. This work shows that the procedure of deriving the standard Lagrangians leads to Lagrangians for which the Euler–Lagrange equation vanishes identically, and that only some of these Lagrangians become the null Lagrangians with the well-defined gauge functions. It is also demonstrated that the non-standard Lagrangians require that the Euler–Lagrange equations are amended by the auxiliary conditions, which is a new phenomenon in the calculus of variations. The existence of the auxiliary conditions has profound implications on the validity of the Helmholtz conditions. The obtained results are used to derive the Lagrangians for the Airy, Bessel, Legendre and Hermite equations. The presented examples clearly demonstrate that the developed Lagrangian formalism is applicable to all considered differential equations, including the Airy (and other similar) equations, and that the regular and modified Bessel equations are the only ones with the gauge functions. Possible implications of the existence of the gauge functions for these equations are discussed.

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2

Xuegang, Yu. "Hyperbolic Lagrangian functions." Applied Mathematics and Mechanics 19, no. 12 (December 1998): 1189–95. http://dx.doi.org/10.1007/bf02456640.

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3

Musielak, Z. E., N. Davachi, and M. Rosario-Franco. "Lagrangians, Gauge Functions, and Lie Groups for Semigroup of Second-Order Differential Equations." Journal of Applied Mathematics 2020 (June 22, 2020): 1–11. http://dx.doi.org/10.1155/2020/3170130.

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A set of linear second-order differential equations is converted into a semigroup, whose algebraic structure is used to generate novel equations. The Lagrangian formalism based on standard, null, and nonstandard Lagrangians is established for all members of the semigroup. For the null Lagrangians, their corresponding gauge functions are derived. The obtained Lagrangians are either new or generalization of those previously known. The previously developed Lie group approach to derive some equations of the semigroup is also described. It is shown that certain equations of the semigroup cannot be factorized, and therefore, their Lie groups cannot be determined. A possible solution of this problem is proposed, and the relationship between the Lagrangian formalism and the Lie group approach is discussed.

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4

ZHANG, LI-WEI, YONG-HONG REN, YUE WU, and XIAN-TAO XIAO. "A CLASS OF NONLINEAR LAGRANGIANS: THEORY AND ALGORITHM." Asia-Pacific Journal of Operational Research 25, no. 03 (June 2008): 327–71. http://dx.doi.org/10.1142/s021759590800178x.

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This paper establishes a theory framework of a class of nonlinear Lagrangians for solving nonlinear programming problems with inequality constraints. A set of conditions are proposed to guarantee the convergence of nonlinear Lagrangian algorithms, to analyze condition numbers of nonlinear Lagrangian Hessians as well as to develop the dual approaches. These conditions are satisfied by well-known nonlinear Lagrangians appearing in literature. The convergence theorem shows that the dual algorithm based on any nonlinear Lagrangian in the class is locally convergent when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions and the error bound solution, depending on the penalty parameter, is also established. The paper also develops the dual problems based on the proposed nonlinear Lagrangians, and the related duality theorem and saddle point theorem are demonstrated. Furthermore, it is shown that the condition numbers of Lagrangian Hessians at optimal solutions are proportional to the controlling penalty parameters. We report some numerical results obtained by using nonlinear Lagrangians.

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5

Musielak, Zdzislaw E. "Nonstandard Null Lagrangians and Gauge Functions for Newtonian Law of Inertia." Physics 3, no. 4 (October 4, 2021): 903–12. http://dx.doi.org/10.3390/physics3040056.

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New null Lagrangians and gauge functions are derived and they are called nonstandard because their forms are different than those previously found. The invariance of the action is used to make the Lagrangians and gauge functions exact. The first exact nonstandard null Lagrangian and its gauge function for the law of inertia are obtained, and their physical implications are discussed.

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6

Chen, Bang-Yen. "Jacobi's elliptic functions and Lagrangian immersions." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 126, no. 4 (1996): 687–704. http://dx.doi.org/10.1017/s0308210500023003.

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First, we establish a sharp inequality between the squared mean curvature and the scalar curvature for a Lagrangian submanifold in a nonflat complex-space-form. Then, by utilising the Jacobi's elliptic functions en and dn, we introduce three families of Lagrangian submanifolds and two exceptional Lagrangian submanifolds Fn, Ln in nonflat complex-space-forms which satisfy the equality case of the inequality. Finally, we obtain the complete classification of Lagrangian submanifolds in nonflat complex-space-forms which satisfy this basic equality.

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7

El-Nabulsi, R. A. "Nonstandard fractional exponential Lagrangians, fractional geodesic equation, complex general relativity, and discrete gravity." Canadian Journal of Physics 91, no. 8 (August 2013): 618–22. http://dx.doi.org/10.1139/cjp-2013-0145.

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Nonstandard Lagrangians are generating functions of different equations of motion. They have gained increasing importance in many different fields. In fact, nonstandard Lagrangians date back to 1978, when Arnold entitled them “non-natural” in his classic book, Mathematical Methods of Classical Mechanics (Springer, New York. 1978). In applied mathematics, most dynamical equations can be obtained by using generating Lagrangian functions (e.g., power-law and exponential Lagrangians), which has been shown by mathematicians, who have also demonstrated that there is an infinite number of such functions. Besides this interesting field, the topic of fractional calculus of variations has gained growing importance because of its wide application in different fields of science. In this paper, we generalize the fractional actionlike variational approach for the case of a nonstandard exponential Lagrangian. To appreciate this new approach, we explore some of its main consequences in Einstein’s general relativity. Some results are revealed and discussed accordingly mainly the transition from general relativity to complex relativity and emergence of a discrete gravitational coupling constant.

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8

Jeffrey, Lisa. "Chern–Simons gauge theory and symplectic quantum mechanics." Canadian Journal of Physics 93, no. 9 (September 2015): 971–73. http://dx.doi.org/10.1139/cjp-2014-0563.

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We describe the relation between the Chern–Simons gauge theory partition function and the partition function defined using the symplectic action functional as the Lagrangian. We show that the partition functions obtained using these two Lagrangians agree, and we identify the semiclassical formula for the partition function defined using the symplectic action functional. We also compute the semiclassical formulas for the partition functions obtained using the two different Lagrangians: the Chern–Simons functional and the symplectic action functional.

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9

Obaidullah, U., and Sameerah Jamal. "pp-wave potential functions: A complete study using Noether symmetries." International Journal of Geometric Methods in Modern Physics 18, no. 07 (March 18, 2021): 2150108. http://dx.doi.org/10.1142/s0219887821501085.

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In this paper, we examine the functional forms of the potentials [Formula: see text] that emerge from the Lagrangian of the pp-wave spacetime. To facilitate this investigation, Noether symmetries are employed as well as their linear combinations and subalgebras. We exploit the geometric fact that Noether point symmetries of geodesic Lagrangians are generated from the Homothetic algebra of spacetimes. Thus, we provide a complete analysis of the potentials of this spacetime, which are split into 14 isometry categories.

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10

Frenk, J. B. G., and G. Kassay. "Lagrangian Duality and Cone Convexlike Functions." Journal of Optimization Theory and Applications 134, no. 2 (June 29, 2007): 207–22. http://dx.doi.org/10.1007/s10957-007-9221-1.

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11

Zybin, K. P., V. A. Sirota, A. S. Il’in, and A. V. Gurevich. "Lagrangian structure functions in hydrodynamic turbulence." Journal of Experimental and Theoretical Physics 107, no. 5 (November 2008): 879–86. http://dx.doi.org/10.1134/s1063776108110198.

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12

Forbes, G. W. "Singularities of multivariate Lagrangian aberration functions." Journal of the Optical Society of America A 3, no. 9 (September 1, 1986): 1370. http://dx.doi.org/10.1364/josaa.3.001370.

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13

Okubo, Katsumi. "Differential geometry of generalized lagrangian functions." Journal of Mathematics of Kyoto University 31, no. 4 (1991): 1095–103. http://dx.doi.org/10.1215/kjm/1250519677.

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14

Brown, J. David. "Singular Lagrangians and the Dirac–Bergmann algorithm in classical mechanics." American Journal of Physics 91, no. 3 (March 2023): 214–24. http://dx.doi.org/10.1119/5.0107540.

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Textbook treatments of classical mechanics typically assume that the Lagrangian is nonsingular; that is, the matrix of second derivatives of the Lagrangian with respect to the velocities is invertible. This assumption ensures that (i) Lagrange's equations can be solved for the accelerations as functions of coordinates and velocities, and (ii) the definitions of the conjugate momenta can be inverted to solve for the velocities as functions of coordinates and momenta. This assumption, however, is unnecessarily restrictive—there are interesting classical dynamical systems with singular Lagrangians. The algorithm for analyzing such systems was developed by Dirac and Bergmann in the 1950s. After a brief review of the Dirac–Bergmann algorithm, several examples are presented using familiar components: point masses connected by massless springs, rods, cords, and pulleys.

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15

WANAS, M. I., and SAMAH A. AMMAR. "SPACETIME STRUCTURE AND ELECTROMAGNETISM." Modern Physics Letters A 25, no. 20 (June 28, 2010): 1705–21. http://dx.doi.org/10.1142/s0217732310032883.

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Two Lagrangian functions are used to construct geometric field theories. One of these Lagrangians depends on the curvature of space, while the other depends on curvature and torsion. It is shown that the theory constructed from the first Lagrangian gives rise to pure gravity, while the theory constructed using the second Lagrangian gives rise to both gravity and electromagnetism. The two theories are constructed in a version of absolute parallelism geometry in which both curvature and torsion are, simultaneously, nonvanishing. One single geometric object, W-tensor, reflecting the properties of curvature and torsion, is defined in this version and is used to construct the second theory. The main conclusion is that a necessary condition for geometric representation of electromagnetism is the presence of a nonvanishing torsion in the geometry used.

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16

Bistagnino, A., G. Boffetta, and A. Mazzino. "Lagrangian velocity structure functions in Bolgiano turbulence." Physics of Fluids 19, no. 1 (January 2007): 011703. http://dx.doi.org/10.1063/1.2432154.

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17

Meng, Fanwen, Gongyun Zhao, Mark Goh, and Robert De Souza. "Lagrangian-Dual Functions and Moreau–Yosida Regularization." SIAM Journal on Optimization 19, no. 1 (January 2008): 39–61. http://dx.doi.org/10.1137/060673746.

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18

Zhou, Y. Y., and X. Q. Yang. "Augmented Lagrangian functions for constrained optimization problems." Journal of Global Optimization 52, no. 1 (February 23, 2011): 95–108. http://dx.doi.org/10.1007/s10898-011-9688-z.

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19

Bhatia, Davinder, and Aparna Mehra. "Lagrangian Duality for Preinvex Set-Valued Functions." Journal of Mathematical Analysis and Applications 214, no. 2 (October 1997): 599–612. http://dx.doi.org/10.1006/jmaa.1997.5599.

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20

GIMÉNEZ, ÁNGEL. "RELATIVISTIC PARTICLES ALONG NULL CURVES IN 3D LORENTZIAN SPACE FORMS." International Journal of Bifurcation and Chaos 20, no. 09 (September 2010): 2851–59. http://dx.doi.org/10.1142/s0218127410027404.

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We study relativistic particles modeled by actions whose Lagrangians are arbitrary functions on the curvature of null paths in (2 + 1)-dimensions backgrounds with constant curvature. We obtain first integrals of the Euler–Lagrange equation by using geometrical methods involving the search for Killing vector fields along critical curves of the action. In the case in which Lagrangian density depends quadratically on Cartan curvature, it is shown that the mechanical system is governed by a stationary Korteweg–De Vries system. Motion equations are completely integrated by quadratures in terms of elliptic and hyperelliptic functions.

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21

Vijayalakshmi, Palanisamy, Zhiheng Jiang, and Xiong Wang. "Lagrangian Formulation of Lorenz and Chen Systems." International Journal of Bifurcation and Chaos 31, no. 04 (March 30, 2021): 2150055. http://dx.doi.org/10.1142/s0218127421500553.

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This paper presents the formulation of Lagrangian function for Lorenz, Modified Lorenz and Chen systems using Lagrangian functions depending on fractional derivatives of differentiable functions, and the estimation of the conserved quantity associated with the respective systems.

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22

Nasiri, Sadollah, and Samira Bahrami. "Reality of the Wigner Functions and Quantization." Research Letters in Physics 2009 (June 1, 2009): 1–5. http://dx.doi.org/10.1155/2009/298790.

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Here we use the extended phase space formulation of quantum statistical mechanics proposed in an earlier work to define an extended lagrangian for Wigner's functions (WFs). The extended action defined by this lagrangian is a function of ordinary phase space variables. The reality condition of WFs is employed to quantize the extended action. The energy quantization is obtained as a direct consequence of the quantized action. The technique is applied to find the energy states of harmonic oscillator, particle in the box, and hydrogen atom as the illustrative examples.

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23

OHYA, TSUTOMU, and YOSHIYUKI WATABIKI. "ONE-PARTICLE-IRREDUCIBLE EFFECTIVE LAGRANGIAN IN THE SIGMA-MODEL APPROACH TO STRING THEORIES." Modern Physics Letters A 04, no. 06 (March 20, 1989): 543–55. http://dx.doi.org/10.1142/s0217732389000678.

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The effective Lagrangian of background fields is studied by using the weak background field expansion. We propose a new method of obtaining the effective Lagrangian from the vanishing of β-functions. Although the β-functions are not one-particle-irreducible at the more than five-string interaction level, we can rewrite them in a one-particle-irreducible form by redefining the background fields. The vanishing of the β-functions is then considered as the equations of motion of the modified background fields. We show explicitly that the resulting effective Lagrangian is one-particle-irreducible up to the five-string interaction level.

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24

Karabegov, Alexander. "Lagrangian fields, Calabi functions, and local symplectic groupoids." Differential Geometry and its Applications 85 (December 2022): 101933. http://dx.doi.org/10.1016/j.difgeo.2022.101933.

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25

Milinković, Darko. "Morse homology for generating functions of Lagrangian submanifolds." Transactions of the American Mathematical Society 351, no. 10 (March 8, 1999): 3953–74. http://dx.doi.org/10.1090/s0002-9947-99-02217-5.

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26

Ding, Yuqiong, and Yonghai Li. "Finite volume element method with Lagrangian cubic functions." Journal of Systems Science and Complexity 24, no. 5 (October 2011): 991–1006. http://dx.doi.org/10.1007/s11424-011-9113-1.

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27

Arnaud, Marie-Claude. "The tiered Aubry set for autonomous Lagrangian functions." Annales de l’institut Fourier 58, no. 5 (2008): 1733–59. http://dx.doi.org/10.5802/aif.2397.

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28

Huang, X. X., and X. Q. Yang. "Duality for Multiobjective Optimization via Nonlinear Lagrangian Functions." Journal of Optimization Theory and Applications 120, no. 1 (January 2004): 111–27. http://dx.doi.org/10.1023/b:jota.0000012735.86699.a1.

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29

Wang, Changyu, Qian Liu, and Biao Qu. "Global saddle points of nonlinear augmented Lagrangian functions." Journal of Global Optimization 68, no. 1 (July 28, 2016): 125–46. http://dx.doi.org/10.1007/s10898-016-0456-y.

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30

Gonen, Amnon, and Mordecai Avriel. "Duality in nonlinear programs using augmented Lagrangian functions." Journal of Mathematical Analysis and Applications 121, no. 1 (January 1987): 39–56. http://dx.doi.org/10.1016/0022-247x(87)90236-8.

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31

Huang, X. X., K. L. Teo, and X. Q. Yang. "Approximate Augmented Lagrangian Functions and Nonlinear Semidefinite Programs." Acta Mathematica Sinica, English Series 22, no. 5 (May 15, 2006): 1283–96. http://dx.doi.org/10.1007/s10114-005-0702-6.

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32

Fukuda, Ellen H., and Bruno F. Lourenço. "Exact augmented Lagrangian functions for nonlinear semidefinite programming." Computational Optimization and Applications 71, no. 2 (June 20, 2018): 457–82. http://dx.doi.org/10.1007/s10589-018-0017-z.

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33

Wang, Han, and Oliver Bühler. "Anisotropic Statistics of Lagrangian Structure Functions and Helmholtz Decomposition." Journal of Physical Oceanography 51, no. 5 (May 2021): 1375–93. http://dx.doi.org/10.1175/jpo-d-20-0199.1.

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AbstractWe present a new method to estimate second-order horizontal velocity structure functions, as well as their Helmholtz decomposition into rotational and divergent components, from sparse data collected along Lagrangian observations. The novelty compared to existing methods is that we allow for anisotropic statistics in the velocity field and also in the collection of the Lagrangian data. Specifically, we assume only stationarity and spatial homogeneity of the data and that the cross covariance between the rotational and divergent flow components is either zero or a function of the separation distance only. No further assumptions are made and the anisotropy of the underlying flow components can be arbitrarily strong. We demonstrate our new method by testing it against synthetic data and applying it to the Lagrangian Submesoscale Experiment (LASER) dataset. We also identify an improved statistical angle-weighting technique that generally increases the accuracy of structure function estimations in the presence of anisotropy.

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34

Saunders, D. J. "Homogeneous variational problems and Lagrangian sections." Communications in Mathematics 24, no. 2 (December 1, 2016): 115–23. http://dx.doi.org/10.1515/cm-2016-0008.

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Abstract We define a canonical line bundle over the slit tangent bundle of a manifold, and define a Lagrangian section to be a homogeneous section of this line bundle. When a regularity condition is satisfied the Lagrangian section gives rise to local Finsler functions. For each such section we demonstrate how to construct a canonically parametrized family of geodesics, such that the geodesics of the local Finsler functions are reparametrizations.

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35

Valluri, S. R., D. R. Lamm, and W. J. Mielniczuk. "Applications of the representation of the Heisenberg–Euler Lagrangian by means of special functions." Canadian Journal of Physics 71, no. 7-8 (July 1, 1993): 389–97. http://dx.doi.org/10.1139/p93-060.

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A convenient series representation for the real part of the Heisenberg–Euler Lagrangian density of quantum electrodynamics for arbitrary nonvanishing electric fields, E, and magnetic fields, B, has been previously provided by Mielniczuk. Using this representation, numerical information for the Lagrangian is presented for the range [Formula: see text] and [Formula: see text] (subscript cr stands for critical) with the electric and magnetic fields parallel and Ecr ≈ 1.7 × 1016 V cm−1 and Bcr ≈ 4.4 × 1013 G. It was found that for a fixed electric field, the Lagrangian is monotonically increasing with increasing magnetic field strength. However, for a fixed magnetic field, the Lagrangian exhibits a positively valued maximum before turning monotonically decreasing with increasing electric field strength. Further, the series representation is extended to the case of vanishing electric or magnetic field. Numerical results for these special cases are in very close agreement with previous results, which indicated a maximum value for the Lagrangian density for B = 0 at E/Ecr ≈ 3. Also, the techniques developed for deriving the real part of the Heisenberg–Euler Lagrangian are applied to the imaginary part to deduce a similar, convenient series representation that agrees with the previous results derived by others for the special case of a vanishing magnetic field. Possible applications of this Lagrangian to quantum chromodynamics are discussed. This series representation will be of use in calculations of a quantum-electrodynamical field energy density in the absence of real charges, and for calculations of polarization and magnetization of the vacuum. More accurate calculations of the cross-section scattering of light by light in the presence of a constant, homogeneous magnetic and (or) electric field are possible with the aid of this series representation.

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36

Kosmas, Odysseas. "Energy Minimization Scheme for Split Potential Systems Using Exponential Variational Integrators." Applied Mechanics 2, no. 3 (June 24, 2021): 431–41. http://dx.doi.org/10.3390/applmech2030024.

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In previous works we developed a methodology of deriving variational integrators to provide numerical solutions of systems having oscillatory behavior. These schemes use exponential functions to approximate the intermediate configurations and velocities, which are then placed into the discrete Lagrangian function characterizing the physical system. We afterwards proved that, higher order schemes can be obtained through the corresponding discrete Euler–Lagrange equations and the definition of a weighted sum of “continuous intermediate Lagrangians” each of them evaluated at an intermediate time node. In the present article, we extend these methods so as to include Lagrangians of split potential systems, namely, to address cases when the potential function can be decomposed into several components. Rather than using many intermediate points for the complete Lagrangian, in this work we introduce different numbers of intermediate points, resulting within the context of various reliable quadrature rules, for the various potentials. Finally, we assess the accuracy, convergence and computational time of the proposed technique by testing and comparing them with well known standards.

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37

Farooq, Muhammad Umar, Chaudry Masood Khalique, and Fazal M. Mahomed. "First Integrals of Two-Dimensional Dynamical Systems via Complex Lagrangian Approach." Symmetry 11, no. 10 (October 4, 2019): 1244. http://dx.doi.org/10.3390/sym11101244.

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The aim of the present work is to classify the Noether-like operators of two-dimensional physical systems whose dynamics is governed by a pair of Lane-Emden equations. Considering first-order Lagrangians for these systems, we construct corresponding first integrals. It is seen that for a number of forms of arbitrary functions appearing in the set of equations, the Noether-like operators also fulfill the classical Noether symmetry condition for the pairs of real Lagrangians and the generated first integrals are reminiscent of those we obtain from the complex Lagrangian approach. We also investigate the cases in which the underlying systems are reducible via quadrature. We derive some interesting results about the nonlinear systems under consideration and also find that the algebra of Noether-like operators is Abelian in a few cases.

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38

Fung, J. C. H. "The Dependence of the Time Scale of Relative Lagrangian Motion on the Initial Separation." Journal of Applied Mechanics 65, no. 1 (March 1, 1998): 204–8. http://dx.doi.org/10.1115/1.2789027.

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Kinematic simulation of homogeneous isotropic turbulence are used to compute Lagrangian statistics of turbulence and, in particular, its time scales. The computed pseudo-Lagrangian velocity autocorrelation functions Rˆ11L(l,t) compare well with theory for a small initial separation l and short time t. We also demonstrate the feasibility of using kinematic simulation as a means of constructing Lagrangian statistics.

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Favretti, Marco. "Lagrangian Submanifolds of Symplectic Structures Induced by Divergence Functions." Entropy 22, no. 9 (September 3, 2020): 983. http://dx.doi.org/10.3390/e22090983.

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Divergence functions play a relevant role in Information Geometry as they allow for the introduction of a Riemannian metric and a dual connection structure on a finite dimensional manifold of probability distributions. They also allow to define, in a canonical way, a symplectic structure on the square of the above manifold of probability distributions, a property that has received less attention in the literature until recent contributions. In this paper, we hint at a possible application: we study Lagrangian submanifolds of this symplectic structure and show that they are useful for describing the manifold of solutions of the Maximum Entropy principle.

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40

Vérine, Alexandre. "Bohr–Sommerfeld Lagrangian submanifolds as minima of convex functions." Journal of Symplectic Geometry 18, no. 1 (2020): 333–53. http://dx.doi.org/10.4310/jsg.2020.v18.n1.a9.

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41

Singh, C., D. Bhatia, and N. Rueda. "Duality in nonlinear multiobjective programming using augmented Lagrangian functions." Journal of Optimization Theory and Applications 88, no. 3 (March 1996): 659–70. http://dx.doi.org/10.1007/bf02192203.

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42

BENZI, R., L. BIFERALE, R. FISHER, D. Q. LAMB, and F. TOSCHI. "Inertial range Eulerian and Lagrangian statistics from numerical simulations of isotropic turbulence." Journal of Fluid Mechanics 653 (June 2, 2010): 221–44. http://dx.doi.org/10.1017/s002211201000056x.

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We present a study of Eulerian and Lagrangian statistics from a high-resolution numerical simulation of isotropic and homogeneous turbulence using the FLASH code, with an estimated Taylor microscale Reynolds number of around 600. Statistics are evaluated over a data set with 18563 spatial grid points and with 2563 = 16.8 million particles, followed for about one large-scale eddy turnover time. We present data for the Eulerian and Lagrangian structure functions up to the tenth order. We analyze the local scaling properties in the inertial range. The Eulerian velocity field results show good agreement with previous data and confirm the puzzling differences previously found between the scaling of the transverse and the longitudinal structure functions. On the other hand, accurate measurements of sixth-and-higher-order Lagrangian structure functions allow us to highlight some discrepancies from earlier experimental and numerical results. We interpret this result in terms of a possible contamination from the viscous scale, which may have affected estimates of the scaling properties in previous studies. We show that a simple bridge relation based on a multifractal theory is able to connect scaling properties of both Eulerian and Lagrangian observables, provided that the small differences between intermittency of transverse and longitudinal Eulerian structure functions are properly considered.

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43

Pommerening, Florian, Gabriele Röger, Malte Helmert, Hadrien Cambazard, Louis-Martin Rousseau, and Domenico Salvagnin. "Lagrangian Decomposition for Optimal Cost Partitioning." Proceedings of the International Conference on Automated Planning and Scheduling 29 (May 25, 2021): 338–47. http://dx.doi.org/10.1609/icaps.v29i1.3496.

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Optimal cost partitioning of classical planning heuristics has been shown to lead to excellent heuristic values but is often prohibitively expensive to compute. Lagrangian decomposition and Lagrangian relaxation are classical tools in mathematical programming that apply to optimization problems with a special block structure. We analyze the application of Lagrangian decomposition to cost partitioning in the context of operator-counting heuristics and interpret Lagrangian multipliers as cost functions for the combined heuristics. This allows us to view the computation of an optimal cost partitioning as an iterative process that can be seeded with any cost partitioning and improves over time. We derive an anytime algorithm to compute an optimal non-negative cost partitioning of abstraction heuristics without involving an LP solver. In each iteration, the computation reduces to independent shortest path problems in all abstractions. Finally, we discuss the extension to general cost functions.

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44

Innami, Nobuhiro. "Natural Lagrangian systems without conjugate points." Ergodic Theory and Dynamical Systems 14, no. 1 (March 1994): 169–80. http://dx.doi.org/10.1017/s0143385700007781.

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AbstractThe variation vector fields through extremals of the variational principles of natural Lagrangian functions satisfy the equation of Jacobi type. By making use of the Jacobi equation we obtain the estimates of measure-theoretic entropy for natural Lagrangian systems without conjugate points.

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45

Ren, Yong-Hong. "Second-Order Multiplier Iteration Based on a Class of Nonlinear Lagrangians." Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/210284.

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Nonlinear Lagrangian algorithm plays an important role in solving constrained optimization problems. It is known that, under appropriate conditions, the sequence generated by the first-order multiplier iteration converges superlinearly. This paper aims at analyzing the second-order multiplier iteration based on a class of nonlinear Lagrangians for solving nonlinear programming problems with inequality constraints. It is suggested that the sequence generated by the second-order multiplier iteration converges superlinearly with order at least two if in addition the Hessians of functions involved in problem are Lipschitz continuous.

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Bektaş, Burcu, Marilena Moruz, Joeri Van der Veken, and Luc Vrancken. "Lagrangian submanifolds of the nearly Kähler 𝕊3 × 𝕊3 from minimal surfaces in 𝕊3." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 149, no. 03 (December 27, 2018): 655–89. http://dx.doi.org/10.1017/prm.2018.43.

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AbstractWe study non-totally geodesic Lagrangian submanifolds of the nearly Kähler 𝕊3 × 𝕊3 for which the projection on the first component is nowhere of maximal rank. We show that this property can be expressed in terms of the so-called angle functions and that such Lagrangian submanifolds are closely related to minimal surfaces in 𝕊3. Indeed, starting from an arbitrary minimal surface, we can construct locally a large family of such Lagrangian immersions, including one exceptional example. We also show that locally all such Lagrangian submanifolds can be obtained in this way.

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McKeon, D. G. C. "A spin-two gauge theory." Canadian Journal of Physics 67, no. 8 (August 1, 1989): 743–46. http://dx.doi.org/10.1139/p89-130.

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We formulate a Lagrangian for a free massless spin-two field hμν that is invariant under the transformation hμν→hμν + ∂μfν + ∂νfμ This is then coupled to a massive scalar field [Formula: see text]. In two dimensions the classical Lagrangian for hμν vanishes identically, and hence the kinetic term in the effective Lagrangian for hμν reduces to the gauge fixing Lagrangian. The theory is renormalizable in two dimensions provided we mix all operators of the form [Formula: see text]. One-loop two-point functions are computed using operator regularization.

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Dittrich, Walter. "The Heisenberg–Euler Lagrangian as an example of an effective field theory." International Journal of Modern Physics A 29, no. 26 (October 16, 2014): 1430052. http://dx.doi.org/10.1142/s0217751x1430052x.

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We review the beginning of the effective Lagrangian in QED that was first introduced in the literature by W. Heisenberg and H. Euler in 1936. Deviating from their way of calculating the one-loop effective correction to the classical Maxwell Lagrangian, we use Green's functions and adopt the Fock–Schwinger proper-time method. The important role of the Heisenberg–Euler effective Lagrangian is explicitly demonstrated for low-energy photon–photon processes.

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Bandyopadhyay, Abhijit, and Anirban Chatterjee. "Realizing interactions between dark matter and dark energy using k-essence cosmology." Modern Physics Letters A 34, no. 27 (September 6, 2019): 1950219. http://dx.doi.org/10.1142/s0217732319502195.

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In this paper, we exploit dynamics of a [Formula: see text]-essence scalar field to realize interactions between dark components of universe resulting in an evolution consistent with observed features of late-time phase of cosmic evolution. Stress–energy tensor corresponding to a [Formula: see text]-essence Lagrangian [Formula: see text] (where [Formula: see text]) is shown to be equivalent to an ideal fluid with two components having same equation of state. Stress–energy tensor of one of the components may be generated from a constant potential [Formula: see text]-essence Lagrangian of form [Formula: see text] ([Formula: see text] constant) and that of other from another Lagrangian of form [Formula: see text] with [Formula: see text]. We have shown that the unified dynamics of dark matter and dark energy described by a single scalar field [Formula: see text] driven by a [Formula: see text]-essence Lagrangian [Formula: see text] may be viewed in terms of diffusive interactions between the two hypothetical fluid components “1” and “2” with stress–energy tensors equivalent to that of Lagrangians [Formula: see text] and [Formula: see text], respectively. The energy transfer between the fluid components is determined by functions [Formula: see text], [Formula: see text] and their derivatives. Such a realization is shown to be consistent with the Supernova Ia data with certain constraints on the temporal behavior of [Formula: see text]-essence potential [Formula: see text]. We have described a methodology to obtain such constraints.

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Konstandin, Lukas, Christoph Federrath, Ralf S. Klessen, and Wolfram Schmidt. "Statistical properties of supersonic turbulence in the Lagrangian and Eulerian frameworks." Journal of Fluid Mechanics 692 (December 19, 2011): 183–206. http://dx.doi.org/10.1017/jfm.2011.503.

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AbstractWe present a systematic study of the influence of different forcing types on the statistical properties of supersonic, isothermal turbulence in both the Lagrangian and Eulerian frameworks. We analyse a series of high-resolution, hydrodynamical grid simulations with Lagrangian tracer particles and examine the effects of solenoidal (divergence-free) and compressive (curl-free) forcing on structure functions, their scaling exponents, and the probability density functions of the gas density and velocity increments. Compressively driven simulations show significantly larger density contrast, more intermittent behaviour, and larger fractal dimension of the most dissipative structures at the same root mean square Mach number. We show that the absolute values of Lagrangian and Eulerian structure functions of all orders in the integral range are only a function of the root mean square Mach number, but independent of the forcing. With the assumption of a Gaussian distribution for the probability density function of the velocity increments for large scales, we derive a model that describes this behaviour.

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