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Artykuły w czasopismach na temat „Lagrangian functions”

Autor: Grafiati

Data publikacji: 4 czerwca 2021

Data aktualizacji: 1 sierpnia 2024

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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

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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

Ambot, Amelia A. P., Herry F. Lalus, and Hartoyo Yudhawardana. "ANALISIS LAGRANGIAN NULL NONSTANDAR DAN FUNGSI GAUGE UNTUK HUKUM INERSIA NEWTON : SEBUAH REVIEW." JOURNAL ONLINE OF PHYSICS 9, no. 1 (November 2, 2023): 6–14. http://dx.doi.org/10.22437/jop.v9i1.25909.

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This paper describes a review of a journal entitled 'Nonstandard Null Lagrangians and Gauge Functions for Newtonian Law of Inertia' which discusses Nonstandard Lagrangian Null solutions for Newton's Law of Inertia. The purpose of this study is to present in detail the Lagrangian Formalism method for generating Nonstandard Lagrangian Null and its Gauge function for Newton's Law of Inertia, as well as the role of action invariant in generating Lagrangian Null and Exact Gauge functions, by deriving a one-dimensional oscillator arm using the basic Lagrangian equations. The Nonstandard Null Lagrang

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4

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

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5

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 le

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6

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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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 functi

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8

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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9

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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10

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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11

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 Lagrangian

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12

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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13

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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14

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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15

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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16

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

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17

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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18

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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19

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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20

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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21

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 e

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22

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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23

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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24

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 u

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25

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 separa

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26

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

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27

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

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28

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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29

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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30

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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31

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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32

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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33

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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34

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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35

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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36

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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37

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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38

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 investi

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39

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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40

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 agr

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41

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 partiti

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42

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

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43

Bektaş, Burcu, Marilena Moruz, Joeri Van der Veken та 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, № 03 (27 грудня 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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44

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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45

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 [

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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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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 describin

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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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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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