Journal articles: 'Lagrangian points'

1

Linton, J. Oliver. "The Lagrangian points." Physics Education 52, no. 2 (2017): 023002. http://dx.doi.org/10.1088/1361-6552/aa568d.

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Hurkat, Pankaj, Nisha Patel, and Pruthul Desai. "Lagrangian Equilibrium Points." Resonance 30, no. 4 (2025): 477–503. https://doi.org/10.1007/s12045-025-1775-4.

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Innami, Nobuhiro. "Natural Lagrangian systems without conjugate points." Ergodic Theory and Dynamical Systems 14, no. 1 (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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Popescu, Mihai. "Optimal transfer from the lagrangian points." Acta Astronautica 12, no. 4 (1985): 225–28. http://dx.doi.org/10.1016/0094-5765(85)90036-0.

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Moyaux, P. M., and L. Vandembroucq. "Lagrangian intersections, critical points and Qcategory." Mathematische Zeitschrift 246, no. 1-2 (2004): 85–103. http://dx.doi.org/10.1007/s00209-003-0583-2.

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Naz, R., I. Naeem, and F. M. Mahomed. "First Integrals for Two Linearly Coupled Nonlinear Duffing Oscillators." Mathematical Problems in Engineering 2011 (2011): 1–14. http://dx.doi.org/10.1155/2011/831647.

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We investigate Noether and partial Noether operators of point type corresponding to a Lagrangian and a partial Lagrangian for a system of two linearly coupled nonlinear Duffing oscillators. Then, the first integrals with respect to Noether and partial Noether operators of point type are obtained explicitly by utilizing Noether and partial Noether theorems for the system under consideration. Moreover, if the partial Euler-Lagrange equations are independent of derivatives, then the partial Noether operators become Noether point symmetry generators for such equations. The difference arises in the gauge terms due to Lagrangians being different for respective approaches. This study points to new ways of constructing first integrals for nonlinear equations without regard to a Lagrangian. We have illustrated it here for nonlinear Duffing oscillators.

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Kosmas, Odysseas. "Energy Minimization Scheme for Split Potential Systems Using Exponential Variational Integrators." Applied Mechanics 2, no. 3 (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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Zhao, Wen Ling, Jing Zhang, and Jin Chuan Zhou. "Local Saddle Points Theory of a New Augmented Lagrangians." Advanced Materials Research 121-122 (June 2010): 123–27. http://dx.doi.org/10.4028/www.scientific.net/amr.121-122.123.

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In connection with Problem (P) with both the equality constraints and inequality constraints, we introduce a new augmented lagrangian function. We establish the existence of local saddle point under the weaker sufficient second order condition, discuss the relationships between local optimal solution of the primal problem and local saddle point of the augmented lagrangian function.

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Xia, Zhihong. "Homoclinic points and intersections of Lagrangian submanifold." Discrete & Continuous Dynamical Systems - A 6, no. 1 (2000): 243–53. http://dx.doi.org/10.3934/dcds.2000.6.243.

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Giuliatti Winter, S. M., O. C. Winter, and D. C. Mourão. "Peculiar trajectories around the Lagrangian equilateral points." Physica D: Nonlinear Phenomena 225, no. 1 (2007): 112–18. http://dx.doi.org/10.1016/j.physd.2006.10.006.

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Kaas, Eigil, Annette Guldberg, and Philippe Lopez. "A Lagrangian Advection Scheme Using Tracer Points." Atmosphere-Ocean 35, sup1 (1997): 171–94. http://dx.doi.org/10.1080/07055900.1997.9687347.

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Eapen, Roshan Thomas, and Ram Krishan Sharma. "Mars interplanetary trajectory design via Lagrangian points." Astrophysics and Space Science 353, no. 1 (2014): 65–71. http://dx.doi.org/10.1007/s10509-014-2012-x.

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13

Zhao, Wenling, Jing Zhang, and Jinchuan Zhou. "Existence of Local Saddle Points for a New Augmented Lagrangian Function." Mathematical Problems in Engineering 2010 (2010): 1–13. http://dx.doi.org/10.1155/2010/324812.

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We give a new class of augmented Lagrangian functions for nonlinear programming problem with both equality and inequality constraints. The close relationship between local saddle points of this new augmented Lagrangian and local optimal solutions is discussed. In particular, we show that a local saddle point is a local optimal solution and the converse is also true under rather mild conditions.

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Bolton, J., C. Rodriguez Montealegre, and L. Vrancken. "Characterizing warped-product Lagrangian immersions in complex projective space." Proceedings of the Edinburgh Mathematical Society 52, no. 2 (2009): 273–86. http://dx.doi.org/10.1017/s0013091507000922.

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AbstractStarting from two Lagrangian immersions and a horizontal curve in S3(1), it is possible to construct a new Lagrangian immersion, which we call a warped-product Lagrangian immersion. In this paper, we find two characterizations of warped-product Lagrangian immersions. We also investigate Lagrangian submanifolds which attain at every point equality in the improved version of Chen's inequality for Lagrangian submanifolds of ℂPn(4) as discovered by Opreaffi We show that, for n≥4, an n-dimensional Lagrangian submanifold in ℂPn(4) for which equality is attained at all points is necessarily minimal.

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15

Montesinos, Matías, Juan Garrido-Deutelmoser, Johan Olofsson, et al. "Dust trapping around Lagrangian points in protoplanetary disks." Astronomy & Astrophysics 642 (October 2020): A224. http://dx.doi.org/10.1051/0004-6361/202038758.

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Aims. Trojans are defined as objects that share the orbit of a planet at the stable Lagrangian points L4 and L5. In the Solar System, these bodies show a broad size distribution ranging from micrometer (μm) to centimeter (cm) particles (Trojan dust) and up to kilometer (km) rocks (Trojan asteroids). It has also been theorized that earth-like Trojans may be formed in extra-solar systems. The Trojan formation mechanism is still under debate, especially theories involving the effects of dissipative forces from a viscous gaseous environment. Methods. We perform hydro-simulations to follow the evolution of a protoplanetary disk with an embedded 1–10 Jupiter-mass planet. On top of the gaseous disk, we set a distribution of μm–cm dust particles interacting with the gas. This allows us to follow dust dynamics as solids get trapped around the Lagrangian points of the planet. Results. We show that large vortices generated at the Lagrangian points are responsible for dust accumulation, where the leading Lagrangian point L4 traps a larger amount of submillimeter (submm) particles than the trailing L5, which traps mostly mm–cm particles. However, the total bulk mass, with typical values of ~Mmoon, is more significant in L5 than in L4, in contrast to what is observed in the current Solar System a few gigayears later. Furthermore, the migration of the planet does not seem to affect the reported asymmetry between L4 and L5. Conclusions. The main initial mass reservoir for Trojan dust lies in the same co-orbital path of the planet, while dust migrating from the outer region (due to drag) contributes very little to its final mass, imposing strong mass constraints for the in situ formation scenario of Trojan planets.

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Prado, Antonio F. B. A. "Traveling between the Lagrangian points and the Earth." Acta Astronautica 39, no. 7 (1996): 483–86. http://dx.doi.org/10.1016/s0094-5765(97)85428-8.

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17

Freire, R. S., M. V. P. Garcia, and F. A. Tal. "Instability of equilibrium points of some Lagrangian systems." Journal of Differential Equations 245, no. 2 (2008): 490–504. http://dx.doi.org/10.1016/j.jde.2008.02.016.

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Ekholm, Tobias, Yakov Eliashberg, Emmy Murphy, and Ivan Smith. "Constructing exact Lagrangian immersions with few double points." Geometric and Functional Analysis 23, no. 6 (2013): 1772–803. http://dx.doi.org/10.1007/s00039-013-0243-6.

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19

Liu, Qian, Wan Mei Tang, and Xin Min Yang. "Properties of saddle points for generalized augmented Lagrangian." Mathematical Methods of Operations Research 69, no. 1 (2008): 111–24. http://dx.doi.org/10.1007/s00186-008-0213-1.

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

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Basto-Gonçalves, J. "Inflection points and asymptotic lines on Lagrangian surfaces." Differential Geometry and its Applications 35 (August 2014): 9–29. http://dx.doi.org/10.1016/j.difgeo.2014.04.012.

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22

COULAUD, O., E. SONNENDRÜCKER, E. DILLON, P. BERTRAND, and A. GHIZZO. "Parallelization of semi-Lagrangian Vlasov codes." Journal of Plasma Physics 61, no. 3 (1999): 435–48. http://dx.doi.org/10.1017/s0022377899007527.

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We describe the parallel implementation of semi-Lagrangian Vlasov solvers, which are an alternative to particle-in-cell (PIC) simulations for the numerical investigation of the behaviour of charged particles in their self-consistent electromagnetic fields. The semi-Lagrangian method, which couples the Lagrangian and Eulerian points of view, is particularly interesting on parallel computers, since the solution is computed on grid points, the number of which remains constant in time on each processor, unlike the number of particles in PIC simulations, and thus greatly simplifies the parallelization process.

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23

Guo, Daniel X. "Semi-Lagrangian forward methods for some time-dependent nonlinear partial differential equations." Electronic Journal of Differential Equations, Conference 26 (August 25, 2022): 97–113. http://dx.doi.org/10.58997/ejde.conf.26.g1.

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In this article, we study one-step Semi-Lagrangian forward method for computing the numerical solutions of time-dependent nonlinear partial differential equations with initial and boundary conditions in one space dimension. Comparing with classic Semi-Lagrangian method, this method is more straight forward to analyze and implement. This method is based on Lagrangian trajectory from the departure points (regular nodes) to the arrival points by Runge-Kutta methods. The arrival points are traced forward from the departure points along the trajectory of the path. Most likely the arrival points are not on the regular grid nodes. However, it is convenient to approximate the high order derivative terms in spatial dimension on regular nodes. The convergence and stability are studied for the explicit methods. The numerical examples show that those methods work very efficient for the time-dependent nonlinear partial differential equations. For more information see https://ejde.math.txstate.edu/conf-proc/26/g1/abstr.html

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Stefanov, Stefan M. "Well-Posedness and Primal-Dual Analysis of Some Convex Separable Optimization Problems." Advances in Operations Research 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/279030.

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We focus on some convex separable optimization problems, considered by the author in previous papers, for which problems, necessary and sufficient conditions or sufficient conditions have been proved, and convergent algorithms of polynomial computational complexity have been proposed for solving these problems. The concepts of well-posedness of optimization problems in the sense of Tychonov, Hadamard, and in a generalized sense, as well as calmness in the sense of Clarke, are discussed. It is shown that the convex separable optimization problems under consideration are calm in the sense of Clarke. The concept of stability of the set of saddle points of the Lagrangian in the sense of Gol'shtein is also discussed, and it is shown that this set is not stable for the “classical” Lagrangian. However, it turns out that despite this instability, due to the specificity of the approach, suggested by the author for solving problems under consideration, it is not necessary to use modified Lagrangians but only the “classical” Lagrangians. Also, a primal-dual analysis for problems under consideration in view of methods for solving them is presented.

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25

Upadhyay, Balendu Bhooshan, Shivani Sain, and Ioan Stancu-Minasian. "Lagrange Duality and Saddle-Point Optimality Conditions for Nonsmooth Interval-Valued Multiobjective Semi-Infinite Programming Problems with Vanishing Constraints." Axioms 13, no. 9 (2024): 573. http://dx.doi.org/10.3390/axioms13090573.

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Thisarticle deals with a class of nonsmooth interval-valued multiobjective semi-infinite programming problems with vanishing constraints (NIMSIPVC). We introduce the VC-Abadie constraint qualification (VC-ACQ) for NIMSIPVC and employ it to establish Karush–Kuhn–Tucker (KKT)-type necessary optimality conditions for the considered problem. Regarding NIMSIPVC, we formulate interval-valued weak vector, as well as interval-valued vector Lagrange-type dual and scalarized Lagrange-type dual problems. Subsequently, we establish the weak, strong, and converse duality results relating to the primal problem, NIMSIPVC, and the corresponding dual problems. Moreover, we introduce the notion of saddle points for the interval-valued vector Lagrangian and scalarized Lagrangian of NIMSIPVC. Furthermore, we derive the saddle-point optimality criteria for NIMSIPVC by establishing the relationships between the solutions of NIMSIPVC and the saddle points of the corresponding Lagrangians of NIMSIPVC, under convexity assumptions. Non-trivial illustrative examples are provided to demonstrate the validity of the established results. The results presented in this paper extend the corresponding results derived in the existing literature from smooth to nonsmooth optimization problems, and we further generalize them for interval-valued multiobjective semi-infinite programming problems with vanishing constraints.

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Ritesh, Arohan, and Krishan Sharma Ram. "Periodic orbits in the planar restricted photo-gravitational problem when the smaller primary is an oblate spheroid." Indian Journal of Science and Technology 13, no. 16 (2020): 1630–40. https://doi.org/10.17485/IJST/v13i16.401.

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Abstract Background/Objectives: This study deals wit h the stationary solutions of the planar circular restricted three-body problem when the more massive primary is a source of radiation and the smaller primary is an oblate spheroid with its equatorial plane coincident with the plane of motion. The objective is to study the location of the Lagrangian points and to find the values of critical mass. Also, to study the periodic orbits around the Lagrangian points. Methods: A new mean motion expression by including the secular perturbation due to oblateness utilized by(1,2) is used in the present studies. The characteristic roots are obtained by linearizing the equation of the motion around the Lagrangian points. Findings: The critical mass parameter µcrit(3,4) , which decreases radiation force, whereas it increases with oblateness when we consider the value of new mean motion. Through special choice of initial conditions, retrograde elliptical periodic orbits exist for the case µ = µcrit, whose eccentricity increases with oblateness and decreases with radiation force for non-zero oblateness, although there is slight variation in L2 location. Keywords: Restricted three body problem; Lagrangian points; Eccentricity; Oblateness; Critical mass; Radiation force; Mean motion.

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Arantza Jency, A., and Ram Krishan Sharma. "Location and stability of the triangular Lagrange points in photo-gravitational elliptic restricted three body problem with the more massive primary as an oblate spheroid." International Journal of Advanced Astronomy 7, no. 2 (2019): 57. http://dx.doi.org/10.14419/ijaa.v7i2.29814.

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The triangular Lagrangian points of the elliptic restricted three-body problem (ERTBP) with oblate and radiating more massive primary are studied. The mean motion equation used here is different from the ones employed in many studies on the perturbed ERTBP. The effect of oblateness on the mean motion equation varies. This change influences the location and stability of the triangular Lagrangian points. The points tend to shift in the y-direction. The influence of the oblateness on the critical mass ratio is also altered. But the eccentricity limit for stability remains the same.

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McGregor, John L. "Economical Determination of Departure Points for Semi-Lagrangian Models." Monthly Weather Review 121, no. 1 (1993): 221–30. http://dx.doi.org/10.1175/1520-0493(1993)121<0221:edodpf>2.0.co;2.

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de la Barre, C. M., W. M. Kaula, and F. Varadi. "A Study of Orbits near Saturn's Triangular Lagrangian Points." Icarus 121, no. 1 (1996): 88–113. http://dx.doi.org/10.1006/icar.1996.0073.

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Zhao, Chen, Ziyan Luo, Weiyue Li, Houduo Qi, and Naihua Xiu. "Lagrangian duality and saddle points for sparse linear programming." Science China Mathematics 62, no. 10 (2019): 2015–32. http://dx.doi.org/10.1007/s11425-018-9546-9.

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31

Wood, Nigel, Andrew Staniforth, and Andy White. "Determining near-boundary departure points in semi-Lagrangian models." Quarterly Journal of the Royal Meteorological Society 135, no. 644 (2009): 1890–96. http://dx.doi.org/10.1002/qj.478.

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32

Goryunov, Victor, and Katy Gallagher. "On planar caustics." Journal of Knot Theory and Its Ramifications 25, no. 12 (2016): 1642004. http://dx.doi.org/10.1142/s0218216516420049.

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We study local invariants of planar caustics, that is, invariants of Lagrangian maps from surfaces to [Formula: see text] whose increments in generic homotopies are determined entirely by diffeomorphism types of local bifurcations of the caustics. Such invariants are dual to trivial codimension 1 cycles supported on the discriminant in the space [Formula: see text] of the Lagrangian maps. We obtain a description of the spaces of the discriminantal cycles (possibly non-trivial) for the Lagrangian maps of an arbitrary surface, both for the integer and mod 2 coefficients. It is shown that all integer local invariants of caustics of Lagrangian maps without corank 2 points are essentially exhausted by the numbers of various singular points of the caustics and the Ohmoto–Aicardi linking invariant of ordinary maps. As an application, we use the discriminantal cycles to establish non-contractibility of certain loops in [Formula: see text].

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Hariandja, Binsar Halomoan. "Finite Element Analysis of Nonlinear Contact Problems with Mixed Eulerian-Lagrangian Description." PRESUNIVE CIVIL ENGINEERING JOURNAL 1, no. 2 (2023): 37. http://dx.doi.org/10.33021/pcej.v1i2.4417.

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This study deals with analysis of structures using mixed Eulerian-Lagrangian description. Apart from generally used Lagrangian description which uses initial configuration as reference, the newly proposed uses one of actual configuration as reference. Therefore, the total displacement is decomposed into two portions, i.e., the portion covering displacement from initial into referential configuration called Eulerian displacement, and the portion covering displacement from referential configuration into current configuration called Lagrangian displacement. The new technique is suitable to be applied to several classes of structures such as frictional contact and tensile structures. Eulerian displacement is used to represent relative displacements between material points paired in a nodal point in which slip mode occurs, while Lagrangian displacement is used to represent mutual displacement of material points paired in a nodal point in which stick mode occurs. The method was applied to certain contact problems, and the results obtained agreed fairly well with existing results found in references.

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ARCOSTANZO, MARC. "The integrability of symplectic twist maps without conjugate points." Ergodic Theory and Dynamical Systems 41, no. 1 (2019): 48–65. http://dx.doi.org/10.1017/etds.2019.49.

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It is proved that a symplectic twist map of the cotangent bundle $T^{\ast }\mathbb{T}^{d}$ of the $d$-dimensional torus that is without conjugate points is $C^{0}$-integrable, that is $T^{\ast }\mathbb{T}^{d}$ is foliated by a family of invariant $C^{0}$ Lagrangian graphs.

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Zhou, Zhi-Ang. "ϵ-Henig Saddle Points and Duality of Set-Valued Optimization Problems in Real Linear Spaces". Scientific World Journal 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/403642.

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We studyϵ-Henig saddle points and duality of set-valued optimization problems in the setting of real linear spaces. Firstly, an equivalent characterization ofϵ-Henig saddle point of the Lagrangian set-valued map is obtained. Secondly, under the assumption of the generalized cone subconvexlikeness of set-valued maps, the relationship between theϵ-Henig saddle point of the Lagrangian set-valued map and theϵ-Henig properly efficient element of the set-valued optimization problem is presented. Finally, some duality theorems are given.

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James Raj, Xavier, and Bhola Ishwar. "Diagonalization of Hamiltonian in the photogravitation-al restricted three body problem with P-R drag." International Journal of Advanced Astronomy 5, no. 2 (2017): 79. http://dx.doi.org/10.14419/ijaa.v5i2.7931.

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In this paper, restricted, three-body problem (RTBP) is generalised to study the non-linear stability of equilibrium points in the photogravitational RTBP with P-R drag. In the present study, both primaries are considered as a source of radiation and effect of P-R drag. Hence the problem will contain four parameters q1, q2, W1 and W2. At first, the Lagrangian and the Hamiltonian of the problem were computed, then the Lagrangian function is expanded in power series of the coordinates of the triangular equilibrium points x and y. Lastly, diagonalized the quadratic term of the Hamiltonian of the problem, which is obtained by expanding original Lagrangian or Hamiltonian by Taylor's series about triangular equilibrium point. Finally, the study concluded that the diagonalizable Hamiltonian is H2=ω1I1-ω2I2.

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Yoshiyasu, Toru. "On Lagrangian embeddings into the complex projective spaces." International Journal of Mathematics 27, no. 05 (2016): 1650044. http://dx.doi.org/10.1142/s0129167x16500440.

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We prove that for any closed orientable connected [Formula: see text]-manifold [Formula: see text] and any Lagrangian immersion of the connected sum [Formula: see text] either into the complex projective [Formula: see text]-space [Formula: see text] or into the product [Formula: see text] of the complex projective line and the complex projective plane, there exists a Lagrangian embedding which is homotopic to the initial Lagrangian immersion. To prove this, we show that Eliashberg–Murphy’s [Formula: see text]-principle for Lagrangian embeddings with a concave Legendrian boundary and Ekholm–Eliashberg–Murphy–Smith’s [Formula: see text]-principle for self-transverse Lagrangian immersions with the minimal or near-minimal number of double points hold for six-dimensional simply connected compact symplectic manifolds.

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Aycan, Cansel, and Simge Şimşek. "Time-Dependent Lagrangian Energy Systems on Supermanifolds with Graph Bundles." Journal of Mathematics 2021 (April 28, 2021): 1–17. http://dx.doi.org/10.1155/2021/5528123.

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The aim of this article is firstly to improve time-dependent Lagrangian energy equations using the super jet bundles on supermanifolds. Later, we adapted this study to the graph bundle. Thus, we created a graph bundle by examining the graph manifold structure in superspace. The geometric structures obtained for the mechanical energy system with superbundle coordinates were reexamined with the graph bundle coordinates. Thus, we were able to calculate the energy that occurs during the motion of a particle when we examine this motion with graph points. The supercoordinates on the superbundle structure of supermanifolds have been given for body and soul and also even and odd dimensions. We have given the geometric interpretation of this property in coordinates for the movement on graph points. Lagrangian energy equations have been applied to the presented example, and the advantage of examining the movement with graph points was presented. In this article, we will use the graph theory to determine the optimal motion, velocity, and energy of the particle, due to graph points. This study showed a physical application and interpretation of supervelocity and supertime dimensions in super-Lagrangian energy equations utilizing graph theory.

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Tartaglia, Angelo, Giampiero Esposito, Emmanuele Battista, Simone Dell’Agnello, and Bin Wang. "Looking for a new test of general relativity in the solar system." Modern Physics Letters A 33, no. 24 (2018): 1850136. http://dx.doi.org/10.1142/s0217732318501365.

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This paper discusses three matter-of-principle methods for measuring the general relativity correction to the Newtonian values of the position of collinear Lagrangian points L1 and L2 of the Sun–Earth-satellite system. All approaches are based on time measurements. The first approach exploits a pulsar emitting signals and two receiving antennas located at L1 and L2, respectively. The second method is based on a relativistic positioning system based on the Lagrangian points themselves. These first two methods depend crucially on the synchronization of clocks at L1 and L2. The third method combines a pulsar and an artificial emitter at the stable points L4 or L5 forming a basis for the positioning of the collinear points L1 and L2. Further possibilities are mentioned and the feasibility of the measurements is considered.

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Haber, Nick, and András Vasy. "Propagation of singularities around a Lagrangian submanifold of radial points." Bulletin de la Société mathématique de France 143, no. 4 (2015): 679–726. http://dx.doi.org/10.24033/bsmf.2702.

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41

Facchinei, F., and C. Kanzow. "On Unconstrained and Constrained Stationary Points of the Implicit Lagrangian." Journal of Optimization Theory and Applications 92, no. 1 (1997): 99–115. http://dx.doi.org/10.1023/a:1022688013571.

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42

Sawon, Justin. "Lagrangian fibrations on Hilbert schemes of points on K3 surfaces." Journal of Algebraic Geometry 16, no. 3 (2007): 477–97. http://dx.doi.org/10.1090/s1056-3911-06-00453-x.

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43

Haber, Nick. "A Normal Form Around a Lagrangian Submanifold of Radial Points." International Mathematics Research Notices 2014, no. 17 (2013): 4804–21. http://dx.doi.org/10.1093/imrn/rnt096.

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44

Liu, Qian, Xinmin Yang, and Heung Wing Joseph Lee. "On saddle points of a class of augmented lagrangian functions." Journal of Industrial & Management Optimization 3, no. 4 (2007): 693–700. http://dx.doi.org/10.3934/jimo.2007.3.693.

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45

RK, Tiwary, Srivastava VK, and Kushvah BS. "Computation of three-dimensional periodic orbits in the sun-earth system." Physics & Astronomy International Journal 2, no. 1 (2018): 81–90. http://dx.doi.org/10.15406/paij.2018.02.00052.

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In this paper, a third-order analytic approximation is described for computing the three-dimensional periodic halo orbits near the collinear L1 and L2 Lagrangian points for the photo gravitational circular restricted three-body problem in the Sun-Earth system. The constructed third-order approximation is chosen as a starting initial guess for the numerical computation using the differential correction method. The effect of the solar radiation pressure on the location of two collinear Lagrangian points and on the shape of the halo orbits is discussed. It is found that the time period of the halo orbit increases whereas the Jacobi constant decreases around both the collinear points taking into account the solar radiation pressure of the Sun for the fixed out-of-plane amplitude.

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EKHOLM, TOBIAS, JOHN ETNYRE, and MICHAEL SULLIVAN. "ORIENTATIONS IN LEGENDRIAN CONTACT HOMOLOGY AND EXACT LAGRANGIAN IMMERSIONS." International Journal of Mathematics 16, no. 05 (2005): 453–532. http://dx.doi.org/10.1142/s0129167x05002941.

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We show how to orient moduli spaces of holomorphic disks with boundary on an exact Lagrangian immersion of a spin manifold into complex n-space in a coherent manner. This allows us to lift the coefficients of the contact homology of Legendrian spin submanifolds of standard contact (2n + 1)-space from ℤ2 to ℤ. We demonstrate how the ℤ-lift provides a more refined invariant of Legendrian isotopy. We also apply contact homology to produce lower bounds on double points of certain exact Lagrangian immersions into ℂn and again including orientations strengthens the results. More precisely, we prove that the number of double points of an exact Lagrangian immersion of a closed manifold M whose associated Legendrian embedding has good DGA is at least half of the dimension of the homology of M with coefficients in an arbitrary field if M is spin and in ℤ2 otherwise.

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47

Bolle, Philippe. "Morse Index of Approximating Periodic Solutions for the Billiard Problem. Application to Existence Results." Canadian Journal of Mathematics 50, no. 3 (1998): 497–524. http://dx.doi.org/10.4153/cjm-1998-027-6.

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AbstractThis paper deals with periodic solutions for the billiard problem in a bounded open set of ℝN which are limits of regular solutions of Lagrangian systems with a potential well. We give a precise link between the Morse index of approximate solutions (regarded as critical points of Lagrangian functionals) and the properties of the bounce trajectory to which they converge.

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48

YAMAGISHI, HIDENAGA. "THE OPERATOR STRUCTURE OF LAGRANGIAN FIELD THEORY." International Journal of Modern Physics A 04, no. 10 (1989): 2591–612. http://dx.doi.org/10.1142/s0217751x8900100x.

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Lagrangian field theory is reformulated as an integrable system for a causal S matrix. Field equations are derived from the action principle, without introducing however ill-defined products of field operators at coincident space-time points.

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49

CONTRERAS, GONZALO, and RENATO ITURRIAGA. "Convex Hamiltonians without conjugate points." Ergodic Theory and Dynamical Systems 19, no. 4 (1999): 901–52. http://dx.doi.org/10.1017/s014338579913387x.

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We construct the Green bundles for an energy level without conjugate points of a convex Hamiltonian. In this case we give a formula for the metric entropy of the Liouville measure and prove that the exponential map is a local diffeomorphism. We prove that the Hamiltonian flow is Anosov if and only if the Green bundles are transversal. Using the Clebsch transformation of the index form we prove that if the unique minimizing measure of a generic Lagrangian is supported on a periodic orbit, then it is a hyperbolic periodic orbit.We also show some examples of differences with the behaviour of a geodesic flow without conjugate points, namely: (non-contact) flows and periodic orbits without invariant transversal bundles, segments without conjugate points but with crossing solutions and non-surjective exponential maps.

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Rădulescu, Iulia-Maria, Alexandru Boicea, Florin Rădulescu, and Daniel-Călin Popeangă. "A Lagrangian Backward Air Parcel Trajectories Clustering Framework." Water 13, no. 24 (2021): 3638. http://dx.doi.org/10.3390/w13243638.

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Many studies concerning atmosphere moisture paths use Lagrangian backward air parcel trajectories to determine the humidity sources for specific locations. Automatically grouping trajectories according to their geographical position simplifies and speeds up their analysis. In this paper, we propose a framework for clustering Lagrangian backward air parcel trajectories, from trajectory generation to cluster accuracy evaluation. We employ a novel clustering algorithm, called DenLAC, to cluster troposphere air currents trajectories. Our main contribution is representing trajectories as a one-dimensional array consisting of each trajectory’s points position vector directions. We empirically test our pipeline by employing it on several Lagrangian backward trajectories initiated from Břeclav District, Czech Republic.

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Journal articles on the topic 'Lagrangian points'

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