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Journal articles on the topic 'Lagrange equation'

Author: Grafiati

Published: 2 June 2025

Last updated: 31 July 2025

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1

Bodaghi, Abasalt, Hossein Moshtagh, and Amir Mousivand. "Characterization and Stability of Multi-Euler-Lagrange Quadratic Functional Equations." Journal of Function Spaces 2022 (October 10, 2022): 1–9. http://dx.doi.org/10.1155/2022/3021457.

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The aim of the current article is to characterize and to prove the stability of multi-Euler-Lagrange quadratic mappings. In other words, it reduces a system of equations defining the multi-Euler-Lagrange quadratic mappings to an equation, say, the multi-Euler-Lagrange quadratic functional equation. Moreover, some results corresponding to known stability (Hyers, Rassias, and Gӑvruta) outcomes regarding the multi-Euler-Lagrange quadratic functional equation are presented in quasi- β -normed and Banach spaces by using the fixed point methods. Lastly, an example for the nonstable multi-Euler-Lagra

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2

Moiseenko, R. P., and O. O. Kondratenko. "LAGRANGIAN METHOD FOR ALGORITHM OPTIMIZATION OF RIBBED THIN PLATES." Vestnik Tomskogo gosudarstvennogo arkhitekturno-stroitel'nogo universiteta. JOURNAL of Construction and Architecture, no. 1 (April 13, 2018): 140–47. http://dx.doi.org/10.31675/1607-1859-2018-20-1-140-147.

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The paper presents two iteration algorithms for the equation solution using the method of Lagrange multipliers. It is shown that these iteration algorithms do not converge. For comparison, we use the optimum parameters of a ribbed plate obtained by other methods. The proposed method is based on the specific properties of optimality of ribbed plates formulated as a result of the Lagrange equation analysis. These optimum parameters satisfy each of Lagrange equations. The solution of these equations shows that optimization of ribbed plates is possible only with the use of specific optimality prop

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3

Hasan, Amna, Hakeem A. Othman, and Sami H. Altoum. "q-Euler Lagrange Equation." American Journal of Applied Sciences 16, no. 9 (2019): 283–88. http://dx.doi.org/10.3844/ajassp.2019.283.288.

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4

Ingtem, Natal’ya V. "THE APPROACH OF LAGRANGE AND RUFFINI TO THE ISSUE OF SOLVING EQUATIONS OF THE 5TH DEGREE IN RADICALS." RSUH/RGGU Bulletin. Series Information Science. Information Security. Mathematics, no. 3 (2021): 96–106. http://dx.doi.org/10.28995/2686-679x-2021-3-96-106.

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The article deals in two points of view on the solution of the equation of the fifth degree. From one side, it is Lagrange’s analysis, in which a new method for investigating the possibility of solving equations is presented. That method consists in using the roots of a given equation to consequently construct functions from them the degree of the equation with respect to each subsequent function will decrease. To implement the method, Lagrange invented a technique which consists in studying the behavior of a function with all possible permutations of roots in it. In the process of analysis he

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5

Wang, Zhao Qing, Jian Jiang, Bing Tao Tang, and Wei Zheng. "Numerical Solution of Bending Problem for Elliptical Plate Using Differentiation Matrix Method Based on Barycentric Lagrange Interpolation." Applied Mechanics and Materials 638-640 (September 2014): 1720–24. http://dx.doi.org/10.4028/www.scientific.net/amm.638-640.1720.

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A differentiation matrix method based on barycentric Lagrange interpolation for numerical analysis of bending problem for elliptical plate is presented. Embedded the elliptical domain into a rectangular, the barycentric Lagrange interpolation in tensor form is used to approximate unknown function. The governing equation of bending plate is discretized by the differentiation matrix derived from barycentric Lagrange interpolation to form a system of algebraic equations. The boundary conditions on curved boundary are directly discretized using barycentric Lagrange interpolation. Combining discret

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6

Zhang, Xiang Mei, An Ping Xu, and Xian Zhou Guo. "Stability Analysis of Fractional Delay Differential Equations by Lagrange Polynomial." Advanced Materials Research 500 (April 2012): 591–95. http://dx.doi.org/10.4028/www.scientific.net/amr.500.591.

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The paper deals with the numerical stability analysis of fractional delay differential equations with non-smooth coefficients using the Lagrange collocation method. In this paper, based on the Grunwald-Letnikov fractional derivatives, we discuss the approximation of fractional differentiation by the Lagrange polynomial. Then we study the numerical stability of the fractional delay differential equations. Finally, the stability of the delayed Mathieu equation of fractional order is studied and examined by Lagrange collocation method.

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7

Rahmah, Arina Alfa, Mahsa Akhdania, and Bayu Setiaji. "STUDI LITERATUR: PENERAPAN KONSEP MEKANIKA LAGRANGE PADA KEHIDUPAN SEHARI-HARI." Charm Sains: Jurnal Pendidikan Fisika 5, no. 2 (2024): 61–67. https://doi.org/10.53682/charmsains.v5i3.341.

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The application of mechanical concepts to everyday life is important in understanding the physical phenomena that occur. One branch of physics is mechanics which studies the behavior of objects influenced by force and displacement. In complex situations, many variables can be involved in the solution, so it is necessary to involve mathematical methods using the Lagrange equation. This article explains the application of Lagrange's equation in various systems such as pulleys, levers, oscillations, and double pendulum systems that are often encountered in everyday life. In addition, this article

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8

Damayanti, Fitria Siska, RA Sania Noviana, Yora Inda Lestari, Hamdi Akhsan, and Ismet Ismet. "Exploring Applications of Lagrange’s Equations in Technology: A Systematic Literature Review." Aceh International Journal of Science and Technology 13, no. 2 (2024): 123–30. https://doi.org/10.13170/aijst.13.2.39380.

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Lagrange's equation is a formula in analytical mechanics used to solve problems with physical system dynamics. It allows mathematical modeling to simplify complex mechanical problems by changing the coordinate system, thus providing a deeper understanding of motion. In this research, a literature study was conducted using the Systemic Literature Review (SLR) method from 30 data sources, 24 of which were indexed by Scopus. A total of 11 articles have been reviewed with a focus on the application of Lagrange's equation in various technologies. The review results show that Lagrange multipliers pr

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9

Dewi, Nila Mutia. "PENERAPAN PERSAMAAN LAGRANGE PADA SISTEM BANDUL GANDA YANG TERHUBUNG OLEH PEGAS." JURNAL PEMBELAJARAN FISIKA 12, no. 4 (2023): 172. http://dx.doi.org/10.19184/jpf.v12i4.44541.

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Lagrange mechanics is an analytical method in classical mechanics that does not consider forces acting on the system. It focuses instead on kinetic and potential energies as the core of the entire system. The primary goal of this study is to derive the equations of motion for the coupled pendulums connected by a spring system using Lagrange equation. Two identical pendulums, each positioned at the end of its respective lengths, were connected by a spring. Formulating the Lagrangian for this system enabled the derivation of the equation of motion through the Lagrange equation. These equations w

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10

Kasap, Zeki. "Weyl–Euler–Lagrange equations on twistor space for tangent structure." International Journal of Geometric Methods in Modern Physics 13, no. 07 (2016): 1650095. http://dx.doi.org/10.1142/s021988781650095x.

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Twistor spaces are certain complex three-manifolds, which are associated with special conformal Riemannian geometries on four-manifolds. Also, classical mechanic is one of the major subfields for mechanics of dynamical system. A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space for classical mechanic. Euler–Lagrange equations are an efficient use of classical mechanics to solve problems using mathematical modeling. On the other hand, Weyl submitted a metric with a conformal transformation for unified theo

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11

Pandey, P. N., та Suresh K. Shukla. "On Almost φ-Lagrange Spaces". ISRN Geometry 2011 (27 грудня 2011): 1–16. http://dx.doi.org/10.5402/2011/505161.

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We initiate a study on the geometry of an almost φ-Lagrange space (APL-space in short). We obtain the expressions for the symmetric metric tensor, its inverse, semispray coefficients, solution curves of Euler-Lagrange equations, nonlinear connection, differential equation of autoparallel curves, coefficients of canonical metrical d-connection, and h- and v-deflection tensors in an APL-space. Corresponding expressions in a φ-Lagrange space and an almost Finsler Lagrange space (AFL-space in short) have also been deduced.

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12

Wu, Guo-Cheng. "Variational Iteration Method forq-Difference Equations of Second Order." Journal of Applied Mathematics 2012 (2012): 1–5. http://dx.doi.org/10.1155/2012/102850.

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Recently, Liu extended He's variational iteration method to strongly nonlinearq-difference equations. In this study, the iteration formula and the Lagrange multiplier are given in a more accurate way. Theq-oscillation equation of second order is approximately solved to show the new Lagrange multiplier's validness.

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13

Brechenmacher, Frédéric. "Lagrange and the secular equation." Lettera Matematica 2, no. 1-2 (2014): 79–91. http://dx.doi.org/10.1007/s40329-014-0051-3.

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14

Salman, Nour, and Muna Mansour Mustfaf. "Numerical Solution of Fractional Volterra-Fredholm Integro-Differential Equation Using Lagrange Polynomials." Baghdad Science Journal 17, no. 4 (2020): 1234. http://dx.doi.org/10.21123/bsj.2020.17.4.1234.

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In this study, a new technique is considered for solving linear fractional Volterra-Fredholm integro-differential equations (LFVFIDE's) with fractional derivative qualified in the Caputo sense. The method is established in three types of Lagrange polynomials (LP’s), Original Lagrange polynomial (OLP), Barycentric Lagrange polynomial (BLP), and Modified Lagrange polynomial (MLP). General Algorithm is suggested and examples are included to get the best effectiveness, and implementation of these types. Also, as special case fractional differential equation is taken to evaluate the validity of the

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15

Bhat, Imtiyaz Ahmad, and Lakshmi Narayan Mishra. "Numerical Solutions of Volterra Integral Equations of Third Kind and Its Convergence Analysis." Symmetry 14, no. 12 (2022): 2600. http://dx.doi.org/10.3390/sym14122600.

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The current work suggests a method for the numerical solution of the third type of Volterra integral equations (VIEs), based on Lagrange polynomial, modified Lagrange polynomial, and barycentric Lagrange polynomial approximations. To do this, the interpolation of the unknown function is considered in terms of the above polynomials with unknown coefficients. By substituting this approximation into the considered equation, a system of linear algebraic equations is obtained. Then, we demonstrate the method’s convergence and error estimations. The proposed approaches retain the possible singularit

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16

Lu, Chang-Na, Sheng-Xiang Chang, Luo-Yan Xie, and Zong-Guo Zhang. "Generation and solutions to the time-space fractional coupled Navier-Stokes equations." Thermal Science 24, no. 6 Part B (2020): 3899–905. http://dx.doi.org/10.2298/tsci2006899l.

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In this paper, a Lagrangian of the coupled Navier-Stokes equations is proposed based on the semi-inverse method. The fractional derivatives in the sense of Riemann-Liouville definition are used to replace the classical derivatives in the Lagrangian. Then the fractional Euler-Lagrange equation can be derived with the help of the fractional variational principles. The Agrawal?s method is devot?ed to lead to the time-space fractional coupled Navier-Stokes equations from the above Euler-Lagrange equation. The solution of the time-space fractional coupled Navier-Stokes equations is obtained by mean

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17

Gavrilyuk, Sergey, and Keh-Ming Shyue. "Hyperbolic approximation of the BBM equation." Nonlinearity 35, no. 3 (2022): 1447–67. http://dx.doi.org/10.1088/1361-6544/ac4c49.

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Abstract It is well known that the Benjamin–Bona–Mahony (BBM) equation can be seen as the Euler–Lagrange equation for a Lagrangian expressed in terms of the solution potential. We approximate the Lagrangian by a two-parameter family of Lagrangians depending on three potentials. The corresponding Euler–Lagrange equations can be then written as a hyperbolic system of conservations laws. The hyperbolic BBM system has two genuinely nonlinear eigenfields and one linear degenerate eigenfield. Moreover, it can be written in terms of Riemann invariants. Such an approach conserves the variational struc

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18

Janev, Marko, Teodor Atanackovic, and Stevan Pilipovic. "Noether’s theorem for Herglotz type variational problems utilizing complex fractional derivatives." Theoretical and Applied Mechanics 48, no. 2 (2021): 127–42. http://dx.doi.org/10.2298/tam210913011j.

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This is a review article which elaborates the results presented in [1], where the variational principle of Herglotz type with a Lagrangian that depends on fractional derivatives of both real and complex orders is formulated and the invariance of this principle under the action of a local group of symmetries is determined. The conservation law for the corresponding fractional Euler Lagrange equation is obtained and a sequence of approximations of a fractional Euler?Lagrange equation by systems of integer order equations established and analyzed.

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19

Petrova, Ludmila. "Qualitative Investigation of Hamiltonian Systems by Application of Skew-Symmetric Differential Forms." Symmetry 13, no. 1 (2020): 25. http://dx.doi.org/10.3390/sym13010025.

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In the present paper, a role of Hamiltonian systems in mathematical and physical formalisms is considered with the help of skew-symmetric differential forms. In classical mechanics the Hamiltonian system is realized from the Euler–Lagrange equation as the integrability condition of the Euler-Lagrange equation and discloses specific features of Lagrange formalism. In the theory of differential equations, the Hamiltonian systems reveals canonical relations that define the integrability conditions of differential equations. The Hamiltonian systems, as a self-independent equations, are an example

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20

Dryl, Monika, and Delfim F. M. Torres. "Necessary Condition for an Euler-Lagrange Equation on Time Scales." Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/631281.

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We prove a necessary condition for a dynamic integrodifferential equation to be an Euler-Lagrange equation. New and interesting results for the discrete and quantum calculus are obtained as particular cases. An example of a second order dynamic equation, which is not an Euler-Lagrange equation on an arbitrary time scale, is given.

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21

Bakke, V. L., and Z. Jackiewicz. "Stability analysis of linear multistep methods for delay differential equations." International Journal of Mathematics and Mathematical Sciences 9, no. 3 (1986): 447–58. http://dx.doi.org/10.1155/s0161171286000583.

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Stability properties of linear multistep methods for delay differential equations with respect to the test equationy′(t)=ay(λt)+by(t), t≥0,0<λ<1, are investigated. It is known that the solution of this equation is bounded if and only if|a|<−band we examine whether this property is inherited by multistep methods with Lagrange interpolation and by parametrized Adams methods.

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22

Alotaibi, Mastourah M., and Sami H. Altoum. "Euler‐Lagrange Equation in Free Coordinates." Journal of Mathematics 2022 (June 24, 2022): 1–6. http://dx.doi.org/10.1155/2022/3860704.

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In this paper, we introduce different equivalent formulations of variational principle. The language of differential forms and manifold has been utilized to deduce Euler–Lagrange equations in free coordinates. Thus, the expression is simple and global.

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23

Lewis, Clinton L. "Explicit gauge covariant Euler–Lagrange equation." American Journal of Physics 77, no. 9 (2009): 839–43. http://dx.doi.org/10.1119/1.3153503.

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24

KAPARULIN, D. S., S. L. LYAKHOVICH, and A. A. SHARAPOV. "ON LAGRANGE STRUCTURE OF UNFOLDED FIELD THEORY." International Journal of Modern Physics A 26, no. 07n08 (2011): 1347–62. http://dx.doi.org/10.1142/s0217751x11052840.

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Any local field theory can be equivalently reformulated in the so-called unfolded form. General unfolded equations are non-Lagrangian even though the original theory is Lagrangian. Making use of the unfolded massless scalar field equations as a basic example, the concept of Lagrange anchor is applied to perform a consistent path-integral quantization of unfolded dynamics. It is shown that the unfolded representation for the canonical Lagrange anchor of the d'Alembert equation inevitably involves an infinite number of space–time derivatives.

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25

Erol, H. "Characteristic equations of longitudinally vibrating rods carrying a tip mass and several viscously damped spring-mass systems in-span." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 218, no. 10 (2004): 1103–14. http://dx.doi.org/10.1243/0954406042369134.

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This paper deals with the determination of two alternative approximate formulations for the frequency equation of a longitudinally vibrating fixed-free elastic rod carrying a tip mass (primary system) to which several spring-mass-damper systems (secondary systems) are attached in-span. The first approximate formulation presented in this study is based upon the assumed-mode method in conjunction with the Lagrange multiplier method. The result is a simple analytical formula for the characteristic equation of the system. Hence, the eigenfrequency parameters of the system are determined by solving

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26

Agilan, P., K. Julietraja, Nabil Mlaiki, and Aiman Mukheimer. "Intuitionistic Fuzzy Stability of an Euler–Lagrange Symmetry Additive Functional Equation via Direct and Fixed Point Technique (FPT)." Symmetry 14, no. 11 (2022): 2454. http://dx.doi.org/10.3390/sym14112454.

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In this article, a new class of real-valued Euler–Lagrange symmetry additive functional equations is introduced. The solution of the equation is provided, assuming the unknown function to be continuous and without any regularity conditions. The objective of this research is to derive the Hyers–Ulam–Rassias stability (HURS) in intuitionistic fuzzy normed spaces (IFNS) by applying the classical direct method and fixed point techniques (FPT). Furthermore, it is proven that the Euler–Lagrange symmetry additive functional equation and the control function, which is the IFNS of the sums and products

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27

Buckholz, R. H. "Effects of Power—Law, Non-Newtonian Lubricants on Load Capacity and Friction for Plane Slider Bearings." Journal of Tribology 108, no. 1 (1986): 86–91. http://dx.doi.org/10.1115/1.3261149.

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The lubrication of a conventional, finite width plane bearing, using a power-law, non-Newtonian lubricant, is studied. The basic assumptions in this analysis are: thin fluid-film, no thermal effects, and a modified Reynolds’ equation for small bearing aspect ratios. Results from this study include bearing pressure, load, and friction formulas. Similar results for the not-so-small bearing aspect ratios are found via an Euler-Lagrange equation. This Euler-Lagrange equation is derived from the optimization integral for the modified Reynolds’ equation. Approximate solutions to the modified Reynold

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28

Rovenski, Vladimir. "Einstein-Hilbert type action on spacetimes." Publications de l'Institut Math?matique (Belgrade) 103, no. 117 (2018): 199–210. http://dx.doi.org/10.2298/pim1817199r.

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The mixed gravitational field equations have been recently introduced for codimension one foliated manifolds, e.g. stably causal and globally hyperbolic spacetimes. These Euler-Lagrange equations for the total mixed scalar curvature (as analog of Einstein-Hilbert action) involve a new kind of Ricci curvature (called the mixed Ricci curvature). In the work, we derive Euler-Lagrange equations of the action for any spacetime, in fact, for a pseudo-Riemannian manifold endowed with a non-degenerate distribution. The obtained equations are presented in the classical form of Einstein field equation w

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29

Mauda, R., and M. Pinchas. "16QAM Blind Equalization via Maximum Entropy Density Approximation Technique and Nonlinear Lagrange Multipliers." Scientific World Journal 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/548714.

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Recently a new blind equalization method was proposed for the 16QAM constellation input inspired by the maximum entropy density approximation technique with improved equalization performance compared to the maximum entropy approach, Godard’s algorithm, and others. In addition, an approximated expression for the minimum mean square error (MSE) was obtained. The idea was to find those Lagrange multipliers that bring the approximated MSE to minimum. Since the derivation of the obtained MSE with respect to the Lagrange multipliers leads to a nonlinear equation for the Lagrange multipliers, the par

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30

Wang, Yuhan, Peiyao Wang, Rongpei Zhang, and Jia Liu. "Solution of the Elliptic Interface Problem by a Hybrid Mixed Finite Element Method." Mathematics 12, no. 12 (2024): 1892. http://dx.doi.org/10.3390/math12121892.

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This paper addresses the elliptic interface problem involving jump conditions across the interface. We propose a hybrid mixed finite element method on the triangulation where the interfaces are aligned with the mesh. The second-order elliptic equation is initially decomposed into two equations by introducing a gradient term. Subsequently, weak formulations are applied to these equations. Scheme continuity is enforced using the Lagrange multiplier technique. Finally, we derive an explicit formula for the entries of the matrix equation representing Lagrange multiplier unknowns resulting from hyb

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31

Ibraheem, Rasha Hassein. "Fuzzy Lagrange Polynomials for Solving Two-Dimensional Fuzzy Fractional Volterra Integro-Differential Equations." Global Journal of Mathematics and Statistics 2, no. 1 (2025): 52–62. https://doi.org/10.61424/gjms.v2i1.291.

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In this study, we present a numerical approach to solve first-order fuzzy fractional Volterra integro-differential equations in two dimensions space, using three different formulations of fuzzy Lagrange polynomials: the fuzzy original Lagrange polynomial (FOLP), the fuzzy barycentric Lagrange polynomial (FBLP), and the fuzzy modified Lagrange polynomial (FMLP). Comprehensive algorithm is constructed to improve the computational efficiency of the proposed method and its effectiveness was tested through numerical application the numerical results demonstrate that the three methods can preserve t

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32

Mohiuddine, Syed Abdul, John Michael Rassias, and Abdullah Alotaibi. "Solution of the Ulam stability problem for Euler–Lagrange k-quintic mappings." Georgian Mathematical Journal 27, no. 4 (2020): 585–92. http://dx.doi.org/10.1515/gmj-2018-0063.

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AbstractThe “oldest quartic” functional equationf(x+2y)+f(x-2y)=4[f(x+y)+f(x-y)]-6f(x)+24f(y)was introduced and solved by the second author of this paper (see J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glas. Mat. Ser. III 34(54) 1999, 2, 243–252). Similarly, an interesting “quintic” functional equation was introduced and investigated by I. G. Cho, D. Kang and H. Koh, Stability problems of quintic mappings in quasi-β-normed spaces, J. Inequal. Appl. 2010 2010, Article ID 368981, in the following form:2f(2x+y)+2f(2x-y)+f(x+2y)+f(x-2y)=20[f(x+y)+f(x-y)]+90f(x).In

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Wcislik, Miroslaw, and Karol Suchenia. "Holonomicity analysis of electromechanical systems." Open Physics 15, no. 1 (2017): 942–47. http://dx.doi.org/10.1515/phys-2017-0115.

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Abstract Electromechanical systems are described using state variables that contain electrical and mechanical components. The equations of motion, both electrical and mechanical, describe the relationships between these components. These equations are obtained using Lagrange functions. On the basis of the function and Lagrange - d’Alembert equation the methodology of obtaining equations for electromechanical systems was presented, together with a discussion of the nonholonomicity of these systems. The electromechanical system in the form of a single-phase reluctance motor was used to verify th

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Park, Hae Yeon, and Jung Hoon Kim. "Model-free control approach to uncertain Euler-Lagrange equations with a Lyapunov-based $ L_\infty $-gain analysis." AIMS Mathematics 8, no. 8 (2023): 17666–86. http://dx.doi.org/10.3934/math.2023902.

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<abstract><p>This paper considers a model-free control approach to Euler-Lagrange equations and proposes a new quantitative performance measure with its Lyapunov-based computation method. More precisely, this paper aims to solve a trajectory tracking problem for uncertain Euler-Lagrange equations by using a model-free controller with a proportional-integral-derivative (PID) control form. The $ L_\infty $-gain is evaluated for the closed-loop systems obtained through the feedback connection between the Euler-Lagrange equation and the model-free controller. To this end, the input-to-

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Rassias, John Michael, Narasimman Pasupathi, Reza Saadati та Manuel de la Sen. "Approximation of Mixed Euler-Lagrange σ -Cubic-Quartic Functional Equation in Felbin’s Type f-NLS". Journal of Function Spaces 2021 (12 лютого 2021): 1–7. http://dx.doi.org/10.1155/2021/8068673.

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In this research paper, the authors present a new mixed Euler-Lagrange σ -cubic-quartic functional equation. For this introduced mixed type functional equation, the authors obtain general solution and investigate the various stabilities related to the Ulam problem in Felbin’s type of fuzzy normed linear space (f-NLS) with suitable counterexamples. This approach leads us to approximate the Euler-Lagrange σ -cubic-quartic functional equation with better estimation.

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Sumin, V. I., and M. I. Sumin. "On regularization of the Lagrange principle in the optimization problems for linear distributed Volterra type systems with operator constraints." Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta 59 (May 2022): 85–113. http://dx.doi.org/10.35634/2226-3594-2022-59-07.

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Regularization of the classical optimality conditions - the Lagrange principle and the Pontryagin maximum principle - in a convex optimal control problem subject to functional equality and inequality constraints is considered. The controlled system is described by a linear functional-operator equation of second kind of the general form in the space $L_2^m$. The main operator on the right-hand side of the equation is assumed to be quasi-nilpotent. The objective functional to be minimized is strongly convex. The derivation of the regularized classical optimality conditions is based on the use of

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Li, Yu, Ming Fu Fu, and Zhi Long Xie. "Generalized Variation Problem of the Fractured Rock Mass under the Linear Elastic Unloading Stage." Applied Mechanics and Materials 170-173 (May 2012): 390–94. http://dx.doi.org/10.4028/www.scientific.net/amm.170-173.390.

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First, the constitutive equation of the linear elastic and the basic equations flexibility of the unloading rock mass were introduced, and then using the equations and the boundary conditions the variation principle was contracture. Prior to construct an appropriate minimum potential energy functional, on this basis, pending the introduction of two Lagrange multipliers, and absorb the variation constraints to establish the new functional. Last will 、、、as an independent variable, considering the variation stationary conditions functional of the new functional can identify the Lagrange multiplie

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Hegedűs, György, and Sándor Apáti. "Investigation and simulation of a scotch yoke mechanism." Multidiszciplináris Tudományok 13, no. 2 (2023): 144–52. http://dx.doi.org/10.35925/j.multi.2023.2.13.

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This paper studies the electromechanical model of a jigsaw mechanism. The jigsaw is powered by a battery, which drives a DC motor. The rotational motion is converted into linear motion through a Scotch Yoke mechanism. The electromechanical equations are based on energy approach using the Lagrange-equation. The Lagrange function contains the energies of the model, i.e., the kinetic co-energy of the kinematic chain and the magnetic co-energy of the inductance. As regard the non-conservative elements their virtual works are written. The formulated equations of the jigsaw model allow us to examine

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Bonfanti, Giovanni, and Arrigo Cellina. "The validity of the Euler-Lagrange equation." Discrete & Continuous Dynamical Systems - A 28, no. 2 (2010): 511–17. http://dx.doi.org/10.3934/dcds.2010.28.511.

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Nester, J. M. "Invariant derivation of the Euler-Lagrange equation." Journal of Physics A: Mathematical and General 21, no. 21 (1988): L1013—L1017. http://dx.doi.org/10.1088/0305-4470/21/21/003.

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Ledzewicz, Urszula, and Heinz Schaettler. "On generalizations of the Euler–Lagrange equation." Nonlinear Analysis: Theory, Methods & Applications 47, no. 1 (2001): 339–50. http://dx.doi.org/10.1016/s0362-546x(01)00181-x.

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Rampho, Gaotsiwe J. "The Schrödinger equation on a Lagrange mesh." Journal of Physics: Conference Series 905 (October 2017): 012037. http://dx.doi.org/10.1088/1742-6596/905/1/012037.

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Rampho, G. J., L. C. Mabunda, and M. Ramantswana. "Few-body integrodifferential equation on Lagrange mesh." Journal of Physics: Conference Series 915 (October 2017): 012005. http://dx.doi.org/10.1088/1742-6596/915/1/012005.

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Toma, Antonela, and Octavian Postavaru. "Fractional Complex Euler–Lagrange Equation: Nonconservative Systems." Fractal and Fractional 7, no. 11 (2023): 799. http://dx.doi.org/10.3390/fractalfract7110799.

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Classical forbidden processes paved the way for the description of mechanical systems with the help of complex Hamiltonians. Fractional integrals of complex order appear as a natural generalization of those of real order. We propose the complex fractional Euler-Lagrange equation, obtained by finding the stationary values associated with the fractional integral of complex order. The complex Hamiltonian obtained from the Lagrangian is suitable for describing nonconservative systems. We conclude by presenting the conserved quantities attached to Noether symmetries corresponding to complex systems

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Vlysidis, Michail, and Yiannis Kaznessis. "Solving Stochastic Reaction Networks with Maximum Entropy Lagrange Multipliers." Entropy 20, no. 9 (2018): 700. http://dx.doi.org/10.3390/e20090700.

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The time evolution of stochastic reaction networks can be modeled with the chemical master equation of the probability distribution. Alternatively, the numerical problem can be reformulated in terms of probability moment equations. Herein we present a new alternative method for numerically solving the time evolution of stochastic reaction networks. Based on the assumption that the entropy of the reaction network is maximum, Lagrange multipliers are introduced. The proposed method derives equations that model the time derivatives of these Lagrange multipliers. We present detailed steps to trans

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Faqiri, A., Yu L. Rutman, and N. V. Ostrovskaya. "Calculation and theoretical evaluation of the Yu. D. Cherepinsky kinematic supports efficiency with consideration of their parameters." Вестник гражданских инженеров 19, no. 1 (2022): 38–47. http://dx.doi.org/10.23968/1999-5571-2022-19-1-38-47.

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The paper presents the equations of the oscillations of seismic-protected object located on the Yu. D. Cherepinsky kinematic support obtained using the Lagrange 1-st kind method. Previously, these equations were obtained by a number of authors using the Lagrange 2-nd kind method. Using the Lagrange1st kind method allows giving the components of the equations a transparent physical meaning. The support model used assumes that the rolling surface of the support, as well as the support itself, remain rigid during oscillations. With small oscillations, the obtained equation can be reduced to the l

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Saeed, Umer, and Muhammad Umair. "A modified method for solving non-linear time and space fractional partial differential equations." Engineering Computations 36, no. 7 (2019): 2162–78. http://dx.doi.org/10.1108/ec-01-2019-0011.

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Purpose The purpose of the paper is to extend the differential quadrature method (DQM) for solving time and space fractional non-linear partial differential equations on a semi-infinite domain. Design/methodology/approach The proposed method is the combination of the Legendre polynomials and differential quadrature method. The authors derived and constructed the new operational matrices for the fractional derivatives, which are used for the solutions of non-linear time and space fractional partial differential equations. Findings The fractional derivative of Lagrange polynomial is a big hurdle

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ШайдуровВ.В., ШайдуровВ В., and ЧередниченкоО М. ЧередниченкоО.М. "Semi-Lagrangian approximations of the convection operator in symmetric form." Вычислительные технологии, no. 3 (June 21, 2023): 101–16. http://dx.doi.org/10.25743/ict.2023.28.3.007.

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Рассмотрены два полулагранжевых численных метода для одномерного (по пространству) уравнения переноса с оператором в симметричной форме: эйлероволагранжев и лагранжево-эйлеров. Оба метода свободны от ограничения Куранта на соотношение шагов по времени и пространству. Причем во втором методе достигнут второй порядок аппроксимации для гладких решений и продемонстрировано отсутствие численной вязкости для разрывных решений. Purpose. The purpose of the study is the development and comparison of two numerical semi-Lagrangian methods with fulfillment of the conservation law at a discrete level. The

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BRACKEN, PAUL. "A STRING MODEL FOR D-DIMENSIONAL DE SITTER SPACE–TIME AND EQUATIONS OF MOTION." International Journal of Modern Physics A 20, no. 26 (2005): 6065–81. http://dx.doi.org/10.1142/s0217751x0502553x.

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De Sitter space–time is considered to be represented by a D-dimensional hyperboloid embedded in (D+1)-dimensional Minkowski space–time. The string equation is derived from a string action which contains a Lagrange multiplier to restrict coordinates to de Sitter space–time. The string system of equations is equivalent to a type of generalized sinh–Gordon equation. The evolution equations for all the variables including the coordinates and their derivatives are obtained for D=2,3 and 4.

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DUAN, ZHISHENG, JINZHI WANG, RONG LI, and LIN HUANG. "A GENERALIZATION OF SMOOTH CHUA'S EQUATIONS UNDER LAGRANGE STABILITY." International Journal of Bifurcation and Chaos 17, no. 09 (2007): 3047–59. http://dx.doi.org/10.1142/s0218127407018853.

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In this paper, smooth Chua's equation is generalized to a higher order system from a special viewpoint of interconnected systems. Simple conditions for Lagrange stability are established. And a detailed Lagrange stable region analysis is given for the canonical Chua's oscillator. In addition, a new nonlinearly coupled Chua's circuit that appeared in the recent literature is also discussed and a Lagrange stability condition is presented. Several examples are presented to illustrate the results.

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