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Journal articles on the topic 'Lagrange equations'

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Published: 4 June 2021

Last updated: 9 June 2025

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1

Derevenskii, V. P. "Lagrange matrix equations." Russian Mathematics 59, no. 12 (November 7, 2015): 10–20. http://dx.doi.org/10.3103/s1066369x15120026.

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2

Potts, Renfrey B. "Discrete Lagrange equations." Bulletin of the Australian Mathematical Society 37, no. 2 (April 1988): 227–33. http://dx.doi.org/10.1017/s0004972700026769.

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3

Ivanov, Anastas Ivanov. "Condense Lagrange Equations." International Journal of Research and Methodology in Social Science 5, no. 3 (September 30, 2019): 30–35. https://doi.org/10.5281/zenodo.3566885.

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In this article, new equations called Condensed Lagrange equations (CLE) are defined. With their help, the rigid body absolute general motion is studied. The rigid body is considered to be homogeneous and unsymmetrical. CLE are similar in structure to the classical Lagrange equations from second type, applied in vector-matrix form, but CLE are differing from them by four indicators. These differences are commented in detail in this article. The use of the CUL is fully equivalent to the application of the theorem, called Theorem for change of the rigid body generalized impulse. Using CLE the differential equations in a matrix form, describing the rigid body absolute general motion, are obtained. CLE enrich the theory of Rigid Body Mechanics. Moreover, CLE represent a second alternative variant of the Theorem for change of the rigid body generalized impulse, they serve for verification it, and finally, they make the study completely.

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4

Koh, Youngmee, and Sangwook Ree. "Lagrange and Polynomial Equations." Journal for History of Mathematics 27, no. 3 (June 30, 2014): 165–82. http://dx.doi.org/10.14477/jhm.2014.27.3.165.

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5

Ellis, David C. P., François Gay-Balmaz, Darryl D. Holm, and Tudor S. Ratiu. "Lagrange–Poincaré field equations." Journal of Geometry and Physics 61, no. 11 (November 2011): 2120–46. http://dx.doi.org/10.1016/j.geomphys.2011.06.007.

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6

Baev, V. K. "Lagrange Equations of Envelopes." Journal of Physics: Conference Series 941 (December 2017): 012087. http://dx.doi.org/10.1088/1742-6596/941/1/012087.

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7

Zeng, Yan. "Urban and Rural Multi-Objective Programming Based on Augmented Lagrange Multiplier Method for Nonlinear Mathematical Equations." Journal of Corrosion and Materials 48, no. 1 (December 2, 2024): 58–64. http://dx.doi.org/10.61336/jcm2023-6.

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Augmented Lagrange multiplier method is an important method to solve constrained optimization problems. In recent years, it has become more important to study the application of augmented Lagrange multiplier method. This paper first introduces the augmented Lagrange multiplier method, which leads to the development of the application of the augmented Lagrange method to nonlinear mathematical equations, and summarizes the augmented Lagrange multiplier. The application of nonlinear mathematical equations. At the same time, the paper specifies the application of the augmented Lagrange method in urban and rural multi-objective programming, and proves the practical application of the augmented Lagrange multiplier nonlinear mathematical equations.

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8

Mingliang, Zheng. "RENORMALIZATION GROUP METHOD FOR A CLASS OF LAGRANGE MECHANICAL SYSTEMS." Journal of the Serbian Society for Computational Mechanics 16, no. 2 (December 1, 2022): 96–104. http://dx.doi.org/10.24874/jsscm.2022.16.02.07.

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Considering the important role of small parameter perturbation term in mechanical systems, the perturbed dynamic differential equations of Lagrange systems are established. The basic idea and method of solving ordinary differential equations by normal renormalization group method are transplanted into a kind of Lagrange mechanical systems, the renormalization group equations of Euler-Lagrange equations are obtained, and the first-order uniformly valid asymptotic approximate solution of Lagrange systems with a single-degree-of-freedom is given. Two examples are used to show the calculation steps of renormalization group method in detail as well as to verify the correctness of the method. The innovative finding of this paper is that for integrable Lagrange systems, its renormalization group equations are also integrable and satisfy the Hamilton system's structure.

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9

Sun, Lanyin, and Chungang Zhu. "B-Spline Solutions of General Euler-Lagrange Equations." Mathematics 7, no. 4 (April 22, 2019): 365. http://dx.doi.org/10.3390/math7040365.

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The Euler-Lagrange equations are useful for solving optimization problems in mechanics. In this paper, we study the B-spline solutions of the Euler-Lagrange equations associated with the general functionals. The existing conditions of B-spline solutions to general Euler-Lagrange equations are given. As part of this work, we present a general method for generating B-spline solutions of the second- and fourth-order Euler-Lagrange equations. Furthermore, we show that some existing techniques for surface design, such as Coons patches, are exactly the special cases of the generalized Partial differential equations (PDE) surfaces with appropriate choices of the constants.

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10

Nowakowski, Andrzej, and Andrzej Rogowski. "Periodic solutions of Lagrange equations." Topological Methods in Nonlinear Analysis 22, no. 1 (September 1, 2003): 167. http://dx.doi.org/10.12775/tmna.2003.034.

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11

Herzallah, Mohamed A. E., and Dumitru Baleanu. "Fractional Euler–Lagrange equations revisited." Nonlinear Dynamics 69, no. 3 (January 18, 2012): 977–82. http://dx.doi.org/10.1007/s11071-011-0319-5.

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12

Burov, A. A., and I. I. Kosenko. "The Lagrange differential-algebraic equations." Journal of Applied Mathematics and Mechanics 78, no. 6 (2014): 587–98. http://dx.doi.org/10.1016/j.jappmathmech.2015.04.006.

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13

Constantinescu, Oana A., and Ebtsam H. Taha. "Alternative Lagrangians obtained by scalar deformations." International Journal of Geometric Methods in Modern Physics 17, no. 04 (March 2020): 2050050. http://dx.doi.org/10.1142/s0219887820500504.

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We study mechanical systems that can be recast into the form of a system of genuine Euler–Lagrange equations. The equations of motions of such systems are initially equivalent to the system of Lagrange equations of some Lagrangian [Formula: see text], including a covariant force field. We find necessary and sufficient conditions for the existence of a differentiable function [Formula: see text] such that the initial system is equivalent to the system of Euler–Lagrange equations of the deformed Lagrangian [Formula: see text].

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14

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-Lagrange quadratic functional equation is indicated.

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15

Hasan, Eyad Hasan. "Fractional Variational Problems of Euler-Lagrange Equations with Holonomic Constrained Systems." Applied Physics Research 8, no. 3 (April 23, 2016): 60. http://dx.doi.org/10.5539/apr.v8n3p60.

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<p class="1Body">In this paper, we examined the fractional Euler-Lagrange equations for Holonomic constrained systems. The Euler-Lagrange equations are derived using the fractional variational problem of Lagrange. In addition, we achieved that the classical results were obtained are agreement when fractional derivatives are replaced with the integer order derivatives. Two physical examples are discussed to demonstrate the formalism.</p>

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16

Fu, Jing-Li, Lijun Zhang, Chaudry Khalique, and Ma-Li Guo. "Circulatory integral and Routh's equations of Lagrange systems with Riemann-Liouville fractional derivatives." Thermal Science 25, no. 2 Part B (2021): 1355–63. http://dx.doi.org/10.2298/tsci200520034f.

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In this paper, the circulatory integral and Routh?s equations of Lagrange systems are established with Riemann-Liouville fractional derivatives, and the circulatory integral of Lagrange systems is obtained by making use of the relationship between Riemann-Liouville fractional integrals and fractional derivatives. Thereafter, the Routh?s equations of Lagrange systems are given based on the fractional circulatory integral. Two examples are presented to illustrate the application of the results.

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17

Djukic, Djordje. "Generalized Lagrange-D’Alembert principle." Publications de l'Institut Math?matique (Belgrade) 91, no. 105 (2012): 49–58. http://dx.doi.org/10.2298/pim1205049d.

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The major issues in the analysis of the motion of a constrained dynamic system are to determine this motion and calculate constraint forces. In the analytical mechanics, only the first of the two problems is analyzed. Here, the problem is solved simultaneously using: 1) Principle of liberation of constraints; 2) Principle of generalized virtual displacement; 3) Idea of ideal constraints; 4) Concept of generalized and ?supplementary" generalized coordinates. The Lagrange-D?Alembert principle of virtual work is generalized introducing virtual displacement as vectorial sum of the classical virtual displacement and virtual displacement in the ?supplementary" directions. From such principle of virtual work we derived Lagrange equations of the second kind and equations of dynamical equilibrium in the ?supplementary" directions. Constrained forces are calculated from the equations of dynamic equilibrium. At the same time, this principle can be used for consideration of equilibrium of system of material particles. This principle simultaneously gives the connection between applied forces at equilibrium state and the constrained forces. Finally, the principle is applied to a few particular problems.

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18

Yuan, Xiaoping. "Lagrange stability for asymmetric Duffing equations." Nonlinear Analysis: Theory, Methods & Applications 43, no. 2 (January 2001): 137–51. http://dx.doi.org/10.1016/s0362-546x(99)00170-4.

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19

Grabowska, Katarzyna, Janusz Grabowski, and Paweł Urbański. "AV-differential geometry: Euler–Lagrange equations." Journal of Geometry and Physics 57, no. 10 (September 2007): 1984–98. http://dx.doi.org/10.1016/j.geomphys.2007.04.003.

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20

Kikuchi, Keiichi. "EXTENDED HARMONIC MAPS AND LAGRANGE EQUATIONS." JP Journal of Geometry and Topology 20, no. 1 (May 6, 2017): 39–60. http://dx.doi.org/10.17654/gt020010039.

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21

Shen, Jianhua, Lu Chen, and Xiaoping Yuan. "Lagrange stability for impulsive Duffing equations." Journal of Differential Equations 266, no. 11 (May 2019): 6924–62. http://dx.doi.org/10.1016/j.jde.2018.11.022.

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22

Yuan, Xiaoping. "Lagrange Stability for Duffing-Type Equations." Journal of Differential Equations 160, no. 1 (January 2000): 94–117. http://dx.doi.org/10.1006/jdeq.1999.3663.

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23

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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24

Gorobtsov, Alexander, Oleg Sychev, Yulia Orlova, Evgeniy Smirnov, Olga Grigoreva, Alexander Bochkin, and Marina Andreeva. "Optimal Greedy Control in Reinforcement Learning." Sensors 22, no. 22 (November 18, 2022): 8920. http://dx.doi.org/10.3390/s22228920.

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We consider the problem of dimensionality reduction of state space in the variational approach to the optimal control problem, in particular, in the reinforcement learning method. The control problem is described by differential algebraic equations consisting of nonlinear differential equations and algebraic constraint equations interconnected with Lagrange multipliers. The proposed method is based on changing the Lagrange multipliers of one subset based on the Lagrange multipliers of another subset. We present examples of the application of the proposed method in robotics and vibration isolation in transport vehicles. The method is implemented in FRUND—a multibody system dynamics software package.

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25

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 (September 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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26

Agrawal, Sunil K., Pana Claewplodtook, and Brian C. Fabien. "Optimal Trajectories of Open-Chain Robot Systems: A New Solution Procedure Without Lagrange Multipliers." Journal of Dynamic Systems, Measurement, and Control 120, no. 1 (March 1, 1998): 134–36. http://dx.doi.org/10.1115/1.2801309.

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For an n d.o.f. robot system, optimal trajectories using Lagrange multipliers are characterized by 4n first-order nonlinear differential equations with 4n boundary conditions at the two end time. Numerical solution of such two-point boundary value problems with shooting techniques is hard since Lagrange multipliers can not be guessed. In this paper, a new procedure is proposed where the dynamic equations are embedded into the cost functional. It is shown that the optimal solution satisfies n fourth-order differential equations. Due to absence of Lagrange multipliers, the two-point boundary-value problem can be solved efficiently and accurately using classical weighted residual methods.

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27

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 discrete algebraic equations of governing equation and boundary conditions to form an over-constraints system of equations, the numerical solutions on rectangular can be obtained by solving it. Then, the numerical solutions on elliptical domain are obtained by interpolating the data on rectangular. Numerical results of elliptical plate with uniform load illustrate the effectiveness and accuracy of the proposed method.

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28

MANOFF, S. "EINSTEIN'S THEORY OF GRAVITATION AS A LAGRANGIAN THEORY FOR TENSOR FIELDS." International Journal of Modern Physics A 13, no. 12 (May 10, 1998): 1941–67. http://dx.doi.org/10.1142/s0217751x98000846.

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Einstein's theory of gravitation (ETG) is considered as a Lagrangian theory of tensor fields over (pseudo) Riemannian spaces without torsion (Vn spaces, n=4) by means of the method of Lagrangians with covariant derivarives (MLCD). In a trivial manner Euler–Lagrange's equations as Einstein's equations are obtained. The corresponding energy–momentum tensors (EMT's) are found for the standard for the ETG Lagrangian invariant on the basis of the covariant Noether identities. The symmetric energy–momentum tensor of Hilbert appears as an element irrelevant to the whole scheme of the considered Lagrangian thoery of tensor fields over Vn spaces despite of the fact that it has some elements of the structure of the variational EMT of Euler–Lagrange. The notion of the active gravitational rest mast density is related to the variational EMT of Euler–Lagrange and on this basis to a certain extent to the EMT of Hilbert.

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29

Askerov, I. "Approximate method of solving one periodic optimal regulated boundary value problem." Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, no. 2 (2019): 66–69. http://dx.doi.org/10.17721/1812-5409.2019/2.7.

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In the present work we considered the solution of one periodic optimal regulated boundary value problem by the asymptotic method. For the solution of the problem with extended functional writing, boundary conditions and Euler-Lagrange equations were found. The approach to the solution of the problem depending on a small parameter by seeking a system of nonlinear differential equations and solving Euler-Lagrange equations, the solution of the general problem in the first approach comes down to solving two nonlinear algebraic equations.

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30

Wcislik, Miroslaw, and Karol Suchenia. "Holonomicity analysis of electromechanical systems." Open Physics 15, no. 1 (December 29, 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 the presented method. Mechanical system was built as a system, which can oscillate as the element of physical pendulum. On the base of the pendulum oscillation, parameters of the electromechanical system were defined. The identification of the motor electric parameters as a function of the rotation angle was carried out. In this paper the characteristics and motion equations parameters of the motor are presented. The parameters of the motion equations obtained from the experiment and from the second order Lagrange equations are compared.

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31

Vol’nikov, Mikhail, and Vladimir Vasilevich Smogunov. "Friction accounting in mathematical models of dissipative systems." Vestnik of Astrakhan State Technical University. Series: Management, computer science and informatics 2022, no. 2 (April 29, 2022): 110–18. http://dx.doi.org/10.24143/2072-9502-2022-2-110-118.

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Obtaining models of mechanical processes with dissipation based on the Euler-Lagrange theory has undoubted advantages over Newton's theory due to the smaller size of the considered vector of variables included in the equations. However, the Euler-Lagrange variation theory is not applicable to the description of the motion of systems with dissipation. The aim of the work is to demonstrate the possibility of using the Euler-Lagrange theory in relation to dissipative systems with different types of friction. Mathematical models of systems with dissipation are based on the superposition of mechanical and thermodynamic Lagrangians. To obtain a mathematical description of dissipative systems it is proposed to use the field theory as applied to the thermodynamics of dissipative processes within the framework of the Lagrange formalism. The Euler-Lagrange equations are obtained for the Stokes and Coulomb friction models. As it was referred to the research results obtained there is possibility of accounting the energy dissipation in the Lagrange formalism. The mathematical models proposed describe dynamic processes in heterogeneous structures with friction based on the Euler-Lagrange theory. There are presented mathematical transformations that allow transition from models based on the Lagrange formalism to models based on Newtonian mechanics

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32

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 properties.

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33

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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34

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 (December 30, 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 provide a powerful tool for optimizing energy flow within complex smart grids. The benefits extend beyond smart grids. Lagrange's equations are a powerful mathematical tool applicable to various engineering challenges that involve finding optimal solutions under constraints.

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35

Storchak, S. N. "The Poincaré variational principle in the Lagrange–Poincaré reduction of mechanical systems with symmetry." International Journal of Geometric Methods in Modern Physics 16, no. 05 (May 2019): 1950068. http://dx.doi.org/10.1142/s0219887819500683.

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The local Lagrange–Poincaré equations (the reduced Euler–Lagrange equations) for the mechanical system describing the motion of a scalar particle on a finite-dimensional Riemannian manifold with a given free isometric smooth action of a compact semi-simple Lie group are obtained. The equations are written in terms of dependent coordinates which are used to represent the local dynamic given on the orbit space of the principal fiber bundle. The derivation of the equations is performed with the help of the variational principle developed by Poincaré for mechanical systems with symmetry.

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36

Herzallah, Mohamed A. E., and Dumitru Baleanu. "Fractional-order Euler–Lagrange equations and formulation of Hamiltonian equations." Nonlinear Dynamics 58, no. 1-2 (March 18, 2009): 385–91. http://dx.doi.org/10.1007/s11071-009-9486-z.

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37

Xenitidis, Pavlos, Frank Nijhoff, and Sarah Lobb. "On the Lagrangian formulation of multidimensionally consistent systems." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467, no. 2135 (July 13, 2011): 3295–317. http://dx.doi.org/10.1098/rspa.2011.0124.

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Multidimensional consistency has emerged as a key integrability property for partial difference equations (PΔEs) defined on the ‘space–time’ lattice. It has led, among other major insights, to a classification of scalar affine-linear quadrilateral PΔEs possessing this property, leading to the so-called Adler–Bobenko–Suris (ABS) list. Recently, a new variational principle has been proposed that describes the multidimensional consistency in terms of discrete Lagrangian multi-forms. This description is based on a fundamental and highly non-trivial property of Lagrangians for those integrable lattice equations, namely the fact that on the solutions of the corresponding PΔE the Lagrange forms are closed, i.e. they obey a closure relation . Here, we extend those results to the continuous case: it is known that associated with the integrable PΔEs there exist systems of partial differential equations (PDEs), in fact differential equations with regard to the parameters of the lattice as independent variables, which equally possess the property of multidimensional consistency. In this paper, we establish a universal Lagrange structure for affine-linear quad-lattices alongside a universal Lagrange multi-form structure for the corresponding continuous PDEs, and we show that the Lagrange forms possess the closure property.

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38

Shalby, Lina, and Noha Ali. "The behavior of a satellite trajectory near the equilibrium points of sun-earth system and its control." Theoretical and Applied Mechanics, no. 00 (2023): 3. http://dx.doi.org/10.2298/tam220816003s.

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In this paper, the behavior of a satellite trajectory near the equilibrium points of the Sun-Earth system is studied. The equations describing the motion of the satellite in the circular restricted three body problem for the Sun-Earth system, are discussed for their ordinary differential equations form, and Lagrange points are determined. Then, the stability is studied at each Lagrange point. The trajectories of a satellite starting its motion near Lagrange points are illustrated, showing the stability and instability behavior. Finally, the unstable trajectory is controlled by using ??2-method at ??1 as an example.

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39

Agrawal, S. K., and T. Veeraklaew. "A Higher-Order Method for Dynamic Optimization of a Class of Linear Systems." Journal of Dynamic Systems, Measurement, and Control 118, no. 4 (December 1, 1996): 786–91. http://dx.doi.org/10.1115/1.2802358.

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This paper deals with optimization of a class of linear dynamic systems with n states and m control inputs, commanded to move between two fixed states in a prescribed final time. This problem is solved conventionally using Lagrange’s multipliers and it is well known that the optimal solution satisfies 2n first-order linear differential equations in the state and Lagrange multiplier variables. In this paper, a new procedure for dynamic optimization is presented that does not use Lagrange multipliers. In this new procedure applied to a class of linear systems with controllability index p = (n/m), optimal solution satisfies m differential equations of order 2p. The boundary conditions on these m variables are computed in terms of higher derivatives (up to p − 1) at the initial and final time. These higher-order differential equations are solved using classical weighted residual methods, methods relatively unknown to controls community but extremely popular with researchers in mechanics. This new procedure for dynamic optimization, higher order necessary condition solved by weighted residual method, is computationally more efficient compared to other conventional procedures, offering benefits for real-time applications.

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40

Shukla, Suresh K., та P. N. Pandey. "Lagrange Spaces with (γ,β)-Metric". Geometry 2013 (30 січня 2013): 1–7. http://dx.doi.org/10.1155/2013/106393.

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We study Lagrange spaces with (γ,β)-metric, where γ is a cubic metric and β is a 1-form. We obtain fundamental metric tensor, its inverse, Euler-Lagrange equations, semispray coefficients, and canonical nonlinear connection for a Lagrange space endowed with a (γ,β)-metric. Several other properties of such space are also discussed.

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41

Kasap, Zeki. "Weyl–Euler–Lagrange equations on twistor space for tangent structure." International Journal of Geometric Methods in Modern Physics 13, no. 07 (July 25, 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 theory of classical mechanic. This paper aims to introduce Euler–Lagrage partial differential equations (mathematical modeling, the equations of motion according to the time) for the movement of objects on twistor space and also to offer a general solution of differential equation system using the Maple software. Additionally, the implicit solution of the equation will be obtained as a result of a special selection of graphics to be drawn.

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42

Colombaro, Ivano, Josep Font-Segura, and Alfonso Martinez. "An Exterior Algebraic Derivation of the Euler–Lagrange Equations from the Principle of Stationary Action." Mathematics 9, no. 18 (September 7, 2021): 2178. http://dx.doi.org/10.3390/math9182178.

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In this paper, we review two related aspects of field theory: the modeling of the fields by means of exterior algebra and calculus, and the derivation of the field dynamics, i.e., the Euler–Lagrange equations, by means of the stationary action principle. In contrast to the usual tensorial derivation of these equations for field theories, that gives separate equations for the field components, two related coordinate-free forms of the Euler–Lagrange equations are derived. These alternative forms of the equations, reminiscent of the formulae of vector calculus, are expressed in terms of vector derivatives of the Lagrangian density. The first form is valid for a generic Lagrangian density that only depends on the first-order derivatives of the field. The second form, expressed in exterior algebra notation, is specific to the case when the Lagrangian density is a function of the exterior and interior derivatives of the multivector field. As an application, a Lagrangian density for generalized electromagnetic multivector fields of arbitrary grade is postulated and shown to have, by taking the vector derivative of the Lagrangian density, the generalized Maxwell equations as Euler–Lagrange equations.

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De la Peña, Luis, Ana María Cetto, and Andrea Valdés-Hernández. "Power and beauty of the Lagrange equations." Revista Mexicana de Física E 17, no. 1 Jan-Jun (January 28, 2020): 47. http://dx.doi.org/10.31349/revmexfise.17.47.

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The Lagrangian formulation of the equations of motion for point particles isusually presented in classical mechanics as the outcome of a series ofinsightful algebraic transformations or, in more advanced treatments, as theresult of applying a variational principle. In this paper we stress two mainreasons for considering the Lagrange equations as a fundamental descriptionof the dynamics of classical particles. Firstly, their structure can benaturally disclosed from the existence of integrals of motion, in a waythat, though elementary and easy to prove, seems to be less popular--or less frequently made explicit-- than others insupport of the Lagrange formulation. The second reason is that the Lagrangeequations preserve their form in \emph{any} coordinate system --even in moving ones, if required. Their covariant nature makes themparticularly suited to deal with dynamical problems in curved spaces orinvolving (holonomic) constraints. We develop the above and related ideas inclear and simple terms, keeping them throughout at the level of intermediatecourses in classical mechanics. This has the advantage of introducing sometools and concepts that are useful at this stage, while they may also serveas a bridge to more advanced courses.

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Ma, Zhichao, and Junxiang Xu. "Lagrange stability for asymptotic linear Duffing equations." Journal of Mathematical Physics 63, no. 10 (October 1, 2022): 102701. http://dx.doi.org/10.1063/5.0044864.

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In this paper, by Moser’s non-twist theorem, we prove the Lagrange stability of asymptotic linear Duffing equations under weaker nonlinear assumptions. Comparing with previous works, we avoid some assumptions, which are usually required to guarantee the twist condition.

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45

Kryński, Wojciech. "The Schwarzian derivative and Euler–Lagrange equations." Journal of Geometry and Physics 182 (December 2022): 104665. http://dx.doi.org/10.1016/j.geomphys.2022.104665.

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Lee, Eun-Hwi, and Young-Seoung Song. "STABILITY OF GENERALIZED EULER-LAGRANGE FUNCTIONAL EQUATIONS." Honam Mathematical Journal 29, no. 1 (March 25, 2007): 61–74. http://dx.doi.org/10.5831/hmj.2007.29.1.061.

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Kikuchi, Keiichi. "Extended Harmonic Mappings and Euler-Lagrange Equations." Geometry, Integrability and Quantization 17 (2016): 284–95. http://dx.doi.org/10.7546/giq-17-2016-284-295.

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Marsden, Jerrold E., Tudor S. Ratiu, and Jürgen Scheurle. "Reduction theory and the Lagrange–Routh equations." Journal of Mathematical Physics 41, no. 6 (June 2000): 3379–429. http://dx.doi.org/10.1063/1.533317.

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Wu, Jing, Zengyuan Guo, and Bai Song. "Application of lagrange equations in heat conduction." Tsinghua Science and Technology 14, S2 (December 2009): 12–16. http://dx.doi.org/10.1016/s1007-0214(10)70023-7.

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50

Vázquez, L., and S. Jiménez. "Conservative numerical schemes for Euler-Lagrange equations." Il Nuovo Cimento A 112, no. 5 (May 1999): 455–59. http://dx.doi.org/10.1007/bf03035857.

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