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Journal articles on the topic 'Lagrangian coordinates'

Author: Grafiati

Published: 4 June 2021

Last updated: 20 June 2025

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1

CAPOZZIELLO, SALVATORE, and RUGGIERO DE RITIS. "SCALE FACTOR DUALITY AND GENERAL TRANSFORMATIONS FOR STRING COSMOLOGY." International Journal of Modern Physics D 02, no. 03 (1993): 367–71. http://dx.doi.org/10.1142/s0218271893000258.

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We adopt a general point of view to obtain the scale factor duality for a class of nonminimally coupled gravitational Lagrangians which comprises the tree-level effective Lagrangian of string-dilaton cosmology. We show that in a new system of coordinates the duality is a reflection and the Lagrangians become cyclic with respect to a coordinate. When this is the situation, the dynamics is simplified and it is easier to obtain exact solutions.

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2

Abrashkin, Anatoly, and Efim Pelinovsky. "Lagrange form of the nonlinear Schrödinger equation for low-vorticity waves in deep water." Nonlinear Processes in Geophysics 24, no. 2 (2017): 255–64. http://dx.doi.org/10.5194/npg-24-255-2017.

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Abstract. The nonlinear Schrödinger (NLS) equation describing the propagation of weakly rotational wave packets in an infinitely deep fluid in Lagrangian coordinates has been derived. The vorticity is assumed to be an arbitrary function of Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. The vorticity effects manifest themselves in a shift of the wave number in the carrier wave and in variation in the coefficient multiplying the nonlinear term. In the case of vorticity dependence on the vertical Lagrangian coordinate only (Gouyon waves), the shift of the wave number and the respective coefficient are constant. When the vorticity is dependent on both Lagrangian coordinates, the shift of the wave number is horizontally inhomogeneous. There are special cases (e.g., Gerstner waves) in which the vorticity is proportional to the squared wave amplitude and nonlinearity disappears, thus making the equations for wave packet dynamics linear. It is shown that the NLS solution for weakly rotational waves in the Eulerian variables may be obtained from the Lagrangian solution by simply changing the horizontal coordinates.

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3

Qian, Yu-Kun, Shiqiu Peng, and Chang-Xia Liang. "Reconciling Lagrangian Diffusivity and Effective Diffusivity in Contour-Based Coordinates." Journal of Physical Oceanography 49, no. 6 (2019): 1521–39. http://dx.doi.org/10.1175/jpo-d-18-0251.1.

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AbstractThe present study reconciles theoretical differences between the Lagrangian diffusivity and effective diffusivity in a transformed spatial coordinate based on the contours of a quasi-conservative tracer. In the transformed coordinate, any adiabatic stirring effect, such as shear-induced dispersion, is naturally isolated from diabatic cross-contour motions. Therefore, Lagrangian particle motions in the transformed coordinate obey a transformed zeroth-order stochastic (i.e., random walk) model with the diffusivity replaced by the effective diffusivity. Such a stochastic model becomes the theoretical foundation on which both diffusivities are exactly unified. In the absence of small-scale diffusion, particles do not disperse at all in the transformed contour coordinate. Besides, the corresponding Lagrangian autocorrelation becomes a delta function and is thus free from pronounced overshoot and negative lobe at short time lags that may be induced by either Rossby waves or mesoscale eddies; that is, particles decorrelate immediately and Lagrangian diffusivity is already asymptotic no matter how small the time lag is. The resulting instantaneous Lagrangian spreading rate is thus conceptually identical to the effective diffusivity that only measures the instantaneous irreversible mixing. In these regards, the present study provides a new look at particle dispersion in contour-based coordinates.

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4

Thiffeault, Jean-Luc. "Advection–diffusion in Lagrangian coordinates." Physics Letters A 309, no. 5-6 (2003): 415–22. http://dx.doi.org/10.1016/s0375-9601(03)00244-5.

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5

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

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

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6

Kopiev, V. F., and S. A. Chernyshev. "LAGRANGIAN FORMALISM IN PROBLEMS OF SMALL OSCILLATIONS OF VORTEX FLOWS AND ITS CONNECTION WITH THE VARIATIONAL PRINCIPLE FOR IDEAL INCOMPRESSIBLE HYDRODYNAMICS OF VORTEX LINES." XXII workshop of the Council of nonlinear dynamics of the Russian Academy of Sciences 47, no. 1 (2019): 74–77. http://dx.doi.org/10.29006/1564-2291.jor-2019.47(1).21.

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The paper discusses the description of vortex flows of an ideal incompressible fluid based on the formalism of Lagrangian mechanics. Using the displacement field of liquid particles as a generalized coordinate, we write out the Lagrangian describing the dynamics of small perturbations (Kopiev, Chernyshev, 2018). The corresponding Lagrange equations are the equation for the displacement field (Drazim, Reid, 1981): This equation is equivalent to the Helmholtz equation for vorticity perturbations. The displacement field is defined as the difference in the positions of liquid particles on trajectories in disturbed and undisturbed flows. Although this definition is given in terms of Lagrangian variables associated with liquid particles, the displacement field itself is an Euler variable, expressed through velocity and vorticity perturbations. An example of using Lagrangian to solve the problem of conservation of the quadrupole moment of a vortex flow is considered. Using the Noether theorem, conditions on a stationary flow are obtained, under which the quadrupole moment of small perturbations of this flow is an integral of motion (Kopiev, Chernyshev, 2018). It is shown that these conditions are satisfied for the jet flows uniform along the longitudinal coordinate. The result obtained is important in aeroacoustics due to the fact that the quadrupole moment of the vortex flow represents the main term of the decomposition of a compact acoustic source in Machnumber (Lighthill, 1952; Crow, 1970; Kopiev, Chernyshev, 1995). The generalization of these results to the nonlinear case is considered. The Lagrangian is obtained for an arbitrary nonlinear displacement field: nowhere Gis Green’s function of the Laplace equation. The corresponding Lagrange equations coincide with the differential equations describing the nonlinear dynamics of the displacement field (Drazin, Reid, 1981). Expansion of the Lagrangian in small perturbations to quadratic terms gives the Lagrangian of the linear system. The question of the relationship of the proposed approach to the description of the dynamics of an incompressible fluid and known approaches based on the formalism of Lagrangian mechanics with the coordinates of liquid particles as generalized coordinates (Chapman, 1978; Goncharov, Pavlov, 2008; Kuznetsov, Ruban, 1998) is considered. It is shown that the transformation of the Lagrangian obtained in (Kuznetsov, Ruban, 1998) to the Lagrangian can be carried out by transforming Lagrangian variables (coordinates of liquid particles) to Eulerian variables (displacement field). This study was supported by the Russian Science Foundation, project No. 17-11-01271.

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7

Meleshko, Sergey V., and Evgeniy I. Kaptsov. "Symmetry Analysis of the Two-Dimensional Stationary Gas Dynamics Equations in Lagrangian Coordinates." Mathematics 12, no. 6 (2024): 879. http://dx.doi.org/10.3390/math12060879.

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This article analyzes the symmetry of two-dimensional stationary gas dynamics equations in Lagrangian coordinates, including the search for equivalence transformations, the group classification of equations, the derivation of group foliations, and the construction of conservation laws. The consideration of equations in Lagrangian coordinates significantly simplifies the procedure for obtaining conservation laws, which are derived using the Noether theorem. The final part of the work is devoted to group foliations of the gas dynamics equations, including for the nonstationary isentropic case. The group foliations approach is usually employed for equations that admit infinite-dimensional groups of transformations (which is exactly the case for the gas dynamics equations in Lagrangian coordinates) and may make it possible to simplify their further analysis. The results obtained in this regard generalize previously known results for the two-dimensional shallow water equations in Lagrangian coordinates.

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8

Thiffeault, Jean-Luc. "The strange eigenmode in Lagrangian coordinates." Chaos: An Interdisciplinary Journal of Nonlinear Science 14, no. 3 (2004): 531–38. http://dx.doi.org/10.1063/1.1759431.

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9

Dubos, Thomas, and Marine Tort. "Equations of Atmospheric Motion in Non-Eulerian Vertical Coordinates: Vector-Invariant Form and Quasi-Hamiltonian Formulation." Monthly Weather Review 142, no. 10 (2014): 3860–80. http://dx.doi.org/10.1175/mwr-d-14-00069.1.

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Abstract The curl form of equations of inviscid atmospheric motion in general non-Eulerian coordinates is obtained. Narrowing down to a general vertical coordinate, a quasi-Hamiltonian form is then obtained in a Lagrangian, isentropic, mass-based or z-based vertical coordinate. In non-Lagrangian vertical coordinates, the conservation of energy by the vertical transport terms results from the invariance of energy under the vertical relabeling of fluid parcels. A complete or partial separation between the horizontal and vertical dynamics is achieved, except in the Eulerian case. The horizontal–vertical separation is especially helpful for (quasi-)hydrostatic systems characterized by vanishing vertical momentum. Indeed for such systems vertical momentum balance reduces to a simple statement: total energy is stationary with respect to adiabatic vertical displacements of fluid parcels. From this point of view the purpose of (quasi-)hydrostatic balance is to determine the vertical positions of fluid parcels, for which no evolution equation is readily available. This physically appealing formulation significantly extends previous work. The general formalism is exemplified for the fully compressible Euler equations in a Lagrangian vertical coordinate and a Cartesian (x, z) slice geometry, and the deep-atmosphere quasi-hydrostatic equations in latitude–longitude horizontal coordinates. The latter case, in particular, illuminates how the apparent intricacy of the time-dependent metric terms and of the additional forces can be absorbed into a proper choice of prognostic variables. In both cases it is shown how the quasi-Hamiltonian form leads straightforwardly to the conservation of energy using only integration by parts. Relationships with previous work and implications for stability analysis and the derivation of approximate sets of equations and energy-conserving numerical schemes are discussed.

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10

Bleck, Rainer, Stan Benjamin, Jin Lee, and Alexander E. MacDonald. "On the Use of an Adaptive, Hybrid-Isentropic Vertical Coordinate in Global Atmospheric Modeling." Monthly Weather Review 138, no. 6 (2010): 2188–210. http://dx.doi.org/10.1175/2009mwr3103.1.

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Abstract This article is one in a series describing the functionality of the Flow-Following, Finite-Volume Icosahedral Model (FIM) developed at NOAA’s Earth System Research Laboratory. Emphasis in this article is on the design of the vertical coordinate—the “flow following” aspect of FIM. The coordinate is terrain-following near the ground and isentropic in the free atmosphere. The spatial transition between the two coordinates is adaptive and is based on the arbitrary Lagrangian–Eulerian (ALE) paradigm. The impact of vertical resolution trade-offs between the present hybrid approach and traditional terrain-following coordinates is demonstrated in a three-part case study.

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11

Abdelmohssin, Faisal A. Y., and Osman M. H. El Mekki. "The Hamiltonian of f(R) gravity." Canadian Journal of Physics 99, no. 9 (2021): 814–19. http://dx.doi.org/10.1139/cjp-2021-0058.

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We derive conjugate momenta variable tensors and the Hamiltonian equation of the source-free f(R) gravity from first principles using the Legendre transformation of these conjugate momenta variable tensors, conjugate coordinates variables — fundamental metric tensor and its first ordinary partial derivatives with respect to space–time coordinates and second ordinary partial derivatives with respect to space–time coordinates — and the Lagrangian of the f(R) gravity. Interpreting the derived Hamiltonian as the energy of the f(R) gravity we have shown that it vanishes for linear Lagrangians in Ricci scalar curvature without source (e.g., Einstein–Hilbert Lagrangian without matter fields), which is the same result obtained using the stress–energy tensor equation derived from variation of the matter field Lagrangian density. The resulting Hamiltonian equation forbids any negative power law model in the dependence of the f(R) gravity on Ricci scalar curvature: f(R) = αR–r, where r and α are positive real numbers; it also forbids any polynomial equation that contains terms with negative powers of the Ricci scalar curvature including a constant term, in which cases the Hamiltonian function in the Ricci scalar and therefore the energy of the f(R) gravity would attain a negative value and would not be bounded from below. The restrictions imposed by the non-negative Hamiltonian have far-reaching consequences as a result of applying f(R) gravity to the study of black holes and the Friedmann–Lemaître–Robertson–Walker model in cosmology.

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12

Andreev, Victor. "Symmetries of Euler Equations in Lagrangian Coordinates." Journal of Nonlinear Mathematical Physics 3, no. 1-2 (1996): 196–201. http://dx.doi.org/10.2991/jnmp.1996.3.1-2.23.

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13

Matarrese, S., and D. Terranova. "Post-Newtonian cosmological dynamics in Lagrangian coordinates." Monthly Notices of the Royal Astronomical Society 283, no. 2 (1996): 400–418. http://dx.doi.org/10.1093/mnras/283.2.400.

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14

Mead, J. L. "The shallow water equations in Lagrangian coordinates." Journal of Computational Physics 200, no. 2 (2004): 654–69. http://dx.doi.org/10.1016/j.jcp.2004.04.014.

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15

Hynd, Ryan. "Lagrangian Coordinates for the Sticky Particle System." SIAM Journal on Mathematical Analysis 51, no. 5 (2019): 3769–95. http://dx.doi.org/10.1137/19m1241775.

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16

Fyrillas, Marios M., and Keiko K. Nomura. "Diffusion and Brownian motion in Lagrangian coordinates." Journal of Chemical Physics 126, no. 16 (2007): 164510. http://dx.doi.org/10.1063/1.2717185.

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17

Raju, M. S., and W. A. Sirignano. "Spray Computations in a Centerbody Combustor." Journal of Engineering for Gas Turbines and Power 111, no. 4 (1989): 710–18. http://dx.doi.org/10.1115/1.3240317.

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A hybrid Eulerian–Lagrangian method is employed to model the reactive flow field of a centerbody combustor. The unsteady two-dimensional gas-phase equations are represented in Eulerian coordinates and liquid-phase equations are formulated in Lagrangian coordinates. The gas-phase equations based on the conservation of mass, momentum, and energy are supplemented by turbulence and combustion models. The vaporization model takes into account the transient effects associated with the droplet heating and liquid-phase internal circulation. The integration scheme is based on the TEACH algorithm for gas-phase equations, the Runge-Kutta method for liquid-phase equations, and linear interpolation between the two coordinate systems. The calculations show that the droplet penetration and recirculation characteristics are strongly influenced by the gas- and liquid-phase interaction in such a way that most of the vaporization process is confined to the wake region of the centerbody, thereby improving the flame stabilization properties of the flow field.

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18

Guan, Chunxia, Kai Yan, and Xuemei Wei. "Lipschitz metric for the modified two-component Camassa–Holm system." Analysis and Applications 16, no. 02 (2018): 159–82. http://dx.doi.org/10.1142/s0219530516500226.

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This paper is devoted to the existence and Lipschitz continuity of global conservative weak solutions in time for the modified two-component Camassa–Holm system on the real line. We obtain the global weak solutions via a coordinate transformation into the Lagrangian coordinates. The key ingredients in our analysis are the energy density given by the positive Radon measure and the proposed new distance functions as well.

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19

Shuh, Jing Ying. "DISCOVERY OF LAGRANGIAN EQUATION FOR FLUID MECHANICS." Mechanical Engineering: An International Journal (MEIJ) 04, no. 1/2/3/4 (2023): 06. https://doi.org/10.5281/zenodo.7759691.

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This paper is intended to build a connection between dynamics and fluid mechanics. Since Lagrangian equations are very useful tools in dynamics now it is discovered that it can be also used in fluid mechanics, so connection can be built through this equation... Here, the lagrangian equation is derived from the momentum equation in fluid mechanics, and then the equation is applied to three different coordinates, Cartesian, cylindrical and spherical. Certainly the application of the equation is not limited to these three different coordinates. This is just for illustration. Many applications are expected.

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20

Bates, Larry M. "Examples for obstructions to action-angle coordinates." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 110, no. 1-2 (1988): 27–30. http://dx.doi.org/10.1017/s0308210500024823.

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SynopsisWe give examples of symplectic manifolds which are also non-trivial principal torus-bundles with Lagrangian fibres. These bundles are examples of spaces with an obstruction to the global existence of action-angle variables.

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21

Gao, Yu, and Jian-Guo Liu. "The modified Camassa-Holm equation in Lagrangian coordinates." Discrete & Continuous Dynamical Systems - B 23, no. 6 (2018): 2545–92. http://dx.doi.org/10.3934/dcdsb.2018067.

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22

Merrill, John T., Rainer Bleck, and Douglas Boudra. "Techniques of Lagrangian Trajectory Analysis in Isentropic Coordinates." Monthly Weather Review 114, no. 3 (1986): 571–81. http://dx.doi.org/10.1175/1520-0493(1986)114<0571:toltai>2.0.co;2.

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23

Osborne, A. R., A. D. Kirwan, A. Provenzale, and L. Bergamasco. "The Korteweg–de Vries equation in Lagrangian coordinates." Physics of Fluids 29, no. 3 (1986): 656. http://dx.doi.org/10.1063/1.865460.

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24

Zamora-Sillero, Elías, and Alexander V. Shapovalov. "Equivalent Lagrangian densities and invariant collective coordinates equations." Journal of Physics A: Mathematical and Theoretical 44, no. 6 (2011): 065204. http://dx.doi.org/10.1088/1751-8113/44/6/065204.

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25

Bira, B., and T. Raja Sekhar. "Exact solutions to magnetogasdynamic equations in Lagrangian coordinates." Journal of Mathematical Chemistry 53, no. 4 (2015): 1162–71. http://dx.doi.org/10.1007/s10910-015-0476-8.

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26

Liu, Xiaodong, Nathaniel R. Morgan, and Donald E. Burton. "Lagrangian discontinuous Galerkin hydrodynamic methods in axisymmetric coordinates." Journal of Computational Physics 373 (November 2018): 253–83. http://dx.doi.org/10.1016/j.jcp.2018.06.073.

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27

Mead, J. L. "Assimilation of simulated float data in Lagrangian coordinates." Ocean Modelling 8, no. 4 (2005): 369–94. http://dx.doi.org/10.1016/j.ocemod.2004.02.003.

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28

Kase, S., and J. Katsui. "Analysis of melt spinning transients in Lagrangian coordinates." Rheologica Acta 24, no. 1 (1985): 34–43. http://dx.doi.org/10.1007/bf01329260.

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29

GEYER, B., D. M. GITMAN, P. M. LAVROV, and P. YU MOSHIN. "SUPERFIELD EXTENDED BRST QUANTIZATION IN GENERAL COORDINATES." International Journal of Modern Physics A 19, no. 05 (2004): 737–49. http://dx.doi.org/10.1142/s0217751x04017604.

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We propose a superfield formalism of Lagrangian BRST–anti-BRST quantization of arbitrary gauge theories in general coordinates with the base manifold of fields and antifields described in terms of both bosonic and fermionic variables.

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30

Bernstein, Dennis S., Ankit Goel, and Omran Kouba. "Deriving Euler’s Equation for Rigid-Body Rotation via Lagrangian Dynamics with Generalized Coordinates." Mathematics 11, no. 12 (2023): 2727. http://dx.doi.org/10.3390/math11122727.

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Euler’s equation relates the change in angular momentum of a rigid body to the applied torque. This paper uses Lagrangian dynamics to derive Euler’s equation in terms of generalized coordinates. This is done by parameterizing the angular velocity vector in terms of 3-2-1 and 3-1-3 Euler angles as well as Euler parameters, that is, quaternions. This paper fills a gap in the literature by using generalized coordinates to parameterize the angular velocity vector and thereby transform the dynamics obtained from Lagrangian dynamics into Euler’s equation for rigid-body rotation.

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31

AYCAN, CANSEL, SEVKET CIVELEK, and SIMGE DAGLI. "IMPROVING LAGRANGIAN ENERGY EQUATIONS ON KÄHLER JET BUNDLES." International Journal of Geometric Methods in Modern Physics 10, no. 07 (2013): 1350026. http://dx.doi.org/10.1142/s0219887813500266.

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This study proposes to improve the Lagrangian energy equations for instructed complex jet bundles on Kähler manifolds. The coordinates on the bundle structure of Kähler manifolds have been given for real and imaginary dimensions. For given bundle structures, all fundamental geometrical properties have been investigated and applications to complex bundle structures are carried out. The energy equations have been applied to the numerical example in order to test its performance. Moreover, velocity and time dimensions for energy movement equations have been presented as a new concept. This study shows some physical applications of those equations and interpretations are made. Results show that the imaginary coordinates are same as real coordinates. One of the interesting conclusion of this study is that the Lagrangian energy of a movement particle is static when the particle moves in a large velocity.

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32

GEYER, B., and P. M. LAVROV. "MODIFIED TRIPLECTIC QUANTIZATION IN GENERAL COORDINATES." International Journal of Modern Physics A 19, no. 10 (2004): 1639–54. http://dx.doi.org/10.1142/s0217751x04018142.

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We present an extension of the previous results1 on the quantization of general gauge theories within the BRST–antiBRST invariant Lagrangian scheme in general coordinates. Namely, we generalize Ref. 1 to the case when the base manifold of fields and antifields is a supermanifold described in terms of both bosonic and fermionic coordinates.

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33

Parsa, Kourosh. "THE LAGRANGIAN DERIVATION OF KANE’S EQUATIONS." Transactions of the Canadian Society for Mechanical Engineering 31, no. 4 (2007): 407–20. http://dx.doi.org/10.1139/tcsme-2007-0029.

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The Lagrangian approach to the development of dynamics equations for a multi-body system, constrained or otherwise, requires solving the forward kinematics of the system at velocity level in order to derive the kinetic energy of the system. The kinetic-energy expression should then be differentiated multiple times to derive the equations of motion of the system. Among these differentiations, the partial derivative of kinetic energy with respect to the system generalized coordinates is specially cumbersome. In this paper, we will derive this partial derivative using a novel kinematic relation for the partial derivative of angular velocity with respect to the system generalized coordinates. It will be shown that, as a result of the use of this relation, the equations of motion of the system are directly derived in the form of Kane’s equations.

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34

Guangbao, Wang, and Ding Guangtao. "The Lagrangian and Hamiltonian for the Two-Dimensional Mathews-Lakshmanan Oscillator." Advances in Mathematical Physics 2020 (August 1, 2020): 1–6. http://dx.doi.org/10.1155/2020/2378989.

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The purpose of this paper is to illustrate the theory and methods of analytical mechanics that can be effectively applied to the research of some nonlinear nonconservative systems through the case study of two-dimensionally coupled Mathews-Lakshmanan oscillator (abbreviated as M-L oscillator). (1) According to the inverse problem method of Lagrangian mechanics, the Lagrangian and Hamiltonian function in the form of rectangular coordinates of the two-dimensional M-L oscillator is directly constructed from an integral of the two-dimensional M-L oscillators. (2) The Lagrange and Hamiltonian function in the form of polar coordinate was rewritten by using coordinate transformation. (3) By introducing the vector form variables, the two-dimensional M-L oscillator motion differential equation, the first integral, and the Lagrange function are written. Therefore, the two-dimensional M-L oscillator is directly extended to the three-dimensional case, and it is proved that the three-dimensional M-L oscillator can be reduced to the two-dimensional case. (4) The two direct integration methods were provided to solve the two-dimensional M-L oscillator by using polar coordinate Lagrangian and pointed out that the one-dimensional M-L oscillator is a special case of the two-dimensional M-L oscillator.

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35

TSUKIOKA, TAKUYA, and YOSHIYUKI WATABIKI. "QUANTIZATION OF BOSONIC STRING MODEL IN (26+2)-DIMENSIONAL SPACE–TIME." International Journal of Modern Physics A 19, no. 12 (2004): 1923–59. http://dx.doi.org/10.1142/s0217751x04017641.

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We investigate the quantization of the bosonic string model which has a local U (1) V × U (1) A gauge invariance as well as the general coordinate and Weyl invariance on the world-sheet. The model is quantized by Lagrangian and Hamiltonian BRST formulations á la Batalin, Fradkin and Vilkovisky and noncovariant light-cone gauge formulation. Upon the quantization the model turns out to be formulated consistently in (26+2)-dimensional background space–time involving two time-like coordinates.

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36

Chaiyasena, A., W. Worapitpong, and S. V. Meleshko. "Generalized Riemann waves and their adjoinment through a shock wave." Mathematical Modelling of Natural Phenomena 13, no. 2 (2018): 22. http://dx.doi.org/10.1051/mmnp/2018027.

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Generalized simple waves of the gas dynamics equations in Lagrangian and Eulerian descriptions are studied in the paper. As in the collision of a shock wave and a rarefaction wave, a flow becomes nonisentropic. Generalized simple waves are applied to describe such flows. The first part of the paper deals with constructing a solution describing their adjoinment through a shock wave in Eulerian coordinates. Even though the Eulerian form of the gas dynamics equations is most frequently used in applications, there are advantages for some problems concerning the gas dynamics equations in Lagrangian coordinates, for example, of being able to be reduced to an Euler–Lagrange equation. Through the technique of differential constraints, necessary and sufficient conditions for the existence of generalized simple waves in the Lagrangian description are provided in the second part of the paper.

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37

Rimrott, F. P. J., and W. M. Szczygielski. "CENTRIFUGAL IMPULSE AS COORDINATE IN THE TABARROKIAN FORMULATION." Transactions of the Canadian Society for Mechanical Engineering 19, no. 3 (1995): 261–69. http://dx.doi.org/10.1139/tcsme-1995-0013.

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While the well-known conventional Lagrange equation, based on kinetic coenergy and potential energy, uses generalized displacements of the inertia (mass) elements of a system as coordinates, the complementary alternative or Tabarrok formulation, is based on kinetic energy and potential coenergy, and uses as coordinates the generalized impulses of the system’s force (spring) elements. A model system specifically selected to be as simple as possible, yet to contain all essential elements for an illustration of the application of the Tabarrokian approach for the case where a centrifugal force is present, has been devised to show that the centrifugal impulse appears as additional coordinate for the complementary Lagrangian, and that the system turns out to be non-Tabarrokian. It is then shown that the centrifugal impulse is related to the other impulse coordinates by a nonholonomic constraint. Eventually the compatibility equations of motion for the model system are obtained.

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38

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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39

Musicki, Djordje. "Extended Lagrangian formalism for rheonomic systems with variable mass." Theoretical and Applied Mechanics 44, no. 1 (2017): 115–32. http://dx.doi.org/10.2298/tam170601006m.

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In this paper the extended Lagrangian formalism for the rheonomic systems (Dj. Musicki, 2004), which began with the modification of the mechanics of such systems (V. Vujicic, 1987), is extended to the systems with variable mass, with emphasis on the corresponding energy relations. This extended Lagrangian formalism is based on the extension of the set of chosen generalized coordinates by new quantities, suggested by the form of nonstationary constraints, which determine the position of the frame of reference in respect to which these generalized coordinates refer. As a consequence, an extended system of the Lagrangian equations is formulated, accommodated to the variability of the masses of particles, where the additional ones correspond to the additional generalized coordinates. By means of these equations, the energy relations of such systems have been studied, where it is demonstrated that here there are four types of energy conservation laws. The obtained energy laws are more complete and natural than the corresponding ones in the usual Lagrangian formulation for such systems. It is demonstrated that the obtained energy laws, are in full accordance with the energy laws in the corresponding vector formulation, if they are expressed in terms of the quantities introduced in this formulation of mechanics. The obtained results are illustrated by an example: the motion of a rocket, which ejects the gasses backwards, while this rocket moves up a straight line on an oblique plane, which glides uniformly in a horizontal direction.

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40

Toy, Michael D. "Incorporating Condensational Heating into a Nonhydrostatic Atmospheric Model Based on a Hybrid Isentropic–Sigma Vertical Coordinate." Monthly Weather Review 139, no. 9 (2011): 2940–54. http://dx.doi.org/10.1175/mwr-d-10-05015.1.

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Using isentropic coordinates in atmospheric models has the advantage of eliminating the cross-coordinate vertical mass flux for adiabatic flow, and virtually eliminating the associated numerical error in the vertical transport. This is a significant benefit since much of the flow in the atmosphere is approximately adiabatic. Nonadiabatic processes, such as condensational heating, result in a nonzero vertical velocity [Formula: see text] in isentropic coordinates. A method for incorporating condensational heating into a nonhydrostatic atmospheric model based on a hybrid isentropic–sigma vertical coordinate is presented. The model is tested with various 2D moist simulations and the results are compared with those using a traditional terrain-following, height-based sigma coordinate. With the hybrid coordinate, there are improvements in the representation of the developing cloud field in a mountain wave experiment. In a simulation of deep convection, the adaptive hybrid coordinate successfully simulates the turbulent nature of the convection, while maintaining the quasi-Lagrangian nature of the isentropic coordinate in the surrounding dry air. The vertical cross-coordinate mass flux is almost zero in the environmental air, as well as in the stratosphere above the convective tower.

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Caicedo, Jose Francisco, C. Klingenberg, Yunguang Lu, and Leonardo Rendon. "Hyperbolic Approximation on System of Elasticity in Lagrangian Coordinates." Natural Science 06, no. 07 (2014): 477–86. http://dx.doi.org/10.4236/ns.2014.67046.

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HSU, HUNG-CHU, YANG-YIH CHEN, JOHN R. C. HSU, and WEN-JER TSENG. "NONLINEAR WATER WAVES ON UNIFORM CURRENT IN LAGRANGIAN COORDINATES." Journal of Nonlinear Mathematical Physics 16, no. 1 (2009): 47–61. http://dx.doi.org/10.1142/s1402925109000054.

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Garrahan, Juan P., Martín Kruczenski, and Daniel R. Bes. "Lagrangian Becchi-Rouet-Stora-Tyutin treatment of collective coordinates." Physical Review D 53, no. 12 (1996): 7176–86. http://dx.doi.org/10.1103/physrevd.53.7176.

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Hsu, Hung-Chu, Yang-Yih Chen, and Cyun-Fu Wang. "Perturbation analysis of short-crested waves in Lagrangian coordinates." Nonlinear Analysis: Real World Applications 11, no. 3 (2010): 1522–36. http://dx.doi.org/10.1016/j.nonrwa.2009.03.014.

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Magazov, F. G., E. S. Shestakovskaya, and M. N. Yakimova. "The shock convergence problem in Euler and Lagrangian coordinates." Journal of Physics: Conference Series 1147 (January 2019): 012029. http://dx.doi.org/10.1088/1742-6596/1147/1/012029.

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46

Osborne, A. R., and G. Boffetta. "The shallow‐water nonlinear Schrödinger equation in Lagrangian coordinates." Physics of Fluids A: Fluid Dynamics 1, no. 7 (1989): 1200–1210. http://dx.doi.org/10.1063/1.857343.

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47

Guz’, A. N. "Identification of Lagrangian and Eulerian coordinates in continuum mechanics." International Applied Mechanics 34, no. 10 (1998): 965–68. http://dx.doi.org/10.1007/bf02701051.

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48

Shashkov, Mikhail, and Burton Wendroff. "A Composite Scheme for Gas Dynamics in Lagrangian Coordinates." Journal of Computational Physics 150, no. 2 (1999): 502–17. http://dx.doi.org/10.1006/jcph.1999.6192.

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49

Künzle, H. P. "Lagrangian formalism for adiabatic fluids on five-dimensional space-time." Canadian Journal of Physics 64, no. 2 (1986): 185–89. http://dx.doi.org/10.1139/p86-032.

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The recently developed Lagrangian formalism on extended five-dimensional space-time that permits a unified description of general relativistic and gravitating nonrelativistic classical fields is applied to a model of an adiabatic perfect fluid described in terms of Lagrangian coordinates. The Lagrangian density is chosen as an arbitrary Lorentz- (or Galilei-) invariant function of the 5-current vector and leads, by variation with respect to frame fields, to a 5-stress-energy tensor, whose additional components are naturally interpreted as an entropy-flux vector.

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Tort, Marine, and Thomas Dubos. "Usual Approximations to the Equations of Atmospheric Motion: A Variational Perspective." Journal of the Atmospheric Sciences 71, no. 7 (2014): 2452–66. http://dx.doi.org/10.1175/jas-d-13-0339.1.

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Abstract The usual geophysical approximations are reframed within a variational framework. Starting from the Lagrangian of the fully compressible Euler equations expressed in a general curvilinear coordinates system, Hamilton’s principle of least action yields Euler–Lagrange equations of motion. Instead of directly making approximations in these equations, the approach followed is that of Hamilton’s principle asymptotics; that is, all approximations are performed in the Lagrangian. Using a coordinate system where the geopotential is the third coordinate, diverse approximations are considered. The assumptions and approximations covered are 1) particular shapes of the geopotential; 2) shallowness of the atmosphere, which allows for the approximation of the relative and planetary kinetic energy; 3) small vertical velocities, implying quasi-hydrostatic systems; and 4) pseudoincompressibility, enforced by introducing a Lagangian multiplier. This variational approach greatly facilitates the derivation of the equations and systematically ensures their dynamical consistency. Indeed, the symmetry properties of the approximated Lagrangian imply the conservation of energy, potential vorticity, and momentum. Justification of the equations then relies, as usual, on a proper order-of-magnitude analysis. As an illustrative example, the asymptotic consistency of recently introduced shallow-atmosphere equations with a complete Coriolis force is discussed, suggesting additional corrections to the pressure gradient and gravity.

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