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Books on the topic 'Lagrangian points'

Author: Grafiati

Published: 4 June 2021

Last updated: 15 June 2025

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1

1959-, Fukaya Kenji, ed. Lagrangian intersection floer theory: Anomaly and obstruction. American Mathematical Society, 2009.

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2

G, Gómez, ed. Dynamics and mission design near libration points. World Scientific, 2001.

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3

Baldomá, Inmaculada. Exponentially small splitting of invariant manifolds of parabolic points. American Mathematical Society, 2004.

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4

Ambrosetti, A. Periodic solutions of singular Lagrangian systems. Birkhäuser, 1993.

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5

Mazzucchelli, Marco. Critical Point Theory for Lagrangian Systems. Springer Basel, 2012. http://dx.doi.org/10.1007/978-3-0348-0163-8.

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6

service), SpringerLink (Online, ed. Critical Point Theory for Lagrangian Systems. Springer Basel AG, 2012.

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7

K, Prasad. Simulation studies on cyclone track prediction by quasi-lagrangian model (QLM) in some historical and recent cases in the Bay of Bengal, using global re-analysis and forecast grid point data sets. SAARC Meteorological Research Centre, 2006.

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8

Salzer, Herbert E., Norman Levine, and Saul Serben. Tables for Lagrangian Interpolation Using Chebyshev Points. Applied Science Pubn, 1988.

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9

Deruelle, Nathalie, and Jean-Philippe Uzan. Lagrangian mechanics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0008.

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This chapter shows how the Newtonian law of motion of a particle subject to a gradient force derived from a ‘potential energy’ can always be obtained from an extremal principle, or ‘principle of least action’. According to Newton’s first law, the trajectory representing the motion of a free particle between two points p1 and p2 is a straight line. In other words, out of all the possible paths between p1 and p2, the trajectory effectively followed by a free particle is the one that minimizes the length. However, even though the use of the principle of extremal length of the paths between two points gives the straight line joining the points, this does not mean that the straight-line path is traced with constant velocity in an inertial frame. Moreover, the trajectory describing the motion of a particle subject to a force is not uniform and rectilinear.

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10

Lagrangian intersection floer theory: Anomaly and obstruction. American Mathematical Society, 2009.

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11

Simo, Carles, J. Llibre, and R. Martinez. Dynamics and Mission Design Near Libration Points, Vol. II: Fundamentals: The Case of Triangular Libration Points. World Scientific Publishing Company, 2001.

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12

Dynamics and Mission Design Near Libration Points, Vol. II: Fundamentals: The Case of Triangular Libration Points. World Scientific Pub Co Inc, 2001.

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13

Jorba, Angel, Carles Simo, and Josep Masdemont. Dynamics and Mission Design Near Libration Points, Vol. III, Advanced Methods for Collinear Points. World Scientific Publishing Company, 2001.

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14

Jorba, Angel, Carles Simo, and Josep Masdemont. Dynamics and Mission Design Near Libration Points, Vol. IV: Advanced Methods for Triangular Points. World Scientific Publishing Company, 2001.

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15

Simo, Carles, J. Llibre, and R. Martinex. Dynamics and Mission Design Near Libration Points, Volume I : Fundamentals : The Case of Collinear Libration Points (World Scientific Monograph Series in Mathematics). World Scientific Publishing Company, 2001.

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16

McDuff, Dusa, and Dietmar Salamon. The arnold conjecture. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198794899.003.0012.

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This chapter contains a proof of the Arnold conjecture for the standard torus, which is based on the discrete symplectic action. The symplectic part of this proof is very easy. However, for completeness of the exposition, one section is devoted to a fairly detailed discussion of the relevant Conley index theory and of Ljusternik–Schnirelmann theory. Closely related to the problem of finding symplectic fixed points is the Lagrangian intersection problem. The chapter outlines a proof of Arnold’s conjecture for cotangent bundles that again uses the discrete symplectic action, this time to construct generating functions for Lagrangian submanifolds. The chapter ends with a brief outline of the construction and applications of Floer homology.

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17

Mann, Peter. Point Transformations in Lagrangian Mechanics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0009.

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This chapter discusses point transformations in Lagrangian mechanics. Sometimes, when solving problems, it is useful to change coordinates in velocity phase space to better suit and simplify the system at hand; this is a requirement of any physical theory. This change is often motivated by some experimentally observed physicality of the system or may highlight new conserved quantities that might have been overlooked using the old description. In the Newtonian formalism, it was a bit of a hassle to change coordinates and the equations of motion will look quite different. In this chapter, point transformations in Lagrangian mechanics are developed and the Euler–Lagrange equation is found to be covariant. The chapter discusses coordinate transformations, parametrisation invariance and the Jacobian of the transform. Re-parametrisations are also included.

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18

Coti-Zelati, V., and A. Ambrosetti. Periodic Solutions of Singular Lagrangian Systems. Birkhauser Verlag, 2012.

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19

Ambrosetti, A. Periodic Solutions of Singular Lagrangian Systems. Springer, 2013.

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20

Mazzucchelli, Marco. Critical Point Theory for Lagrangian Systems. Springer Basel AG, 2014.

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21

Mazzucchelli, Marco. Critical Point Theory for Lagrangian Systems. Springer, 2011.

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22

Critical Point Theory for Lagrangian Systems. Birkhäuser Boston, 2014.

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23

Kachelriess, Michael. Classical mechanics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198802877.003.0001.

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This chapter reviews those concepts of classical mechanics which are essential for progressing towards quantum theory. The Lagrangian and Hamiltonian formulation of classical mechanics are derived from action principles. The Green function method is illustrated and the action of a relativistic point particle recalled.

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24

Mann, Peter. Poisson Brackets & Angular Momentum. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0017.

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This chapter discusses canonical transformations and gauge transformations and is divided into three sections. In the first section, canonical coordinate transformations are introduced to the reader through generating functions as the extension of point transformations used in Lagrangian mechanics, with the harmonic oscillator being used as an example of a canonical transformation. In the second section, gauge theory is discussed in the canonical framework and compared to the Lagrangian case. Action-angle variables, direct conditions, symplectomorphisms, holomorphic variables, integrable systems and first integrals are examined. The third section looks at infinitesimal canonical transformations resulting from functions on phase space. Ostrogradsky equations in the canonical setting are also detailed.

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25

Zeitlin, Vladimir. Vortex Dynamics on the f and beta Plane and Wave Radiation by Vortices. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198804338.003.0006.

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Quasi-geostrophic dynamics being essentially the vortex dynamics, the main notions of vortex dynamics in the plane are introduced in this chapter. Dynamics of vorticity is treated both in Eulerian and Lagrangian descriptions. Dynamics of point vortices and vortex patches (contour dynamics) are recalled, as well as discretisations of the vorticity equation preserving Casimir invariants, which reflect Lagrangian conservation of vorticity. The influence of the beta effect upon vortices is illustrated, and exact modon solutions of the QG equations on the f and beta planes are constructed. Basic notions of turbulence and specific features of two dimensional turbulence are reviewed for future use. Lighthill radiation of gravity waves by vortices is illustrated on the example of a pair of point vortices, and back-reaction of the radiation upon the vortex system is demonstrated and analysed. Influence of rotation upon the Lighthill radiation is explained. Construction of the Kirchhoff vortex solution is proposed as a problem.

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26

Sorrentino, Alfonso. The Hamilton-Jacobi Equation and Weak KAM Theory. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691164502.003.0005.

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This chapter describes another interesting approach to the study of invariant sets provided by the so-called weak KAM theory, developed by Albert Fathi. This approach can be considered as the functional analytic counterpart of the variational methods discussed in the previous chapters. The starting point is the relation between KAM tori (or more generally, invariant Lagrangian graphs) and classical solutions and subsolutions of the Hamilton–Jacobi equation. It introduces the notion of weak (non-classical) solutions of the Hamilton–Jacobi equation and a special class of subsolutions (critical subsolutions). In particular, it highlights their relation to Aubry–Mather theory.

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27

Kachelriess, Michael. Quantum mechanics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198802877.003.0002.

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After a brief review of the operator approach to quantum mechanics, Feynmans path integral, which expresses a transition amplitude as a sum over all paths, is derived. Adding a linear coupling to an external source J and a damping term to the Lagrangian, the ground-state persistence amplitude is obtained. This quantity serves as the generating functional Z[J] for n-point Green functions which are the main target when studying quantum field theory. Then the harmonic oscillator as an example for a one-dimensional quantum field theory is discussed and the reason why a relativistic quantum theory should be based on quantum fields is explained.

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28

Deruelle, Nathalie, and Jean-Philippe Uzan. Hamiltonian mechanics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0009.

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This chapter gives a brief overview of Hamiltonian mechanics. The complexity of the Newtonian equations of motion for N interacting bodies led to the development in the late 18th and early 19th centuries of a formalism that reduces these equations to first-order differential equations. This formalism is known as Hamiltonian mechanics. This chapter shows how, given a Lagrangian and having constructed the corresponding Hamiltonian, Hamilton’s equations amount to simply a rewriting of the Euler–Lagrange equations. The feature that makes the Hamiltonian formulation superior is that the dimension of the phase space is double that of the configuration space, so that in addition to point transformations, it is possible to perform more general transformations in order to simplify solving the equations of motion.

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29

Deruelle, Nathalie, and Jean-Philippe Uzan. The law of gravitation. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0011.

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This chapter embarks on the study of Newton’s law of gravitation. It first discusses gravitational mass and inertial mass, a measure of the ‘resistance’ of the point particle to an applied force. The numerical value of the inertial mass of a body can in principle be obtained from collision experiments by assigning to a reference body a unit inertial mass of one kilogram or, more rigorously, one ‘inertial kilogram’. Next, the chapter considers the ratio of gravitational and inertial masses. It considers that, in the absence of friction, all objects, no matter what their inertial mass, or the nature of their constituents, or the internal energy or cohesive forces of their constituents, fall in the same way in an external gravitational field. Finally, this chapter studies Newton’s gravitational force and field, as well as the Poisson equation and the gravitational Lagrangian.

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30

Mann, Peter. Near-Integrable Systems. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0024.

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This chapter extends the now familiar Lagrangian formulation to a field theory and covers elementary material in this new setting. The motion of systems with a very large number of degrees of freedom makes it necessary to specify an almost infinite number of discrete coordinates. It is possible to simplify the situation by taking the continuum limit, which replaces the individual coordinates with a continuous function that describes a displacement field, which assigns a displacement vector to each position the system could occupy relative to an equilibrium configuration. The field thus takes a point in the spacetime manifold and assigns it a value corresponding to whatever the field represents. In this chapter, many interdisciplinary examples are solved and pedagogical models are discussed. The chapter also discusses Lagrange density, the Lagrange field equation, instantons, the Klein–Gordon equation, Fourier transforms and the Korteweg–de Vries equation.

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31

Nolte, David D. Introduction to Modern Dynamics. Oxford University Press, 2019. http://dx.doi.org/10.1093/oso/9780198844624.001.0001.

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Introduction to Modern Dynamics: Chaos, Networks, Space and Time (2nd Edition) combines the topics of modern dynamics—chaos theory, dynamics on complex networks and the geometry of dynamical spaces—into a coherent framework. This text is divided into four parts: Geometric Mechanics, Nonlinear Dynamics, Complex Systems, and Relativity. These topics share a common and simple mathematical language that helps students gain a unified physical intuition. Geometric mechanics lays the foundation and sets the tone for the rest of the book by emphasizing dynamical spaces, like state space and phase space, whose geometric properties define the set of all trajectories through those spaces. The section on nonlinear dynamics has chapters on chaos theory, synchronization, and networks. Chaos theory provides the language and tools to understand nonlinear systems, introducing fixed points that are classified through stability analysis and nullclines that shepherd system trajectories. Synchronization and networks are central paradigms in this book because they demonstrate how collective behavior emerges from the interactions of many individual nonlinear elements. The section on complex systems contains chapters on neural dynamics, evolutionary dynamics, and economic dynamics. The final section contains chapters on metric spaces and the special and general theories of relativity. In the second edition, sections on conventional topics, like applications of Lagrangians, have been strengthened, as well as being updated to provide a modern perspective. Several of the introductory chapters have been rearranged for improved logical flow and there are expanded homework problems at the end of each chapter.

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