classical dynamics

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1: 23.21 Physical Applications

§23.21 Physical Applications

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§23.21(i) Classical Dynamics

►In §22.19(ii) it is noted that Jacobian elliptic functions provide a natural basis of solutions for problems in Newtonian classical dynamics with quartic potentials in canonical form $(1-x^2)(1-k^2{x}^2)$. …

2: William P. Reinhardt

… ►This is closely connected with his interests in classical dynamical “chaos,” an area where he coauthored a book, Chaos in atomic physics with Reinhold Blümel. …

3: 22.19 Physical Applications

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§22.19(i) Classical Dynamics: The Pendulum

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§22.19(ii) Classical Dynamics: The Quartic Oscillator

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§22.19(v) Other Applications

►Numerous other physical or engineering applications involving Jacobian elliptic functions, and their inverses, to problems of classical dynamics, electrostatics, and hydrodynamics appear in Bowman (1953, Chapters VII and VIII) and Lawden (1989, Chapter 5). …

4: 18.39 Applications in the Physical Sciences

… ►As in classical dynamics this sum is the total energy of the one particle system. …

5: 32.16 Physical Applications

§32.16 Physical Applications

… ►Statistical physics, especially classical and quantum spin models, has proved to be a major area for research problems in the modern theory of Painlevé transcendents. … ►

Integrable Continuous Dynamical Systems

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6: 9.16 Physical Applications

… ►Airy functions are applied in many branches of both classical and quantum physics. … ►The frequent appearances of the Airy functions in both classical and quantum physics is associated with wave equations with turning points, for which asymptotic (WKBJ) solutions are exponential on one side and oscillatory on the other. …Within classical physics, they appear prominently in physical optics, electromagnetism, radiative transfer, fluid mechanics, and nonlinear wave propagation. … ►In fluid dynamics, Airy functions enter several topics. … ►The KdV equation and solitons have applications in many branches of physics, including plasma physics lattice dynamics, and quantum mechanics. …

7: Bibliography B

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Moment equations for probability distributions in classical and quantum mechanics. Phys. Rev. A 58 (3), pp. 1799–1809.

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External Links:

Document

Referenced by:

§24.18.

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A monotonicity property involving ${}_3{}^{}F_2^{}$ and comparisons of the classical approximations of elliptical arc length. SIAM J. Math. Anal. 32 (2), pp. 403–419.

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External Links:

ISSN 1095-7154, Document, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§19.9(i), §19.9(i).

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Electromagnetic Fields and Interactions. Vol. I, Blaisdell, New York.

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Notes:

Replaces Abraham & Becker, Classical Theory of Electricity and Magnetism.

Referenced by:

§19.33(iii), §19.34.

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Algebro-geometric Approach to Nonlinear Integrable Problems. Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin.

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ISBN 3-540-50265-3, ZentralBlatt

Referenced by:

§21.1, 1st item, §21.6(ii), §21.7(i), §21.7(i), §21.9, Ch.21, Profile Alexander I. Bobenko, Profile Alexander A. Its, Notations T.

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8: Bibliography

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The classical moment problem and some related questions in analysis. Classics in Applied Mathematics, Vol. 82, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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Notes:

Reprint of the 1965 edition [ 0184042], Translated by N. Kemmer, With a foreword by H. J. Landau

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ISBN 978-1-611976-38-0, MathReview Entry

Referenced by:

§18.40(ii).

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Classical Orthogonal Polynomials. In Orthogonal Polynomials and Applications, C. Brezinski, A. Draux, A. P. Magnus, P. Maroni, and A. Ronveaux (Eds.), Lecture Notes in Math., Vol. 1171, pp. 36–62.

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External Links:

MathReview, ZentralBlatt

Referenced by:

(18.27.12_5), (18.27.6).

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The Riemann Hypothesis and the Hamiltonian of a Quantum Mechanical System. In Number Theory and Dynamical Systems (York, 1987), M. M. Dodson and J. A. G. Vickers (Eds.), London Math. Soc. Lecture Note Ser., Vol. 134, pp. 153–172.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§25.17.

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Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, Vol. 60, Springer-Verlag, New York.

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Notes:

Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein, Corrected reprint of the second (1989) edition

External Links:

ISBN 0-387-96890-3, MathReview

Referenced by:

§21.5(i).

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Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Classics in Applied Mathematics, Vol. 13, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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Notes:

Corrected reprint of the 1988 original

External Links:

ISBN 0-89871-354-4, MathReview, ZentralBlatt

Referenced by:

§3.7(iii).

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9: Bibliography D

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Pole dynamics for elliptic solutions of the Korteweg-de Vries equation. Math. Phys. Anal. Geom. 3 (1), pp. 49–74.

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External Links:

ISSN 1385-0172, Document, MathReview (Giulio Soliani), ZentralBlatt

Referenced by:

§23.21(ii).

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An Introduction to Chaotic Dynamical Systems. The Benjamin/Cummings Publishing Co. Inc., Menlo Park, CA.

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External Links:

ISBN 0-8053-1601-9, MathReview (Richard C. Churchill), ZentralBlatt

Referenced by:

§3.8(viii).

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Sharp bounds for the extreme zeros of classical orthogonal polynomials. J. Approx. Theory 162 (10), pp. 1793–1804.

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ISSN 0021-9045, Document, FullText, MathReview (Francisco Marcellán), ZentralBlatt

Referenced by:

(18.16.12), (18.16.13), §18.16(ii), §18.16(ii), §18.16(iv).

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Inequalities for extreme zeros of some classical orthogonal and $q$-orthogonal polynomials. Math. Model. Nat. Phenom. 8 (1), pp. 48–59.

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ISSN 0973-5348, Document, FullText, MathReview (Ana Foulquié Moreno), ZentralBlatt

Referenced by:

§18.16(ii), §18.16(iv), §18.16(ii), §18.16(iv), §18.27(iv), §18.27(iv), §18.27(v), §18.27(v).

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Linear operators. Part II. Wiley Classics Library, John Wiley & Sons, Inc., New York.

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Notes:

Spectral theory. Selfadjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1963 original, A Wiley-Interscience Publication

External Links:

ISBN 0-471-60847-5, MathReview Entry

Referenced by:

(1.18.68), §1.18(ix), §1.18(ix), §1.18(ix), §1.18(x).

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10: Bibliography W

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Elliptic Functions According to Eisenstein and Kronecker. Classics in Mathematics, Springer-Verlag, Berlin.

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Notes:

Reprint of the 1976 original

External Links:

ISBN 3-540-65036-9, MathReview, ZentralBlatt

Referenced by:

§22.12, §23.8(ii).

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A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. 4th edition, Cambridge University Press, Cambridge.

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Notes:

Reprinted in 1988.

External Links:

ISBN 0-521-35883-3, MathReview, ZentralBlatt

Referenced by:

§22.19(i), §22.19(iv), §22.19(v), §23.21(i).

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