不急，一点一点来
先贴个Arnold推荐的，这个列表是Arnold在Dynamical Systems III Mathematical Aspects of Classical and Celestial Mechanics[Encyclopaedia of Mathematical Sciences3]V.I.Arnold(1988,Springer Berlin Heidelberg)里推荐的
Recommended Reading
[1] Abraham, R.; Marsden, J.E.: Foundations of mechanics. 2nd ed. Reading, Mass.: The
Benjamin/Cummings Publishing Company, Inc. m-XVI, XXII, 806 p. (1978). Zbl. 397.70001
[2] Alekseev, V.M.: Quasirandom dynamical systems. I, 11, III Mat. Sb., Nov. Sero 76, No. 1,
72-134 (1968) (Russian); 77, No. 4, 545-601 (1968) (Russian); 78, No. 1, 3-50 (1968) (Russian);
English transI.: Math. USSR, Sb. 5, No. 1, 73-128 (1968); 6, No. 4, 505-560 (1968);
7, No. 1, 1-43 (1969). Zbl. 198,569; Zbl. 198,570; Zbl. 198,570
[3] Alekseev, V.M.: Final motions in the three-body problem and symbolic dynamics Usp.
Mat. Nauk 36, No. 4, 161-176 (1981) (Russian); English transI.: Russ. Math. Surv. 36,
No. 4, 181-200 (1981). Zbl. 503.70006
[4] Anosov, D. V.: Geodesic flows on c10sed Riemannian manifolds of negative curvature
Tr. Mat. Inst. Steklova 90, 210 p. (1967) (Russian); English transI.: Proc. Steklov Inst.
Math. 90 (1967). Zbl. 163,436
[5] Appell, P.: Traite de mecanique rationnelle. Tomes 1-2. 4e M. Paris: Gauthier-Villars
(1919/1924)
[6] Arnol'd, V. 1.: Proof of A. N. Kolmogorov's theorem on the preservation of quasi-periodic
motions under small perturbations of the Hamiltonian. Usp. Mat. Nauk 18, No. 5, 13-40
(1963) (Russian); English transI.: Russ. Math. Surv. 18, No. 5, 9-36 (1963). Zbl. 129, 166
[7] Arnol'd, V.I.: Small denominators and problems of stability of motion in c1assical and
celestial mechanics. Usp. Mat. Nauk 18, No. 6,91-192 (1963) (Russian); English transI.:
Russ. Math. Surv. 18, No. 6, 85-192 (1963). Zbl. 135,427
[8] Arnol'd, V. I.: Mathematical methods of classical mechanics. Moskva: Nauka. 431 p. (1974).
(Russian); English trans!.: New York-Heidelberg-Berlin: Springer-Verlag. X, 462 p. (1978).
Zbl. 386.70001
[9] Arnol'd, V.I.: Geometrical methods in the theory of ordinary differential equations. Moskva:
Nauka. (1978) (Russian); English trans!.: New York-Heidelberg-Berlin: SpringerVerlag.
XI, 334 p. (1983). Zbl. 507.34003
[10] Birkhoff, G.D.: Dynamical systems. Am. Math. Soc. Colloq. Publ. IX. New York: American
Mathematical Society. VIII, 295 p. (1927).
[11] Bogolyubov, N.N.: On some statistical methods in mathematical physics. L'vov: Akad.
Nauk Ukr. SSR. 139 p. (1945). (Russian)
[12] Bogolyubov, N.N.; Mitropol'skij, Yu.A.: Asymptotic methods in the theory ofnonlinear
oscillations. 2nd ed. Moskva: Nauka. 408 p. (1958). (Russian); English transI.: Delhi: Hindustan
Publ. Corp.; New York: Gordon and Breach Science Publ. V, 537 p. (1961). Zbl.
83,81
[13] Cartan, E.: Leyons sur les invariants integraux. Paris: Hermann. X, 210 p. (1922). Jrb.
48,538
[14] Chaplygin, S.A.: Investigations in the dynamics of nonholonomic systems. Moskva-Leningrad.
(1949). (Russian)
[15] Charlier, c.L.: Die Mechanik des Himmels. Bd. I, 11. 2. Aufl. Berlin: Walter de Gruyter.
VIII, 488 p.; VIII, 478 p. (1927). Jrb. 53, 892
[16] Dirac, P.A.M.: On generalized Hamiltonian dynamics. Can. J. Math. 2, No. 2, 129-148
(1950). Zbl. 36, 141
[17] Hertz, H.: Die Prinzipien der Mechanik in neuem Zusammenhange dargestellt. Ges. Werke,
Bd.3. Leipzig: Barth. (1910). English transI.: New York: Dover Publications, Inc. 274 p.
(1956). Zbl. 74, 388
Recommended Reading 277
[18] Jacobi, c.G.J.: Vorlesungen über Dynamik. Berlin: G. Reimer Verlag (1884)
[19] Karapetyan, A.V.; Rumyantsev, V.V.: Stability of conservative and dissipative systems.
Itogi Nauki Tekh., Sero Obshch. Mekh. 6,132 p. (1983). (Russian). Zbl. 596.70024
[20] Kozlov, V.V.: Methods of qualitative analysis in the dynamics of a rigid body. Moskva:
Izdatel'stvo Moskovskogo Universiteta. 232 p. (1980). (Russian). Zbl. 557.70009
[21] Kozlov, V.V.: Integrability and non-integrability in Hamiltonian mechanics. Usp. Mal.
Nauk 38, No. 1, 3-67 (1983) (Russian); English transI.: Russ. Math. Surv. 38, No. 1, 1-76
(1983). Zbl. 525.70023
[22] Kolmogorov, AN.: On conservation of conditionally periodic motions under small perturbations
of the Hamiltonian. Dokl. Akad. Nauk SSSR 98, No. 4, 527-530 (1954). (Russian).
Zbl. 56, 315
[23] Kolmogorov, AN.: General theory of dynamical systems and classical mechanics. Proc.
Int. Congr. Math, 1954, Amsterdam 1,315-333 (1957). (Russian); English trans!.: Appendix
in [1]. Zbl. 95, 171
[24] Lagrange, 1.L.: Mecanique analytique. (Euvres de Lagrange, Vols. 11-12 Paris: GauthierVillars
(1888-1889)
[25] Moiseev, N.N.: Asymptotic methods of nonlinear mechanics. 2nd ed. Moskva: Nauka.
400 p. (1981). (Russian). Zbl. 527.70024
[26] Moser, 1.: Convergent series expansions for quasi-periodic motions. Math. Ann. 169, 136-
176 (1967). Zbl. 149,299
[27] Moser, 1.: Lectures on Hamiltonian systems. Mem. Am. Math. Soc. 81, 60 p. (1968). Zbl.
172,114
[28] Moser, 1.: Stable and random motions in dynamical systems. Ann. Math. Stud. 77, VIII,
199 p. (1973). Zbl. 271.70009
[29] Moser, 1.: Various aspects of integrable Hamiltonian systems. Dynamical systems,
C. 1. M. E. Lect., Bressanone 1978. Prog. Math. 8, 233-290 (1980). Zbl. 468.58011
[30] Nejmark, Yu.1.; Fufaev, N.A: Dynamics of nonholonomic systems. Moskva: Nauka.
520 p. (1967). (Russian); English transI.: Transl. Math. Monogr., Am. Math. Soc. 33. IX,
518 p. (1972). Zbl. 171,455
[31] Nekhoroshev, N. N.: An exponential estimate of the time of stability of nearly-integrable
Hamiltonian systems. Usp. Mat. Nauk 32, No. 6, 5-66 (1977) (Russian); English trans!.:
Russ. Math. Surv. 32, No. 6,1-65 (1977). Zbl. 383.70023
[32] Nitecki, Z.: Differentiable dynamies. An introduction to the orbit structure of diffeomorphisms.
Cambridge, Mass.-London: The M.1.T. Press. XV, 282 p. (1971). Zbl. 246.58012
[33] Poincare, H.: Les methodes nouvelles de la mecanique celeste. Vols. 1-3. Paris: GauthierVillars.
(1892/1893/1899); New York: Dover Publications, Inc. Vol. I: 382 p.; Vol. 11:
479 p.; Vol. 111: 414 p. (1957). Zbl. 79, 238
[34] Siegel, C. L.: On the integrals of canonical systems, Ann. Math., 11. Sero 42, No. 3, 806-822
(1941). Zbl. 25, 265
[35] Siegel, c.L.: Uber die Existenz einer Normalform analytischer Hamiltonscher Differentialgleichungen
in der Nahe einer Gleichgewichtslosung. Math. Ann. 128, 144-170 (1954).
Zbl. 57, 320
[36] Siegel, c.L.; Moser, J.: Lectures on celestial mechanics. Berlin-Heidelberg-New York:
Springer-Verlag. XII, 290 p. (1971). Zbl. 312.70017
[37] Smale, S.: Topology and mechanics. I, 11. Invent. Math. 10, No. 6, 305-331 and 11, No. 1,
45-64 (1970). Zbl. 202, 232; Zbl. 203, 261
[38] Variational principles ofmechanics. Collection ofpapers. Moskva: Fizmatgiz. 932 p. (1959).
(Russian). Zbl. 87, 170
[39] Whittaker, E. T.: A treatise on the analytical dynamics of particles and rigid bodies. 4th
ed. Cambridge: Cambridge University Press XIV, 456 p. (1960). Zbl. 91, 164
[40] Williamson, 1.: On the algebraic problem concerning the normal forms oflinear dynamical
systems. Am. 1. Math. 58, No. I, 141-163 (1936). Zbl. 13,284
[41] Williamson, J.: The exponential representation of canonical matrices, Am. J. Math. 61,
No. 4,897-911 (1939). Zbl. 22,100
[42] Wintner, A: The analytical foundations of celestial mechanics. Princeton: Princeton University
Press. XII, 448 p. (1941). Zbl. 26, 23
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FROM 211.161.243.*
Richard Montgomery的推荐：
https://people.ucsc.edu/~rmont/goodbooks.html
Recommended Math and Physics Books.
Bundles with Connections
Start with:
``Vector Bundles with a Connection''' Chern, p. 1-27 in `Global Differential Geometry',MAA Studies in Mathematics, vol 27
move to:
``Lecture Notes on Elementary Topology and Geometry'' , chapter 7, Singer and Thorpe; phenomenal re Gauss-Bonnet on surfaces from perspective of said surfaces unit tangent bundle as a principal circle bundle.
follow up with :
Spivak, vol. 2. Find the commutative diagram that fills up a whole page. Decipher it.
If you want to keep going, try:
Gauge Theory : Freed and Uhlenbeck; the amazing story of Donaldson's exotic R^4's
Gauge Theories , vol 5 of Atiyah's collected works. lecture;
Vortices and Monopoles, Jaffe and Taubes:
An encyclopaedic pre-Donaldson, not physics inspired reference:
Kobayashi-Nomizu, ch . 1 and 2. encyclopaedic. dry.
Riemannian Geometry and Analysis on Manifolds
Quick. To the point. Beautiful.
1. `Morse Theory' -Milnor, just ch. 7 ``A Rapid Course in Differential Geometry'
2. ``Mathematical Methods of Classical Mechanics'' -Arnol'd. just appendix 1. An excellent intuitive description of curvature
Detailed. Intense.
``Comparision Theorems in Riemannian Geometry''- Cheeger and Ebin:
added, april 2020 (during COVID) 60 books in diff'l geom., G.R., and gauge theory, compared
Classics
An Introduction to Differentiable Manifolds and Riemannian Geometry (2nd ed) - William M. Boothby
An Introduction to Manifolds - Loring Tu pdf here
With a physics bent; where I began ...
"Gravitation'' (a.k.a. `The Big Black Book')- Misner, Thorne, and Wheeler. good pictures. philosophy
`Foundations of Mechanics', - Abraham and Marsden.
Classic surface theory
H. Hopf. Springer vol. 1000. Beautiful. Beautiful problems.
Gray. `Volume of Tubes'. Why char. classes appear in looking at the expansion of the volume of a tube about a submanifold.
Differential and Alg. Topology
''Topology from the Differential Viewpoint'' - Milnor
`Morse Theory' -Milnor, again! (shoot, dang near anything by Milnor !)
Bott & Tu Differential forms in algebraic topology.
`Introduction to Topology' -Vassiliev
Algebraic Topology, - Marvin Greenberg. get the1st edition, not the 2nd ! In the 2nd edition Marvin caught the familiar disease of trying to say everything he knows.
Homotopic Topology by Fuchs and Fomenko. Hard to get in English! horrible mess of an index, worth the pain. CW complexes and homology, done right.
Analysis: Simmons' Introduction to Topology and Modern Analysis This is an excellent book, at a level between the usual undergrad analysis, and grad analysis courses.
Measure theory: Royden;
Fractals, Hausdorff Measure (for us beginner's): Falconer.
PDE Linear Differential Operators , by Cornelius Lanczos. (Alex Castro suggested, fall 2011!)
Lectures on Partial Differential Equations (ISBN: 3540404481) by V. I. Arnol'd. Wow! read what Maxwell did for spherical harmonics
Stokes' theorem, beginning calc. on mfds: Spivak's ``Calculus on Manifolds''
Algebra: Herstein
Algebra , Artin
Mechanics: Arnol'd's Mathematical Methods in Classical Mechanics'
Landau and Lifshitz
Abraham and Marsden `Foundations of Mechanics', encyclopaedic; start above, use this as a ref.
Celestial Mechanics
Wintner. Siegel-Moser. notes by Chenciner
Goroff's intro to his translation of Poincare's Les Nouvelles Methodes de Mecanique Celeste.
Quantum Mechanics Dirac's Principles of Quantum Mechanics
George Mackey's mathematical foundations of Quantum Mechanics is excellent for understanding the mathematical structure
UNDERGRAD .
Ordinary Differential Equations 1. Hirsch and Smale `Differential Equations, Dynamical Systems, and Linear Algebra' (preferred; but out-of-print). Or Hirsch, Smale, and Devaney `Differential Equaions, Dynamical Systems, and ..' (expanded version of Hirsch-SSmale) w some `chaos' added. Arnol'd. `Ordinary Differential Equations'. Harder.
Geometry: anything by Stillwell. anything by Coxeter. `Geometry and the Imagination', by Hilbert and Cohn-Vossen.
Celestial Mechanics : Pollard's `Celestial Mechanics'.
Stephanie Singer's `Symmetry in Mechanics'. GENERAL ADVICE: Almost All Schaum's Outlines Schaum's Outlines on `Vector Analysis', `Linear Algebra', `Real Analysis' b
the Schaum style is a short telegraphic section on theory , 1 to 3 pages, followed by scores of worked problems. Each volume has 100s of worked problems. These do-it-yrself books provide a good, quick and dirty way to learn lots of math.
vector calculus, differential forms:
Schaum's Outline on Vector Analysis (
Div Grad Curl are Dead -- by W. Burke.
Linear algebra : Hoffmann and Kunze
Relativity: Einstein's The meaning of relativity is the best book for learning special relativity. *** generally speaking it is best to learn a subject from the person who invented it.
Alex Castro recommendations.
(I have not perused these in depth, but Alex has good taste.)
Mathematical Omnibus: Thirty Lectures on Classic Mathematics [Hardcover] Dmitry Fuchs (Author), Serge Tabachnikov (Author)
Modern Geometry - Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields (Graduate Texts in Mathematics) (Pt. 1)[Hardcover] B.A. Dubrovin (Author), A.T. Fomenko (Author), S.P. Novikov (Author), R.G. Burns (Translator)
Modern Geometric Structures And Fields (Graduate Studies in Mathematics)[Hardcover] S. P. Novikov; I. A. Taimanov (Author)
** new version of Modern Geometry vol. 1 -- Novikov has had some tension with Fomenko since the latter became a "historian", and is trying to get rid of any clues which affiliates them. The new book has cleaner notation, and shorter proofs.
Mathematical Analysis I and II (Universitext) [Paperback] V. A. Zorich (Author), R. Cooke (Translator)
Vinberg's "A Course in Algebra" ** I used it for my prelim
Gilbert Strang's "Calculus" (Free) and "Intro to Linear Algebra **awesome books
Tales of Mathematicians and Physicists [Paperback] Simon Gindikin (Author), Alan Shuchat (Translator)
** in the same spirit of Arnold's biography of Newton
Riemann, Topology, and Physics (Modern BirkhC$user Classics) [Paperback] Michael I. Monastyrsky (Author) ** a lot of fun -- bedtime stuff
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FROM 103.142.140.*