THU’? lI!I|'KH| I Glll TEIFIN BIEINIS TFHJE EIUFIN VQT LY ‘ Mark LEVI N uA xu K I B A M T at i THU , CU KHI TURN HUI: THE MATHEMATICAL MECHANIC: USING PHYSICAL REASONING T0 SOLVE PROBLEMS Copyright © 2009 Princeton University Press All rights reserved Ban tiéhg Viét © NXB Tré, 201 I No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without permission in writing from the Publisher BIEU our BII-IN Muc TRUGC xur BAN nuoc Tnuc iniin not rnu vn§:N KIITH TPIICM Levi, Mark Tho ed tozin hoc — Gizii loan bang - T.P H?) lrt_1'c quan vat ly I Mark Levi ; Huy Nguyen dich Chi Minh : Tré, 2011 240 tr ; 20cm - (Canh elm m6 rong) Nguyen ban : The mathematical mechanic l Vail 1y loan hoe I Huy Nguyen II Ts: The mathematical mechanic 510 dc 22 L664 |3BN 978-6044 -01 274-5 934 97 O THU , CU Kl-II TUFIN HEIC The Mathematical Mechanic sufn TUFIN BIEING T|=:L_r|: EIUFIN MarkLEV| lluy Nguyin NHA xulh d_/‘ch BAN IRE VFIT LY Muc lL_1c THIEU G101 DINH LY PYTHAGORAS cuc Tlu BAT DANG THUC c110 B61 DO/KN MACH TAM 1111111 s1’J PHUONG TRiNH EULER-LAGRANGE THONG QUA NHUNG LO x0 KEO CANG THAU KINH, KINH VIEN VQNG, VA co HOC HAMILTON 10 1/A 1c, téi cia tinh ct‘) gap thy vat lv ca minh Ong da héi vé cat: dL_r tinh cila téi cho hoe ky mtla thu “Bat dau chuyén nganh toan ca em", téi (mp “G1 co? Toan? Anh khvllng r61!” Ong dap lai Téi coi dc’) nhu' m{)t 101 khen (va 06 lé xac nhan quan diém cla éng) 'l‘ht_1'c 1.2 ra, Quyén sach néi vé diéu gi Day khéng phai la “mtjt nhfrng cu6n bia mém,t0, (lay, dL’1 dé giét thri gian qua hai mtta git’), ma né'u du'Qc ném lhfnng tay thi sé khién mét trau nude khuyu g6i” (Nancy Banks-Smith, nha phé binh truyén hinh ngudi Anh) Véi kich th uéc ca n6, cu6n sach sé khéng guc duqc ai, it khéng thé gL_1c bang tac déng vat Iv ca né 'l‘uy nhién, cu6n sach thuc sL_1' la mét dén giang tré — hay cé thé chi lit mét cL'1 chich ch6ng lai quan niém cho rang toan h(_>c la day té ca vat Iv Trong cu6n sach nay, vat Iv bi dat vao vi tri phuc Vl_l toan hc_>c, va t6 la mét day tér cé nang luc (xin l6i cac nha vat Iv) Nhfrng 3? tuéng vat Iv cé thé la v tuéng khai mi’) thuc thu va gqi l(‘7i giai cue kv gian don cho mét bai toan toan hQC Hai ch thé gan bé khang khit dffn n6i ca hai sé chiu t6n that néu bi tach rfyi Su (T161 vai cé thé rat hiéu qua, nhu cu6n sach minh chirng Hoan toan co thé tranh cai xem viéc tach hai bi) mon co la mot eai gi qua nhan tao hay khong* qua Ijch sti Cach giai toan bang true quan vat ly it nhat co ttr thoi Archimedes (khoang nam 287 tr CN — D1'é'm khoang nam 212 tr CN) Cng da chtrng minh dinh ly tich phan n5i tiéng vé the: tich hinh trL_i, hinh cau va hinh non bang cach sir dung mat cai can thang bang gia tuong Ban tom tat ciia dinh ly duoc khae bia ma ca ong Cach tiép can ca Archimedes co thé duoc tim thay cuon [P] Doi voi Newton, hai chL"1 dé v6n1a mat Cac cu6n [U] va [BB] trinh bay nhfrng loi giai vat ly rat dep cho cac bai toan toan hoe Rat nhiéu nhfrng phat kién toan hoe co ban (nhu Hamilton, Riemann, Lagrange, Iaeobi, Mobius, Grassman, Poincare) da duoc dan dat tu nhung suy xét vat 1y C6 hay khéng mét céng thiic ph6' quat cho each t1'é'p can vatIy?Nhu voi cong cu bat ky, vat chat hay tinh than, each tiép can co t6t va co khong Kho khan chinh la “nhin" ban chat vat ly ca bai toan** Mot s6 bai toan phii hop voi each giai nay, mat 56 khac thi khong (c6 nhién, eu6n sach chi bao gom dang thL'r nhat) Tim mo “Toan hoc la mat nhanh ea vat ly ly thuyé't noi ma phan thuc nghiém la ré tién" (V Arnold [ARN]) Khong chi cac thi nghiéin lrong cu6n sach la ré lién — tham chi mién phi, ma thirc chat la cac lhuc nghiem gia tubing (bai toan 2.2; 3.3; 3.13, vii (hire hau heft cac bai loan lrongcuan sach nay) "* Day la each tie-'p can di nguoc trao liru chung: thong thuong mat nguoi bat dau bang mat bai toan vat ly, roi trién khai no mg‘)! bai toan toan hoe; bday chiing ta lam nguqc lai * phéng vefit 1}? cho mcfat béli Loém cu thé cé dé dé1ng,vé1 cé khéng; ngudi dcc cé thé cé 3? kién riéng ca minh sau ludt qua nhfrng trang séch néy Mét béi hcc mé mcfn sinh vién cé thé rL’1t ttr viéc (1‘Qc cu6n séch nély lé tim kiém mét 37 nghia vzfxt 1y toén hQc 151 rt cé ich chgit ché clia toén hpc Lép luém vét 13? ca chtlng ta sé khéng hoén toém cheflt ché Nhfrng leflp luzfm néy chi 151 phéc théo ca nhfmg chtrng minh cheflt ché, C1‘u'Qc di?-zn dat being ngén ngfr vét 1y Téi cé chuyén ngfr“chL'mg minh” vét 13? thélnh chirng minh tozin hqc cho mcfmt véli béli toén chQn1(_>c Lém viéc néy métxcéch ct’) th6ng sé bién quyé'n séch thimh mét séch“t0, dély vél chéln ngzit" Téi hy vqng ngudi dcpc sé nh2_"1n hinh mu dé nt truc quan séu s€ic ~ hai buéc di truéc tfnh chzflt ché cfla toén h(_>c Nhu Archimedes dé viét, “Duong nhién viéc thiét lép mét chtrng minh sé dé délng hon nhiéu néu tru'()c C16 da cé ngudi n€im duqc khéi niém s0 khéi ca béli toézn” ([ARC], tr 8) SL1 Métcéch tiéjv cén r6 réng Thay vi phién dich “chL'1'ng minh" vét 13? thémh chtrng minh ch2f1tcl1é,viécthié'tlé1p cé th6ng “célc tién dé thL_1'c cht" cé lé sé 151 m(_”)t du z'1nthL'1 vi Déy sé 151 mét tefxp hqp czic tién dé thuc ché'tcL"1a co h(_>c, tuong tu nhu célc tién dé hinh hoc/s6 hoc ca Euclid mél dé céc chtmg minh duqc cho cu6n séch néxy tré nén chit ché Ta thé tufmg tuqng mét nén vén minh ngoéxi tréi déit mix dé ngudi La phét trién co hQc trufyc, nhu mét bi) mén cheflt ché vé thun my mang tinh tién dé Trong thé' giéi song hénh nziy, m(f)t ngudi néo dc’) Git dé vié't mét cu6n szich vé vif;-c sir dung hinh hoe C16 chtrng minh céac dinh 1y co hoe C6 thé b€1ihQc déy 121 ngudi khéng nén hoén toéln télp trung v€10 czich tiép céln hay céch tiép can kia, m nén coi dé nhu hai mzfit ca m(f)t d6ng xu Cu6n séch néay véi 151 mét phén (mg ch6nglz_1i su thé khé ph6 bi6n d6i khia canh vefzt 13? cla toén hqc tém Iyhpc Nhng céch giéi vét 13? cu6n séch nély cé thé ducyc dién dich ngén ngfr toén hqc Tuy vefly, khéng thé trénh khéi thifiu sét quél trinh dién dich True giéc 00 hqc 121 m()t thuéc tinh co bén ca So'1uQ'c vé tri tué ngufyi, cng co bén nhu khé néng tuéng tuqng hinh hqc, khéng sir dung chnng lé léing phi mt m6 phéng dang co, cng dfm dau khéng — lé chain m(f)t céi nhm vélo dinh biing céi méng tay M(f)t su giém t6c |'n F = /111)‘, diéu cé the‘-5 gian qua 161 lixm dau Mét véi thiét bi phét dién dimg chuéng reo dé baflt tic bL'1a dao déng va vélo chuéng, céng téic lién tL_1c hét m6 r6i lai déng v51 ngufyi sir dung biét duqc thé nélo 121 mét cL'1 dién gizfxt lién héi Déy lé kéft thL'1c béli h(_)c dau dc'>'n ca chflng ta vé df) tro Khi A.19 B6 m6 phéng dién-bom Tit cé khéi niém ké trén - V, I, q, R,C vii L — cf) mét m6 phéng don gién th6ng 6ng nude (nhu lé th6ng 6ng nuéc nhil chng ha_1n!) Hinh A.9 tém luqc sL_r tucmg d6ng nély vi nhn ring néu céng tic chim dirt su tié'p x11c,d6ng (:6 thé ti6p t\_1c truyén qua khéng kl1inh\rm(>lliali'ra dién Mét hifgu irng cé lién quan duqc gc.N5nglLrr_1ng nhng ion trim lin hoém toém vfri czic ion dziy téc, trb thénh nhiét v51 énh séng * Céc chL'1ngm€it va chgun véi 235 'rA| |_|$u THAM KH/1|: [ARC] Archimedes, Geometrical Solutions Derived from Mechanics, I L Heiberg dich, Chicago: NhaXua'tbé'1n Open Court, 1909 Tap tin PDF hien co tai dia chi http://books.google.c0m/books?id=suYGAAAAYAA] [ARC1] Ban thao cua Archimedes, http://www.archimedes palimpsestorg/ [ARN] V I Arnold, Mathematical Methods of Classical Mechanics, K Vogtmann va A Weinstein dich, New York: Springer-Verlag, 1978 [BB] M B Balk va V G Boltyanskii, Geometriya mass (Tieng Nga) [Hinh hoc chat diém], Bibliotechka Kvant [Thu vién Kvant], 61, Moscow: Nauka, 1987 [CG] H.S.G Coxeter va S L Greitzer, Geometry/Revisite-cl, Washington, DC: Hiep hoi Toan hoc My, 1967 [CH] Courant va D Hilbert, Methods ofMathematical Phycics, t 2, Partial Differential Equations, In lai cua ban g6c 1962, Tu séch Tinh hoa Wiley, New York: Wiley-Interscience, 1989 [CL] Coddington va N Levinson, Theory of Ordinary Differential Equations, New York: McGraw~Hill, 1955 [D] M M Day, Polygons circumscribed about closed convex curves Trans Am Math Soc 62 (1947), tr 315-319 [DO] M DoCarmo, Differential Geometrical of Curves and Surfaces, Englewood Cliffs, N]: Prentice-Hall, 1976 R Z36 Doyle va I L Snell Random Walks and Electric Networks, Washington, DC: Hiep hoi Toan hoc My [DS] P G [Fe] R P [F0] R L Foote, GP] I M Gelfand va S V Fomin, Calculus ofltzriations, Feynman, QED, Princeton, N]: Princeton University Press, 1985 Geometry of the Prytz planimeter, Rep Math Phys 42(1-2), tr 249-271 Englewood Cliffs, N]: Prentice-Hall, 1963 HZ] H Hofer va E Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkauser Advanced Texts/ Basler Lehrbucher, Basel: Birkéiuser Verlag, 1994 [K] B Yu Kogan, The Applications o Mechanics to Geometry, Chicago: University of Chicago Press, 1974 [L1] [L2] , Minimal perimeter triangles, Am Math M0nthlylO9 (2002), tr 890-899 M Levi, A “bicycle wheel” proof of the GaussBonnet theorem, dual cones and some mechanical manifestiation of the Berry phase, Expo Math 12 (1994), tr 145-164 [LS] Yu I Luybich va L A Shor, The Kinemtic Method in Geometrical Problems, V Shokurov dich, Moscow: ban Mir, 1980 N112 Xuéit LW] M Levi va W Weckesser, Non-holonomic effects in averaging, Erg Th &Dynam Sys 22 (2002), tr 14971506 237 M] I Milnor, Morse Theory, Annals of Mathematics Studies, S6 51, Princeton, NI: Princeton University Press, 1963 {NP} R N evanlinna va V Paatero In troduction to Complex Analysis Providence, RI: AMS Chelsea Publishing, 2007 Mathematics and Plausible Reasoning, t Princeton, NI: Princeton University Press, 1990 [P] G Polya [Sp] M R Spiegel, Complex Variables, Schaum’s Outline Series, New York: McGraw-Hill, 1968 [St] I Stewart, Calculus: Concepts and Contexts, Pacific Grove, CA: Brooks/Cole, 2001 Ta] A E Taylor, Ageometric theorem and its applications 1, to biorthogonal systems, Bull Am Math Soc 53 (1974), tr 614-616 TO] Math Monthly 105 To] in geometry, Am tr 697-703 T F Tokieda, Mechanical ideas (8) (1998), L F Toth, Lagerungen in der Ebene auf der Kugel und im Raum, Berlin: Springer-Verlag, 1953 [U] V A Uspenski, Some Applications 0fMechanics to Mathematics, New York: Pergamon Press, 1961 2'58 THQ co KHI TQAN HOC MARK LEVI Huy Nguyén dich Chill mic/1 nhiém mi: bdn: NGUYEN MINH NHUT Chin lrdc/1 nhiém "[...]... énh h0'n, ta cho gradient vs = 0 Béy gifr ta tinh VS Ta cé: l|x— a| = A‘/(x ax ax vél mcfn — a|)2 + (y — cu): = (x —a.) /\ }(. >c — a|)2 + (y— a1)2 céch tuong tu: %|x—a| =(y—a2)/,/(x—a,)2 +(y—a2)2 Theo dé, V|x — a| = (x — a) / |x — a|lé1m ... néng P(x) Luc técdung F(;¢) = -P’(x) P(x) 121 cue tiéu :> F(x) = O (trang théi s6f(x) Dao h€1mf(x) f(x) nhzil => f’(x) = cén bng) Métllru 5? vé hié'u b1'é't co bén Ném duqc giéii tich vé hinh h(_)c... = A‘/(x ax ax vél mcfn — a|)2 + (y — cu): = (x —a.) / }(. >c — a|)2 + (y— a1)2 céch tuong tu: %|x—a| =(y—a2)/,/(x—a,)2 +(y—a2)2 Theo dé, V|x — a| = (x — a) / |x — a|lé1m