\(\newcommand{\footnotename}{footnote}\) \(\def \LWRfootnote {1}\) \(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\) \(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\) \(\let \LWRorighspace \hspace \) \(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\) \(\newcommand {\TextOrMath }[2]{#2}\) \(\newcommand {\mathnormal }[1]{{#1}}\) \(\newcommand \ensuremath [1]{#1}\) \(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \) \(\newcommand {\setlength }[2]{}\) \(\newcommand {\addtolength }[2]{}\) \(\newcommand {\setcounter }[2]{}\) \(\newcommand {\addtocounter }[2]{}\) \(\newcommand {\arabic }[1]{}\) \(\newcommand {\number }[1]{}\) \(\newcommand {\noalign }[1]{\text {#1}\notag \\}\) \(\newcommand {\cline }[1]{}\) \(\newcommand {\directlua }[1]{\text {(directlua)}}\) \(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\) \(\newcommand {\protect }{}\) \(\def \LWRabsorbnumber #1 {}\) \(\def \LWRabsorbquotenumber "#1 {}\) \(\newcommand {\LWRabsorboption }[1][]{}\) \(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\) \(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\) \(\def \mathcode #1={\mathchar }\) \(\let \delcode \mathcode \) \(\let \delimiter \mathchar \) \(\def \oe {\unicode {x0153}}\) \(\def \OE {\unicode {x0152}}\) \(\def \ae {\unicode {x00E6}}\) \(\def \AE {\unicode {x00C6}}\) \(\def \aa {\unicode {x00E5}}\) \(\def \AA {\unicode {x00C5}}\) \(\def \o {\unicode {x00F8}}\) \(\def \O {\unicode {x00D8}}\) \(\def \l {\unicode {x0142}}\) \(\def \L {\unicode {x0141}}\) \(\def \ss {\unicode {x00DF}}\) \(\def \SS {\unicode {x1E9E}}\) \(\def \dag {\unicode {x2020}}\) \(\def \ddag {\unicode {x2021}}\) \(\def \P {\unicode {x00B6}}\) \(\def \copyright {\unicode {x00A9}}\) \(\def \pounds {\unicode {x00A3}}\) \(\let \LWRref \ref \) \(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\) \( \newcommand {\multicolumn }[3]{#3}\) \(\require {textcomp}\) \(\def \LWRtensorindicesthreesub #1#2{{_{#2}}\LWRtensorindicesthree }\) \(\def \LWRtensorindicesthreesup #1#2{{^{#2}}\LWRtensorindicesthree }\) \(\newcommand {\LWRtensorindicesthreenotsup }{}\) \(\newcommand {\LWRtensorindicesthreenotsub }{ \ifnextchar ^ \LWRtensorindicesthreesup \LWRtensorindicesthreenotsup }\) \(\newcommand {\LWRtensorindicesthree }{ \ifnextchar _ \LWRtensorindicesthreesub \LWRtensorindicesthreenotsub }\) \(\newcommand {\LWRtensorindicestwo }{ \ifstar \LWRtensorindicesthree \LWRtensorindicesthree }\) \(\newcommand {\indices }[1]{\LWRtensorindicestwo #1}\) \(\newcommand {\LWRtensortwo }[3][]{{}\indices {#1}{#2}\indices {#3}}\) \(\newcommand {\tensor }{\ifstar \LWRtensortwo \LWRtensortwo }\) \(\newcommand {\LWRnuclidetwo }[2][]{{\vphantom {\mathrm {#2}}{}^{\LWRtensornucleonnumber }_{#1}\mathrm {#2}}}\) \(\newcommand {\nuclide }[1][]{\def \LWRtensornucleonnumber {#1}\LWRnuclidetwo }\) \(\newcommand {\intertext }[1]{\text {#1}\notag \\}\) \(\let \Hat \hat \) \(\let \Check \check \) \(\let \Tilde \tilde \) \(\let \Acute \acute \) \(\let \Grave \grave \) \(\let \Dot \dot \) \(\let \Ddot \ddot \) \(\let \Breve \breve \) \(\let \Bar \bar \) \(\let \Vec \vec \) \(\require {cancel}\)

Chapter 14 To be added

geodesic motion on the surface of the sphere as introduction/application to the chapter “A Lagrangian of sufficient generality”

mostly track 2 chapters

  • • Hamilton-Jacobi theory

  • • normal modes

  • • full proof of Frobenius theorem (inverse)

  • • Lagrangian approach to two body problem

  • • reduction of order and reduced variational principle “Routhian”, Lagrangian and Hamiltonian version

  • • Maupertius, time-reparametrization invariance, apply to go from Lagrangan formulation of brachystocrone problem, to one where the action becomes the time of fall

  • • more details on Lagrangian field theories and Poincaré invariance

Bibliography

  • [1] G. Mayné. “Cours de mécanique analytique. 2ème candidature Sciences ULB. Mathématique et Physique”. unpublished. 1988.

  • [2] P. Gaspard. ULB MATHF204: Mécanique analytique, Notes de cours. ULB, 2014.

  • [3] P. Spindel. Mécanique. Volume 1: Mécanique newtonienne. Gordon and Breach, 2004.

  • [4] P. Spindel. Mécanique. Volume 2: Mécanique analytique. Gordon and Breach, 2002.

  • [5] H. Goldstein, C. Poole, and J. Safko. Classical Mechanics. Addison Wesley, 2002. ISBN: 9780201657029. URL: https://books.google.lu/books?id=tJCuQgAACAAJ.

  • [6] V. I. Arnold. Mathematical Methods of Classical Mechanics. Second. Graduate Texts in Mathematics. Springer, 1989.

  • [7] E. Sudarshan and N. Mukunda. Classical Dynamics: A Modern Perspective. John Wiley & Sons, 1974.

  • [8] C. Lanczos. The Variational Principles of Mechanics. Dover Books On Physics. Dover Publications, 1986. ISBN: 9780486650678. URL: https://books.google.lu/books?id=ZWoYYr8wk2IC.

  • [9] H. B. Callen. Thermodynamics and an introduction to thermostatistics. Second. John Wiley & Sons, 1985.

  • [10] B. Schutz. Geometrical methods of mathematical physics. Cambridge University Press, 1980.

  • [11] M. Henneaux and C. Teitelboim. Quantization of Gauge Systems. Princeton University Press, 1992.

  • [12] C. Bunster. “The Principle of Least Action, from the "Vis-viva" to Quantum Black Hole”. Sixteenth Arnold Sommerfeld Lecture Series, available online at https://www.theorie.physik.uni-muenchen.de/activities/lectures/sixteenth_series/bunster_public/video_bunster/index.html. 2014. URL: https://www.theorie.physik.uni-muenchen.de/activities/lectures/sixteenth_series/bunster_public/video_bunster/index.html.