{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T04:43:27Z","timestamp":1771562607665,"version":"3.50.1"},"publisher-location":"New York, NY","reference-count":182,"publisher":"Springer New York","isbn-type":[{"value":"9780387758886","type":"print"},{"value":"9780387304403","type":"electronic"}],"license":[{"start":{"date-parts":[[2009,1,1]],"date-time":"2009-01-01T00:00:00Z","timestamp":1230768000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2009,1,1]],"date-time":"2009-01-01T00:00:00Z","timestamp":1230768000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2009]]},"DOI":"10.1007\/978-0-387-30440-3_326","type":"book-chapter","created":{"date-parts":[[2009,6,17]],"date-time":"2009-06-17T16:05:42Z","timestamp":1245254742000},"page":"5482-5510","source":"Crossref","is-referenced-by-count":3,"title":["Mechanical Systems: Symmetries and Reduction"],"prefix":"10.1007","author":[{"given":"Jerrold E.","family":"Marsden","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tudor S.","family":"Ratiu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","reference":[{"key":"326_CR1_326","doi-asserted-by":"crossref","unstructured":"Abraham R, Marsden JE (2008) Foundations of Mechanics, 2nd edn. AMS Chelsea Publ,Providence. Orginally published in 1967; second edition revised and enlarged with the assistance of Tudor Ratiu and Richard Cushman, 1978","DOI":"10.1090\/chel\/364"},{"key":"326_CR2_326","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-1029-0","volume-title":"Manifolds, Tensor Analysis and Applications, 2nd edn. Applied Mathematical Sciences","author":"R Abraham","year":"1988","unstructured":"Abraham R, Marsden JE, Ratiu T (1988) Manifolds, Tensor Analysis andApplications, 2nd edn. Applied Mathematical Sciences, vol\u00a075. Springer, New York"},{"key":"326_CR3_326","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/S0167-2789(98)00127-4","volume":"123","author":"MS Alber","year":"1998","unstructured":"Alber MS, Luther GG, Marsden JE, Robbins JM (1998) Geometric phases, reductionand Lie-Poisson structure for the resonant three-wave interaction. Physica D 123:271\u2013290","journal-title":"Physica D"},{"key":"326_CR4_326","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/978-1-4613-9725-0_4","volume-title":"The Geometry of Hamiltonian systems. MSRI Series","author":"JM Arms","year":"1991","unstructured":"Arms JM, Cushman RH, Gotay M (1991) A\u00a0universal reduction procedure forHamiltonian group actions. In: Ratiu T (ed) The Geometry of Hamiltonian systems. MSRI Series, vol\u00a022. Springer, New York, pp\u00a033\u201352"},{"key":"326_CR5_326","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1007\/BF02046759","volume":"78","author":"JM Arms","year":"1981","unstructured":"Arms JM, Marsden JE, Moncrief V (1981) Symmetry and bifurcations of momentummappings. Comm Math Phys 78:455\u2013478","journal-title":"Comm Math Phys"},{"key":"326_CR6_326","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/0003-4916(82)90105-1","volume":"144","author":"JM Arms","year":"1982","unstructured":"Arms JM, Marsden JE, Moncrief V (1982) The structure of the space solutions ofEinstein\u2019s equations: II Several Killing fields and the Einstein\u2013Yang\u2013Mills equations. Ann Phys 144:81\u2013106","journal-title":"Ann Phys"},{"key":"326_CR7_326","doi-asserted-by":"crossref","first-page":"319","DOI":"10.5802\/aif.233","volume":"16","author":"VI Arnold","year":"1966","unstructured":"Arnold VI (1966) Sur la g\u00e9om\u00e9trie differentielle des groupes de Lie de dimensoninfinie et ses applications \u00e0 l\u2019hydrodynamique des fluids parfaits. Ann Inst Fourier Grenoble 16:319\u2013361","journal-title":"Ann Inst Fourier Grenoble"},{"key":"326_CR8_326","first-page":"267","volume":"79","author":"VI Arnold","year":"1969","unstructured":"Arnold VI (1969) On an a\u00a0priori estimate in the theory of hydrodynamicalstability. Am Math Soc Transl 79:267\u2013269","journal-title":"Am Math Soc Transl"},{"key":"326_CR9_326","volume-title":"Mathematical Methods of Classical Mechanics, 1st edn 1978, 2nd edn 1989. Graduate Texts in Math","author":"VI Arnold","year":"1989","unstructured":"Arnold VI (1989) Mathematical Methods of Classical Mechanics, 1st edn 1978, 2ndedn 1989. Graduate Texts in Math, vol\u00a060. Springer, New York"},{"key":"326_CR10_326","volume-title":"Encyclopedia of Mathematics","author":"VI Arnold","year":"1988","unstructured":"Arnold VI, Koslov VV, Neishtadt AI (1988) Dynamical Systems III. In:Encyclopedia of Mathematics, vol\u00a03. Springer, New York"},{"key":"326_CR11_326","first-page":"523","volume":"308","author":"M Atiyahf","year":"1982","unstructured":"Atiyahf M, Bott R (1982) The Yang-Mills equations over Riemann surfaces. PhilTrans R Soc Lond A 308:523\u2013615","journal-title":"Phil Trans R Soc Lond A"},{"key":"326_CR12_326","doi-asserted-by":"crossref","first-page":"201","DOI":"10.2140\/pjm.1997.181.201","volume":"181","author":"L Bates","year":"1997","unstructured":"Bates L, Lerman E (1997) Proper group actions and symplectic stratifiedspaces. Pac J Math 181:201\u2013229","journal-title":"Pac J Math"},{"key":"326_CR13_326","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/0034-4877(93)90073-N","volume":"32","author":"L Bates","year":"1993","unstructured":"Bates L, Sniatycki J (1993) Nonholonomic reduction. Report Math Phys32:99\u2013115","journal-title":"Report Math Phys"},{"key":"326_CR14_326","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1016\/j.jde.2005.06.003","volume":"216","author":"P Birtea","year":"2005","unstructured":"Birtea P, Puta M, Ratiu TS, Tudoran R (2005) Symmetry breaking for toralactions in simple mechanical systems. J\u00a0Differ Eq 216:282\u2013323","journal-title":"J Differ Eq"},{"key":"326_CR15_326","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/S0034-4877(01)90006-0","volume":"47","author":"G Blankenstein","year":"2001","unstructured":"Blankenstein G, Van Der Schaft AJ (2001) Symmetry and reduction in implicitgeneralized Hamiltonian systems. Rep Math Phys 47:57\u2013100","journal-title":"Rep Math Phys"},{"key":"326_CR16_326","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1016\/S0926-2245(00)00014-0","volume":"12","author":"AD Blaom","year":"2000","unstructured":"Blaom AD (2000) Reconstruction phases via Poisson reduction. Differ Geom Appl12:231\u2013252","journal-title":"Differ Geom Appl"},{"issue":"727","key":"326_CR17_326","first-page":"xviii+112","volume":"153","author":"AD Blaom","year":"2001","unstructured":"Blaom AD (2001) A\u00a0geometric setting for Hamiltonian perturbation theory.Mem Am Math Soc 153(727):xviii+112","journal-title":"Mem Am Math Soc"},{"key":"326_CR18_326","doi-asserted-by":"crossref","DOI":"10.1007\/b97376","volume-title":"Nonholonomic mechanics and control. Interdisciplinary Applied Mathematics \u2013 Systems and Control","author":"AM Bloch","year":"2003","unstructured":"Bloch AM (2003) Nonholonomic mechanics and control. Interdisciplinary AppliedMathematics\u00a0\u2013 Systems and Control, vol\u00a024. Springer, New York. With the collaboration of Baillieul J, Crouch P, Marsden J, with scientificinput from Krishnaprasad PS, Murray RM, Zenkov D"},{"key":"326_CR19_326","doi-asserted-by":"crossref","first-page":"1309","DOI":"10.1088\/0951-7715\/15\/4\/316","volume":"15","author":"AM Bloch","year":"2002","unstructured":"Bloch AM, Crouch P, Marsden JE, Ratiu T (2002) The symmetric representation ofthe rigid body equations and their discretization. Nonlinearity 15:1309\u20131341","journal-title":"Nonlinearity"},{"key":"326_CR20_326","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1007\/BF02199365","volume":"136","author":"AM Bloch","year":"1996","unstructured":"Bloch AM, Krishnaprasad PS, Marsden JE, Murray R (1996) Nonholonomicmechanical systems with symmetry. Arch Ration Mech Anal 136:21\u201399","journal-title":"Arch Ration Mech Anal"},{"key":"326_CR21_326","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02101622","volume":"175","author":"AM Bloch","year":"1996","unstructured":"Bloch AM, Krishnaprasad PS, Marsden JE, Ratiu T (1996) The Euler-Poincar\u00e9equations and double bracket dissipation. Commun Math Phys 175:1\u201342","journal-title":"Commun Math Phys"},{"key":"326_CR22_326","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1007\/BF01257419","volume":"122","author":"AI Bobenko","year":"1989","unstructured":"Bobenko AI, Reyman AG, Semenov-Tian-Shansky MA (1989) The Kowalewski Top 99years later: A\u00a0Lax pair, generalizations and explicit solutions. Commun Math Phys 122:321\u2013354","journal-title":"Commun Math Phys"},{"key":"326_CR23_326","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1023\/A:1007654605901","volume":"49","author":"AI Bobenko","year":"1999","unstructured":"Bobenko AI, YB Suris (1999) Discrete Lagrangian reduction, discreteEuler-Poincar\u00e9 equations, and semidirect products. Lett Math Phys 49:79\u201393","journal-title":"Lett Math Phys"},{"key":"326_CR24_326","volume-title":"Lie groups and Lie algebras. In: Elements of Mathematics","author":"N Bourbaki","year":"1998","unstructured":"Bourbaki N (1998) Lie groups and Lie algebras. In: Elements of Mathematics.Springer, Berlin, Chap 1\u20133, No MR1728312, 2001g:17006. Translated from the French, Reprint of the 1989 Englishtranslation"},{"key":"326_CR25_326","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1017\/S0022112070001660","volume":"44","author":"FP Bretherton","year":"1970","unstructured":"Bretherton FP (1970) A\u00a0note on Hamilton\u2019s principle for perfect fluids.J\u00a0Fluid Mech 44:19\u201331","journal-title":"J Fluid Mech"},{"key":"326_CR26_326","volume-title":"Le\u00e7ons sur les Invariants Int\u00e9graux, 1971 edn","author":"E Cartan","year":"1922","unstructured":"Cartan E (1922) Le\u00e7ons sur les Invariants Int\u00e9graux, 1971 edn. Hermann,Paris"},{"key":"326_CR27_326","doi-asserted-by":"crossref","unstructured":"Castrill\u00f3n-L\u00f3pez M, Garcia P\u00e9rez PL, Ratiu TS (2001) Euler-Poincar\u00e9 reductionon principal bundles. Lett Math Phys 58:167\u2013180","DOI":"10.1023\/A:1013303320765"},{"key":"326_CR28_326","doi-asserted-by":"crossref","unstructured":"Castrill\u00f3n-L\u00f3pez M, Marsden JE (2003) Some remarks on Lagrangian and Poissonreduction for field theories. J\u00a0Geom Phys 48:52\u201383","DOI":"10.1016\/S0393-0440(03)00025-1"},{"key":"326_CR29_326","doi-asserted-by":"crossref","unstructured":"Castrill\u00f3n-L\u00f3pez M, Ratiu T (2003) Reduction in principal bundles: covariantLagrange\u2013Poincar\u00e9 equations. Comm Math Phys 236:223\u2013250","DOI":"10.1007\/s00220-003-0797-5"},{"key":"326_CR30_326","doi-asserted-by":"crossref","unstructured":"Castrill\u00f3n-L\u00f3pez M, Ratiu T, Shkoller S (2000) Reduction in principal fiberbundles: Covariant Euler-Poincar\u00e9 equations. Proc Amer Math Soc 128:2155\u20132164","DOI":"10.1090\/S0002-9939-99-05304-6"},{"key":"326_CR31_326","doi-asserted-by":"crossref","first-page":"3138","DOI":"10.1063\/1.532244","volume":"39","author":"H Cendra","year":"1998","unstructured":"Cendra H, Holm DD, Hoyle MJW, Marsden JE (1998) The Maxwell\u2013Vlasovequations in Euler-Poincar\u00e9 form. J\u00a0Math Phys 39:3138\u20133157","journal-title":"J Math Phys"},{"key":"326_CR32_326","first-page":"1","volume":"186","author":"H Cendra","year":"1998","unstructured":"Cendra H, Holm DD, Marsden JE, Ratiu T (1998) Lagrangian Reduction, theEuler\u2013Poincar\u00e9 Equations and Semidirect Products. Amer Math Soc Transl 186:1\u201325","journal-title":"Amer Math Soc Transl"},{"key":"326_CR33_326","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/0393-0440(87)90026-X","volume":"4","author":"H Cendra","year":"1987","unstructured":"Cendra H, Ibort A, Marsden JE (1987) Variational principal fiber bundles:a\u00a0geometric theory of Clebsch potentials and Lin constraints. J\u00a0Geom Phys 4:183\u2013206","journal-title":"J Geom Phys"},{"key":"326_CR34_326","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/0167-2789(87)90005-4","volume":"27","author":"H Cendra","year":"1987","unstructured":"Cendra H, Marsden JE (1987) Lin constraints, Clebsch potentials andvariational principles. Physica D 27:63\u201389","journal-title":"Physica D"},{"key":"326_CR35_326","first-page":"1","volume":"722","author":"H Cendra","year":"2001","unstructured":"Cendra H, Marsden JE, Ratiu TS (2001) Lagrangian reduction by stages. MemAmer Math Soc 722:1\u2013108","journal-title":"Mem Amer Math Soc"},{"key":"326_CR36_326","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1007\/978-3-642-56478-9_10","volume-title":"Mathematics Unlimited-2001 and Beyond","author":"H Cendra","year":"2001","unstructured":"Cendra H, Marsden JE, Ratiu T (2001) Geometric mechanics, Lagrangian reductionand nonholonomic systems. In: Enquist B, Schmid W (eds) Mathematics Unlimited-2001 and Beyond. Springer, New York,pp\u00a0221\u2013273"},{"key":"326_CR37_326","doi-asserted-by":"crossref","first-page":"833","DOI":"10.17323\/1609-4514-2003-3-3-833-867","volume":"3","author":"H Cendra","year":"2003","unstructured":"Cendra H, Marsden JE, Pekarsky S, Ratiu TS (2003) Variational principles forLie-Poisson and Hamilton-Poincar\u00e9 equations. Mosc Math J 3:833\u2013867","journal-title":"Mosc Math J"},{"key":"326_CR38_326","first-page":"393","volume":"8","author":"D Chang","year":"2002","unstructured":"Chang D, Bloch AM, Leonard N, Marsden JE, Woolsey C (2002) The equivalence ofcontrolled Lagrangian and controlled Hamiltonian systems. Control Calc Var (special issue) 8:393\u2013422","journal-title":"Control Calc Var (special issue)"},{"key":"326_CR39_326","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0073665","volume-title":"Properties of Infinite Dimensional Hamiltonian systems. Lecture Notes in Mathematics","author":"PR Chernoff","year":"1974","unstructured":"Chernoff PR, Marsden JE (1974) Properties of Infinite Dimensional Hamiltoniansystems. Lecture Notes in Mathematics, vol\u00a0425. Springer, New York"},{"key":"326_CR40_326","first-page":"253","volume":"5","author":"NG Chetayev","year":"1941","unstructured":"Chetayev NG (1941) On the equations of Poincar\u00e9. J\u00a0Appl Math Mech5:253\u2013262","journal-title":"J Appl Math Mech"},{"key":"326_CR41_326","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1016\/S0196-8858(02)00503-1","volume":"31","author":"P Chossat","year":"2003","unstructured":"Chossat P, Lewis D, Ortega JP, Ratiu T (2003) Bifurcation of relativeequilibria in mechanical systems with symmetry. Adv Appl Math 31:10\u201345","journal-title":"Adv Appl Math"},{"key":"326_CR42_326","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s002050200182","volume":"163","author":"P Chossat","year":"2002","unstructured":"Chossat P, Ortega JP, Ratiu T (2002) Hamiltonian Hopf bifurcation withsymmetry. Arch Ration Mech Anal 163:1\u201333; 167:83\u201384","journal-title":"Arch Ration Mech Anal"},{"key":"326_CR43_326","volume-title":"Geometrie du moment","author":"M Condevaux","year":"1988","unstructured":"Condevaux M, Dazord P, Molino P (1988) Geometrie du moment. SeminaireSud-Rhodanien, Lyon"},{"key":"326_CR44_326","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-0348-8891-2","volume-title":"Global Aspects of Classical Integrable Systems","author":"R Cushman","year":"1997","unstructured":"Cushman R, Bates L (1997) Global Aspects of Classical Integrable Systems.Birkh\u00e4user, Boston"},{"key":"326_CR45_326","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/0167-2789(82)90008-2","volume":"6","author":"R Cushman","year":"1982","unstructured":"Cushman R, Rod D (1982) Reduction of the semi-simple 1:1 resonance. Physica D6:105\u2013112","journal-title":"Physica D"},{"key":"326_CR46_326","doi-asserted-by":"crossref","unstructured":"Cushman R, Sniatycki J (2002) Nonholonomic reduction for free and properactions. Regul Chaotic Dyn 7:61\u201372","DOI":"10.1070\/RD2002v007n01ABEH000196"},{"key":"326_CR47_326","unstructured":"Cushman R, Sniatycki J (1999) Hamiltonian mechanics on principal bundles.C R Math Acad Sci Soc R Can 21:60\u201364"},{"key":"326_CR48_326","volume-title":"Lie Groups","author":"J Duistermaat","year":"1999","unstructured":"Duistermaat J, Kolk J (1999) Lie Groups. Springer, New York"},{"key":"326_CR49_326","doi-asserted-by":"crossref","first-page":"102","DOI":"10.2307\/1970699","volume":"92","author":"DG Ebin","year":"1970","unstructured":"Ebin DG, Marsden JE (1970) Groups of diffeomorphisms and the motion of anincompressible fluid. Ann Math 92:102\u2013163","journal-title":"Ann Math"},{"key":"326_CR50_326","volume-title":"Quantum Mechanics and Path Integrals","author":"R Feynman","year":"1965","unstructured":"Feynman R, Hibbs AR (1965) Quantum Mechanics and Path Integrals. McGraw-Hill,Murray Hill"},{"key":"326_CR51_326","first-page":"147","volume":"33","author":"AE Fischer","year":"1980","unstructured":"Fischer AE, Marsden JE, Moncrief V (1980) The structure of the space ofsolutions of Einstein\u2019s equations, I: One Killing field. Ann Ins H Poincar\u00e9 33:147\u2013194","journal-title":"Ann Ins H Poincar\u00e9"},{"key":"326_CR52_326","volume-title":"Bifurcation Theory, vol 2. Applied Mathematical Sciences","author":"M Golubitsky","year":"1988","unstructured":"Golubitsky M, Stewart I, Schaeffer D (1988) Singularities and Groups inBifurcation Theory, vol\u00a02. Applied Mathematical Sciences, vol\u00a069. Springer, New York"},{"key":"326_CR53_326","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1016\/j.crma.2004.01.029","volume":"338","author":"F Grabsi","year":"2004","unstructured":"Grabsi F, Montaldi J, Ortega JP (2004) Bifurcation and forced symmetrybreaking in Hamiltonian systems. C R Acad Sci Paris S\u00e9r I Math 338:565\u2013570","journal-title":"C R Acad Sci Paris S\u00e9r I Math"},{"key":"326_CR54_326","first-page":"329","volume":"40","author":"A Guichardet","year":"1984","unstructured":"Guichardet A (1984) On rotation and vibration motions of molecules. Ann InstH Poincar\u00e9 40:329\u2013342","journal-title":"Ann Inst H Poincar\u00e9"},{"key":"326_CR55_326","first-page":"1","volume":"1","author":"V Guillemin","year":"1978","unstructured":"Guillemin V, Sternberg S (1978) On the equations of motions of a\u00a0classicparticle in a\u00a0Yang-Mills field and the principle of general covariance. Hadronic J 1:1\u201332","journal-title":"Hadronic J"},{"key":"326_CR56_326","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1016\/0003-4916(80)90155-4","volume":"1278","author":"V Guillemin","year":"1980","unstructured":"Guillemin V, Sternberg S (1980) The moment map and collective motion. AnnPhys 1278:220\u2013253","journal-title":"Ann Phys"},{"key":"326_CR57_326","volume-title":"Symplectic Techniques in Physics","author":"V Guillemin","year":"1984","unstructured":"Guillemin V, Sternberg S (1984) Symplectic Techniques in Physics. CambridgeUniversity Press, Cambridge"},{"key":"326_CR58_326","first-page":"1","volume":"50","author":"G Hamel","year":"1904","unstructured":"Hamel G (1904) Die Lagrange\u2013Eulerschen Gleichungen der Mechanik. Z MathPhys 50:1\u201357","journal-title":"Z Math Phys"},{"key":"326_CR59_326","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-88463-4","volume-title":"Theoretische Mechanik","author":"G Hamel","year":"1949","unstructured":"Hamel G (1949) Theoretische Mechanik. Springer, Heidelberg"},{"key":"326_CR60_326","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/s00332-004-0544-x","volume":"15","author":"A Hernandez","year":"2005","unstructured":"Hernandez A, Marsden JE (2005) Regularization of the amended potential and thebifurcation of relative equilibria. J\u00a0Nonlinear Sci 15:93\u2013132","journal-title":"J Nonlinear Sci"},{"key":"326_CR61_326","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1006\/aima.1998.1721","volume":"137","author":"DD Holm","year":"1998","unstructured":"Holm DD, Marsden JE, Ratiu T (1998) The Euler\u2013Poincar\u00e9 equations andsemidirect products with applications to continuum theories. Adv Math 137:1\u201381","journal-title":"Adv Math"},{"key":"326_CR62_326","first-page":"251","volume-title":"Large-Scale Atmosphere-Ocean Dynamics II: Geometric Methods and Models","author":"DD Holm","year":"2002","unstructured":"Holm DD, Marsden JE, Ratiu T (2002) The Euler\u2013Poincar\u00e9 equations ingeophysical fluid dynamics. In: Norbury J, Roulstone I (eds) Large-Scale Atmosphere-Ocean Dynamics II: Geometric Methods and Models. Cambridge UnivPress, Cambridge, pp\u00a0251\u2013300"},{"key":"326_CR63_326","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0370-1573(85)90028-6","volume":"123","author":"DD Holm","year":"1985","unstructured":"Holm DD, Marsden JE, Ratiu T, Weinstein A (1985) Nonlinear stability of fluidand plasma equilibria. Phys Rep 123:1\u2013196","journal-title":"Phys Rep"},{"key":"326_CR64_326","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1007\/BF01457962","volume":"104","author":"H Hopf","year":"1931","unstructured":"Hopf H (1931) \u00dcber die Abbildungen der dreidimensionalen Sph\u00e4re auf dieKugelfl\u00e4che. Math Ann 104:38\u201363","journal-title":"Math Ann"},{"key":"326_CR65_326","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/S0022-4049(96)00133-8","volume":"126","author":"J Huebschmann","year":"1998","unstructured":"Huebschmann J (1998) Smooth structures on certain moduli spaces for bundles ona\u00a0surface. J\u00a0Pure Appl Algebra 126:183\u2013221","journal-title":"J Pure Appl Algebra"},{"key":"326_CR66_326","first-page":"199","volume":"47","author":"T Iwai","year":"1987","unstructured":"Iwai T (1987) A\u00a0geometric setting for classical molecular dynamics. AnnInst Henri Poincar\u00e9 Phys Theor 47:199\u2013219","journal-title":"Ann Inst Henri Poincar\u00e9 Phys Theor"},{"key":"326_CR67_326","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/0375-9601(90)90889-V","volume":"149","author":"T Iwai","year":"1990","unstructured":"Iwai T (1990) On the Guichardet\/Berry connection. Phys Lett A149:341\u2013344","journal-title":"Phys Lett A"},{"key":"326_CR68_326","doi-asserted-by":"crossref","first-page":"5521","DOI":"10.1088\/0305-4470\/39\/19\/S12","volume":"39","author":"S Jalnapurkar","year":"2006","unstructured":"Jalnapurkar S, Leok M, Marsden JE, West M (2006) Discrete Routh reduction. J\u00a0Phys A Math Gen 39:5521\u20135544","journal-title":"J Phys A Math Gen"},{"key":"326_CR69_326","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1080\/713603744","volume":"15","author":"S Jalnapurkar","year":"2000","unstructured":"Jalnapurkar S, Marsden J (2000) Reduction of Hamilton\u2019s variational principle.Dyn Stab Syst 15:287\u2013318","journal-title":"Dyn Stab Syst"},{"key":"326_CR70_326","doi-asserted-by":"crossref","first-page":"1295","DOI":"10.1002\/1097-0207(20001210)49:10<1295::AID-NME993>3.0.CO;2-W","volume":"49","author":"C Kane","year":"2000","unstructured":"Kane C, Marsden JE, Ortiz M, West M (2000) Variational integrators and theNewmark algorithm for conservative and dissipative mechanical systems. Int J Num Math Eng 49:1295\u20131325","journal-title":"Int J Num Math Eng"},{"key":"326_CR71_326","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1007\/s00332-004-0650-9","volume":"15","author":"E Kanso","year":"2005","unstructured":"Kanso E, Marsden JE, Rowley CW, Melli-Huber J (2005) Locomotion of articulatedbodies in a\u00a0perfect fluid. J\u00a0Nonlinear Sci 15:255\u2013289","journal-title":"J Nonlinear Sci"},{"key":"326_CR72_326","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1002\/cpa.3160310405","volume":"31","author":"D Kazhdan","year":"1978","unstructured":"Kazhdan D, Kostant B, Sternberg S (1978) Hamiltonian group actions anddynamical systems of Calogero type. Comm Pure Appl Math 31:481\u2013508","journal-title":"Comm Pure Appl Math"},{"key":"326_CR73_326","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1080\/02681119608806228","volume":"11","author":"V Kirk","year":"1996","unstructured":"Kirk V, Marsden JE, Silber M (1996) Branches of stable three-tori usingHamiltonian methods in Hopf bifurcation on a\u00a0rhombic lattice. Dyn Stab Syst 11:267\u2013302","journal-title":"Dyn Stab Syst"},{"key":"326_CR74_326","volume-title":"Foundations of Differential Geometry","author":"S Kobayashi","year":"1963","unstructured":"Kobayashi S, Nomizu K (1963) Foundations of Differential Geometry.Wiley, New York"},{"key":"326_CR75_326","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1007\/BF00375092","volume":"118","author":"J Koiller","year":"1992","unstructured":"Koiller J (1992) Reduction of some classical nonholonomic systems withsymmetry. Arch Ration Mech Anal 118:113\u2013148","journal-title":"Arch Ration Mech Anal"},{"key":"326_CR76_326","doi-asserted-by":"crossref","first-page":"901","DOI":"10.1137\/S0363012995290367","volume":"35","author":"WS Koon","year":"1997","unstructured":"Koon WS, Marsden JE (1997) Optimal control for holonomic and nonholonomicmechanical systems with symmetry and Lagrangian reduction. SIAM J Control Optim 35:901\u2013929","journal-title":"SIAM J Control Optim"},{"key":"326_CR77_326","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/S0034-4877(98)80007-4","volume":"42","author":"WS Koon","year":"1998","unstructured":"Koon WS, Marsden JE (1998) The Poisson reduction of nonholonomic mechanicalsystems. Rep Math Phys 42:101\u2013134","journal-title":"Rep Math Phys"},{"key":"326_CR78_326","volume-title":"Proc US-Japan Seminar on Diff Geom","author":"B Kostant","year":"1966","unstructured":"Kostant B (1966) Orbits, symplectic structures and representation theory. In:Proc US-Japan Seminar on Diff Geom, vol\u00a077. Nippon Hyronsha, Kyoto"},{"key":"326_CR79_326","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1512\/iumj.1981.30.30022","volume":"30","author":"M Kummer","year":"1981","unstructured":"Kummer M (1981) On the construction of the reduced phase space ofa\u00a0Hamiltonian system with symmetry. Indiana Univ Math J 30:281\u2013291","journal-title":"Indiana Univ Math J"},{"key":"326_CR80_326","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1016\/0022-0396(90)90057-V","volume":"83","author":"M Kummer","year":"1990","unstructured":"Kummer M (1990) On resonant classical Hamiltonians with n frequencies. J\u00a0Diff Eqns 83:220\u2013243","journal-title":"J Diff Eqns"},{"key":"326_CR81_326","volume-title":"M\u00e9canique Analytique","author":"JL Lagrange","year":"1788","unstructured":"Lagrange JL (1788) M\u00e9canique Analytique. Chez la Veuve Desaint, Paris"},{"key":"326_CR82_326","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/0393-0440(94)00034-2","volume":"15","author":"NP Landsman","year":"1995","unstructured":"Landsman NP (1995) Rieffel induction as generalized quantum Marsden-Weinsteinreduction. J\u00a0Geom Phys 15:285\u2013319. Erratum: J Geom Phys 17:298","journal-title":"J Geom Phys"},{"key":"326_CR83_326","doi-asserted-by":"crossref","first-page":"298","DOI":"10.1016\/0393-0440(95)00040-O","volume":"17","author":"NP Landsman","year":"1998","unstructured":"Landsman NP (1998) Mathematical topics between classical and quantummechanics. J\u00a0Geom Phys 17:298","journal-title":"J Geom Phys"},{"key":"326_CR84_326","first-page":"127","volume-title":"Symplectic Geometry. London Math Soc Lecture Note Ser","author":"E Lerman","year":"1993","unstructured":"Lerman E, Montgomery R, Jamaar RS (1993) Examples of singular reduction. In:Symplectic Geometry. London Math Soc Lecture Note Ser, vol\u00a0192. Cambridge Univ Press, Cambridge, pp\u00a0127\u2013155"},{"key":"326_CR85_326","doi-asserted-by":"crossref","first-page":"1637","DOI":"10.1088\/0951-7715\/11\/6\/012","volume":"11","author":"E Lerman","year":"1998","unstructured":"Lerman E, Singer SF (1998) Stability and persistence of relative equilibria atsingular values of the moment map. Nonlinearity 11:1637\u20131649","journal-title":"Nonlinearity"},{"key":"326_CR86_326","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1016\/S0764-4442(99)80181-5","volume":"328","author":"E Lerman","year":"1999","unstructured":"Lerman E, Tokieda T (1999) On relative normal modes. C R Acad Sci Paris S\u00e9r IMath 328:413\u2013418","journal-title":"C R Acad Sci Paris S\u00e9r I Math"},{"key":"326_CR87_326","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1002\/nme.958","volume":"60","author":"A Lew","year":"2004","unstructured":"Lew A, Marsden JE, Ortiz M, West M (2004) Variational time integration formechanical systems. Int J Num Meth Eng 60:153\u2013212","journal-title":"Int J Num Meth Eng"},{"key":"326_CR88_326","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1016\/0167-2789(86)90207-1","volume":"18","author":"D Lewis","year":"1986","unstructured":"Lewis D, Marsden JE, Montgomery R, Ratiu T (1986) The Hamiltonian structurefor dynamic free boundary problems. Physica D 18:391\u2013404","journal-title":"Physica D"},{"key":"326_CR89_326","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-009-3807-6","volume-title":"Symplectic Geometry and Analytical Mechanics","author":"P Libermann","year":"1987","unstructured":"Libermann P, Marle CM (1987) Symplectic Geometry and Analytical Mechanics.Kluwer, Dordrecht"},{"key":"326_CR90_326","volume-title":"Theorie der Transformationsgruppen. Zweiter Abschnitt","author":"S Lie","year":"1890","unstructured":"Lie S (1890) Theorie der Transformationsgruppen. Zweiter Abschnitt. Teubner,Leipzig"},{"key":"326_CR91_326","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1007\/978-94-010-1508-0_22","volume-title":"Differential Geometry and Relativity","author":"CM Marle","year":"1976","unstructured":"Marle CM (1976) Symplectic manifolds, dynamical groups and Hamiltonianmechanics. In: Cahen M, Flato M (eds) Differential Geometry and Relativity. Reidel, Boston, pp\u00a0249\u2013269"},{"key":"326_CR92_326","volume-title":"Hamiltonian Reduction by Stages. Springer Lecture Notes in Mathematics","author":"J Marsden","year":"2007","unstructured":"Marsden J, Ek GM, Ortega JP, Perlmutter M, Ratiu T (2007) HamiltonianReduction by Stages. Springer Lecture Notes in Mathematics, vol\u00a01913. Springer, Heidelberg"},{"key":"326_CR93_326","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/S0926-2245(98)00021-7","volume":"9","author":"J Marsden","year":"1998","unstructured":"Marsden J, Misiolek G, Perlmutter M, Ratiu T (1998) Symplectic reduction forsemidirect products and central extensions. Diff Geom Appl 9:173\u2013212","journal-title":"Diff Geom Appl"},{"key":"326_CR94_326","volume-title":"Lectures on Geometric Methods in Mathematical Physics","author":"JE Marsden","year":"1981","unstructured":"Marsden JE (1981) Lectures on Geometric Methods in Mathematical Physics.SIAM, Philadelphia"},{"key":"326_CR95_326","volume-title":"Lectures on Mechanics. London Mathematical Society Lecture Notes Series","author":"JE Marsden","year":"1992","unstructured":"Marsden JE (1992) Lectures on Mechanics. London Mathematical Society LectureNotes Series, vol\u00a0174. Cambridge University Press, Cambridge"},{"key":"326_CR96_326","volume-title":"Mathematical Foundations of Elasticity","author":"JE Marsden","year":"1983","unstructured":"Marsden JE, Hughes TJR (1983) Mathematical Foundations of Elasticity.Prentice Hall, Engelwood Cliffs. Reprinted 1994 by Dover"},{"key":"326_CR97_326","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/0003-4916(86)90157-0","volume":"169","author":"JE Marsden","year":"1986","unstructured":"Marsden JE, Montgomery R, Morrison PJ, Thompson WB (1986) Covariant Poissonbrackets for classical fields. Ann Phys 169:29\u201348","journal-title":"Ann Phys"},{"key":"326_CR98_326","volume-title":"mechanics. Memoirs of the AMS","author":"JE Marsden","year":"1990","unstructured":"Marsden JE, Montgomery R, Ratiu T (1990) Reduction, symmetry and phases inmechanics. Memoirs of the AMS, vol\u00a0436. American Mathematical Society, Providence"},{"key":"326_CR99_326","unstructured":"Marsden JE, Ostrowski J (1996) Symmetries in motion: Geometric foundationsof motion control. Nonlinear Sci Today http:\/\/link.springer-ny.com"},{"key":"326_CR100_326","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1007\/s002200050505","volume":"199","author":"JE Marsden","year":"1998","unstructured":"Marsden JE, Patrick GW, Shkoller S (1998) Multisymplectic geometry,variational integrators and nonlinear PDEs. Comm Math Phys 199:351\u2013395","journal-title":"Comm Math Phys"},{"key":"326_CR101_326","doi-asserted-by":"crossref","first-page":"1647","DOI":"10.1088\/0951-7715\/12\/6\/314","volume":"12","author":"JE Marsden","year":"1999","unstructured":"Marsden JE, Pekarsky S, Shkoller S (1999) Discrete Euler\u2013Poincar\u00e9 andLie\u2013Poisson equations. Nonlinearity 12:1647\u20131662","journal-title":"Nonlinearity"},{"key":"326_CR102_326","first-page":"33","volume":"22","author":"JE Marsden","year":"2000","unstructured":"Marsden JE, Perlmutter M (2000) The orbit bundle picture of cotangent bundlereduction. C R Math Acad Sci Soc R Can 22:33\u201354","journal-title":"C R Math Acad Sci Soc R Can"},{"key":"326_CR103_326","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1007\/BF00398428","volume":"11","author":"JE Marsden","year":"1986","unstructured":"Marsden JE, Ratiu T (1986) Reduction of Poisson manifolds. Lett Math Phys11:161\u2013170","journal-title":"Lett Math Phys"},{"key":"326_CR104_326","volume-title":"Applied Mathematics, vol 17. (1999) 2nd edn","author":"JE Marsden","year":"1994","unstructured":"Marsden JE, Ratiu T (1994) Introduction to Mechanics and Symmetry. Texts inApplied Mathematics, vol\u00a017. (1999) 2nd edn. Springer, New York"},{"key":"326_CR105_326","doi-asserted-by":"crossref","first-page":"3379","DOI":"10.1063\/1.533317","volume":"41","author":"JE Marsden","year":"2000","unstructured":"Marsden JE, Ratiu T, Scheurle J (2000) Reduction theory and theLagrange-Routh equations. J\u00a0Math Phys 41:3379\u20133429","journal-title":"J Math Phys"},{"key":"326_CR106_326","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1090\/S0002-9947-1984-0719663-1","volume":"281","author":"JE Marsden","year":"1984","unstructured":"Marsden JE, Ratiu T, Weinstein A (1984) Semi-direct products and reductionin mechanics. Trans Amer Math Soc 281:147\u2013177","journal-title":"Trans Amer Math Soc"},{"key":"326_CR107_326","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1090\/conm\/028\/751975","volume":"28","author":"JE Marsden","year":"1984","unstructured":"Marsden JE, Ratiu T, Weinstein A (1984) Reduction and Hamiltonian structureson duals of semidirect product Lie Algebras. Contemp Math 28:55\u2013100","journal-title":"Contemp Math"},{"key":"326_CR108_326","first-page":"17","volume":"44","author":"JE Marsden","year":"1993","unstructured":"Marsden JE, Scheurle J (1993) Lagrangian reduction and the double sphericalpendulum. ZAMP 44:17\u201343","journal-title":"ZAMP"},{"key":"326_CR109_326","first-page":"139","volume":"1","author":"JE Marsden","year":"1993","unstructured":"Marsden JE, Scheurle J (1993) The reduced Euler\u2013Lagrange equations.Fields Inst Comm 1:139\u2013164","journal-title":"Fields Inst Comm"},{"key":"326_CR110_326","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/0034-4877(74)90021-4","volume":"5","author":"JE Marsden","year":"1974","unstructured":"Marsden JE, Weinstein A (1974) Reduction of symplectic manifolds withsymmetry. Rep Math Phys 5:121\u2013130","journal-title":"Rep Math Phys"},{"key":"326_CR111_326","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1016\/0167-2789(82)90043-4","volume":"4","author":"JE Marsden","year":"1982","unstructured":"Marsden JE, Weinstein A (1982) The Hamiltonian structure of theMaxwell\u2013Vlasov equations. Physica D 4:394\u2013406","journal-title":"Physica D"},{"key":"326_CR112_326","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/0167-2789(83)90134-3","volume":"7","author":"JE Marsden","year":"1983","unstructured":"Marsden JE, Weinstein A (1983) Coadjoint orbits, vortices and Clebschvariables for incompressible fluids. Physica D 7:305\u2013323","journal-title":"Physica D"},{"key":"326_CR113_326","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1017\/S096249290100006X","volume":"10","author":"JE Marsden","year":"2001","unstructured":"Marsden JE, West M (2001) Discrete mechanics and variational integrators.Acta Numerica 10:357\u2013514","journal-title":"Acta Numerica"},{"key":"326_CR114_326","unstructured":"Marsden J, Weinstein A, Ratiu T, Schmid R, Spencer R (1982) Hamiltoniansystems with symmetry, coadjoint orbits and plasma physics. In: Proc. IUTAM-IS1MM Symposium on Modern Developments in Analytical Mechanics, Torino, vol\u00a0117. Atti della Acad della Sc di Torino, pp\u00a0289\u2013340"},{"key":"326_CR115_326","first-page":"536","volume":"251","author":"JL Martin","year":"1959","unstructured":"Martin JL (1959) Generalized classical dynamics and the \u201cclassicalanalogue\u201d of a\u00a0Fermi oscillation. Proc Roy Soc A 251:536","journal-title":"Proc Roy Soc A"},{"key":"326_CR116_326","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198511779.001.0001","volume-title":"Introduction to Symplectic Topology","author":"D Mcduff","year":"1995","unstructured":"Mcduff D, Salamon D (1995) Introduction to Symplectic Topology. OxfordUniversity Press, Oxford"},{"key":"326_CR117_326","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/B978-0-12-550350-1.50025-4","volume-title":"Dynamical Systems","author":"KR Meyer","year":"1973","unstructured":"Meyer KR (1973) Symmetries and integrals in mechanics. In: Peixoto M (ed)Dynamical Systems. Academic Press, New York, pp\u00a0259\u2013273"},{"key":"326_CR118_326","volume-title":"Hamiltonian and lagrangian flows on center manifolds, with applications to elliptic variational problems. Lecture Notes in Mathematics","author":"A Mielke","year":"1991","unstructured":"Mielke A (1991) Hamiltonian and lagrangian flows on center manifolds, withapplications to elliptic variational problems. Lecture Notes in Mathematics, vol\u00a01489. Springer, Heidelberg"},{"key":"326_CR119_326","doi-asserted-by":"crossref","first-page":"121","DOI":"10.2977\/prims\/1195175328","volume":"24","author":"K Mikami","year":"1988","unstructured":"Mikami K, Weinstein A (1988) Moments and reduction for symplectic groupoidactions. Publ RIMS Kyoto Univ 24:121\u2013140","journal-title":"Publ RIMS Kyoto Univ"},{"key":"326_CR120_326","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1007\/BF00420042","volume":"8","author":"R Montgomery","year":"1984","unstructured":"Montgomery R (1984) Canonical formulations of a\u00a0particle ina\u00a0Yang\u2013Mills field. Lett Math Phys 8:59\u201367","journal-title":"Lett Math Phys"},{"key":"326_CR121_326","volume-title":"The Bundle Picture in Mechanics. Ph?D thesis","author":"R Montgomery","year":"1986","unstructured":"Montgomery R (1986) The Bundle Picture in Mechanics. Ph?D thesis, Universityof California Berkeley"},{"key":"326_CR122_326","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1007\/BF01217966","volume":"120","author":"R Montgomery","year":"1988","unstructured":"Montgomery R (1988) The connection whose holonomy is the classical adiabaticangles of Hannay and Berry and its generalization to the non-integrable case. Comm Math Phys 120:269\u2013294","journal-title":"Comm Math Phys"},{"key":"326_CR123_326","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1007\/BF02096874","volume":"128","author":"R Montgomery","year":"1990","unstructured":"Montgomery R (1990) Isoholonomic problems and some applications. Comm MathPhys 128:565\u2013592","journal-title":"Comm Math Phys"},{"key":"326_CR124_326","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1007\/978-1-4613-9725-0_15","volume-title":"The Geometry of Hamiltonian Systems","author":"R Montgomery","year":"1991","unstructured":"Montgomery R (1991) Optimal control of deformable bodies and Its relation togauge theory. In: Ratiu T (ed) The Geometry of Hamiltonian Systems. Springer, New York, pp\u00a0403\u2013438"},{"key":"326_CR125_326","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1119\/1.16514","volume":"59","author":"R Montgomery","year":"1991","unstructured":"Montgomery R (1991) How much does a\u00a0rigid body rotate? A\u00a0Berry\u2019sphase from the 18th century. Amer J Phys 59:394\u2013398","journal-title":"Amer J Phys"},{"key":"326_CR126_326","first-page":"193","volume":"1","author":"R Montgomery","year":"1993","unstructured":"Montgomery R (1993) Gauge theory of the falling cat. Fields Inst Commun1:193\u2013218","journal-title":"Fields Inst Commun"},{"key":"326_CR127_326","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1090\/conm\/028\/751976","volume-title":"Fluids and plasmas: geometry and dynamics. Boulder, 1983","author":"R Montgomery","year":"1984","unstructured":"Montgomery R, Marsden JE, Ratiu T (1984) Gauged Lie-Poisson structures. In:Fluids and plasmas: geometry and dynamics. Boulder, 1983. American Mathematical Society, Providence, pp\u00a0101\u2013114"},{"key":"326_CR128_326","doi-asserted-by":"crossref","first-page":"790","DOI":"10.1103\/PhysRevLett.45.790","volume":"45","author":"PJ Morrison","year":"1980","unstructured":"Morrison PJ, Greene JM (1980) Noncanonical Hamiltonian density formulationof hydrodynamics and ideal magnetohydrodynamics. Phys Rev Lett 45:790\u2013794. (1982) errata 48:569","journal-title":"Phys Rev Lett"},{"key":"326_CR129_326","doi-asserted-by":"crossref","first-page":"2405","DOI":"10.1103\/PhysRevD.7.2405","volume":"7","author":"Y Nambu","year":"1973","unstructured":"Nambu Y (1973) Generalized Hamiltonian dynamics. Phys Rev D7:2405\u20132412","journal-title":"Phys Rev D"},{"key":"326_CR130_326","unstructured":"Ortega JP (1998) Symmetry, Reduction, and Stability in Hamiltonian Systems.Ph?D thesis, University of California, Santa Cruz"},{"key":"326_CR131_326","doi-asserted-by":"crossref","first-page":"999","DOI":"10.1016\/S1631-073X(02)02394-4","volume":"334","author":"JP Ortega","year":"2002","unstructured":"Ortega JP (2002) The symplectic reduced spaces of a\u00a0Poisson action. CR Acad Sci Paris S\u00e9r I Math 334:999\u20131004","journal-title":"C R Acad Sci Paris S\u00e9r I Math"},{"key":"326_CR132_326","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1017\/S0308210500002602","volume":"133","author":"JP Ortega","year":"2003","unstructured":"Ortega JP (2003) Relative normal modes for nonlinear Hamiltonian systems.Proc Royal Soc Edinb Sect A 133:665\u2013704","journal-title":"Proc Royal Soc Edinb Sect A"},{"key":"326_CR133_326","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.geomphys.2004.01.002","volume":"52","author":"JP Ortega","year":"2004","unstructured":"Ortega JP, Planas-Bielsa V (2004) Dynamics on Leibniz manifolds. J\u00a0GeomPhys 52:1\u201327","journal-title":"J Geom Phys"},{"key":"326_CR134_326","doi-asserted-by":"crossref","first-page":"1107","DOI":"10.1016\/S0764-4442(97)88714-9","volume":"325","author":"JP Ortega","year":"1997","unstructured":"Ortega JP, Ratiu T (1997) Persistence and smoothness of critical relativeelements in Hamiltonian systems with symmetry. C R Acad Sci Paris S\u00e9r I Math 325:1107\u20131111","journal-title":"C R Acad Sci Paris S\u00e9r I Math"},{"key":"326_CR135_326","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1016\/S0393-0440(99)00024-8","volume":"32","author":"JP Ortega","year":"1999","unstructured":"Ortega JP, Ratiu T (1999) Non-linear stability of singular relative periodicorbits in Hamiltonian systems with symmetry. J\u00a0Geom Phys 32:160\u2013188","journal-title":"J Geom Phys"},{"key":"326_CR136_326","doi-asserted-by":"crossref","first-page":"693","DOI":"10.1088\/0951-7715\/12\/3\/315","volume":"12","author":"JP Ortega","year":"1999","unstructured":"Ortega JP, Ratiu T (1999) Stability of Hamiltonian relative equilibria.Nonlinearity 12:693\u2013720","journal-title":"Nonlinearity"},{"key":"326_CR137_326","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1007\/0-387-21791-6_11","volume-title":"Geometry, Mechanics and Dynamics","author":"JP Ortega","year":"2002","unstructured":"Ortega JP, Ratiu T (2002) The optimal momentum map. In: Newton P, Holmes P,Weinstein A (eds) Geometry, Mechanics and Dynamics. Springer, New York, pp\u00a0329\u2013362"},{"key":"326_CR138_326","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-3811-7","volume-title":"Momentum maps and Hamiltonian reduction. Progress in Mathematics","author":"JP Ortega","year":"2004","unstructured":"Ortega JP, Ratiu T (2004) Momentum maps and Hamiltonian reduction. Progressin Mathematics, vol\u00a0222. Birkh\u00e4user, Boston, pp\u00a0xxxiv+497"},{"key":"326_CR139_326","doi-asserted-by":"crossref","first-page":"1407","DOI":"10.1098\/rspa.2003.1213","volume":"460","author":"JP Ortega","year":"2004","unstructured":"Ortega JP, Ratiu T (2004) Relative equilibria near stable and unstableHamiltonian relative equilibria. Proc Royal Soc Lond Ser A 460:1407\u20131431","journal-title":"Proc Royal Soc Lond Ser A"},{"key":"326_CR140_326","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1007\/s10455-006-9017-9","volume":"30","author":"JP Ortega","year":"2006","unstructured":"Ortega JP, Ratiu T (2006) The reduced spaces of a\u00a0symplectic Lie groupaction. Ann Glob Analysis Geom 30:335\u2013381","journal-title":"Ann Glob Analysis Geom"},{"key":"326_CR141_326","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/S0034-4877(06)80040-6","volume":"58","author":"JP Ortega","year":"2006","unstructured":"Ortega JP, Ratiu T (2006) The stratified spaces of a\u00a0symplectic Liegroup action. Rep Math Phys 58:51\u201375","journal-title":"Rep Math Phys"},{"key":"326_CR142_326","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1016\/B0-12-512666-2\/00503-4","volume-title":"Encyclopedia of Mathematical Physics","author":"JP Ortega","year":"2006","unstructured":"Ortega JP, Ratiu T (2006) Symmetry and symplectic reduction. In:Fran\u00e7oise JP, Naber G, Tsun TS (eds) Encyclopedia of Mathematical Physics. Elsevier, New York, pp\u00a0190\u2013198"},{"key":"326_CR143_326","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/0393-0440(87)90022-2","volume":"4","author":"M Otto","year":"1987","unstructured":"Otto M (1987) A\u00a0reduction scheme for phase spaces with almost K\u00e4hlersymmetry. Regularity results for momentum level sets. J\u00a0Geom Phys 4:101\u2013118","journal-title":"J Geom Phys"},{"key":"326_CR144_326","volume-title":"A global formulation of the Lie theory of transformation groups. Mem Am Math Soc","author":"RS Palais","year":"1957","unstructured":"Palais RS (1957) A\u00a0global formulation of the Lie theory oftransformation groups. Mem Am Math Soc, vol\u00a022. American Mathematical Society, Providence, pp\u00a0iii+123"},{"key":"326_CR145_326","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/0393-0440(92)90015-S","volume":"9","author":"G Patrick","year":"1992","unstructured":"Patrick G (1992) Relative equilibria in Hamiltonian systems: The dynamicinterpretation of nonlinear stability on a\u00a0reduced phase space. J\u00a0Geom and Phys 9:111\u2013119","journal-title":"J Geom and Phys"},{"key":"326_CR146_326","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1007\/s00205-004-0322-9","volume":"174","author":"G Patrick","year":"2004","unstructured":"Patrick G, Roberts M, Wulff C (2004) Stability of Poisson equilibria andHamiltonian relative equilibria by energy methods. Arch Ration Mech An 174:301\u2013344","journal-title":"Arch Ration Mech An"},{"key":"326_CR147_326","doi-asserted-by":"crossref","first-page":"648","DOI":"10.1007\/BF02815288","volume":"X","author":"W Pauli","year":"1953","unstructured":"Pauli W (1953) On the Hamiltonian structure of non-local field theories. IlNuovo Cim X:648\u2013667","journal-title":"Il Nuovo Cim"},{"key":"326_CR148_326","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1007\/BF00750836","volume":"35","author":"M Pedroni","year":"1995","unstructured":"Pedroni M (1995) Equivalence of the Drinfelcd\u2013Sokolov reductionto a\u00a0bi-Hamiltonian reduction. Lett Math Phys 35:291\u2013302","journal-title":"Lett Math Phys"},{"key":"326_CR149_326","unstructured":"Perlmutter M, Ratiu T (2005) Gauged Poisson structures. Preprint"},{"key":"326_CR150_326","doi-asserted-by":"crossref","unstructured":"Perlmutter M, Rodr\u00edguez-Olmos M, Dias MS (2006) On the geometry of reducedcotangent bundles at zero momentum. J\u00a0Geom Phys 57:571\u2013596","DOI":"10.1016\/j.geomphys.2006.05.003"},{"key":"326_CR151_326","doi-asserted-by":"crossref","unstructured":"Perlmutter M, Rodr\u00edguez-Olmos M, Dias MS (2007) On the symplectic normalspace for cotangent lifted actions. Diff Geom Appl 26:277\u2013297","DOI":"10.1016\/j.difgeo.2007.11.020"},{"key":"326_CR152_326","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1016\/S0034-4877(05)00002-9","volume":"54","author":"V Planas-Bielsa","year":"2004","unstructured":"Planas-Bielsa V (2004) Point reduction in almost symplectic manifolds. RepMath Phys 54:295\u2013308","journal-title":"Rep Math Phys"},{"key":"326_CR153_326","unstructured":"Poincar\u00e9 H (1901) Sur une forme nouvelle des \u00e9quations de la m\u00e9chanique. CR Acad Sci 132:369\u2013371"},{"key":"326_CR154_326","unstructured":"Ratiu T (1980) The Euler\u2013Poisson equations and integrability. Ph?Dthesis, University of California at Berkeley"},{"key":"326_CR155_326","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1007\/BFb0092027","volume-title":"Geometric Methods in Mathematical Physics. Lecture Notes in Mathematics","author":"T Ratiu","year":"1980","unstructured":"Ratiu T (1980) Involution theorems. In: Kaiser G, Marsden J (eds) GeometricMethods in Mathematical Physics. Lecture Notes in Mathematics, vol\u00a0775. Springer, Berlin, pp\u00a0219\u2013257"},{"key":"326_CR156_326","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1512\/iumj.1980.29.29046","volume":"29","author":"T Ratiu","year":"1980","unstructured":"Ratiu T (1980) The motion of the free n-dimensional rigid body. Indiana Univ Math J 29:609\u2013629","journal-title":"Indiana Univ Math J"},{"key":"326_CR157_326","doi-asserted-by":"crossref","first-page":"1327","DOI":"10.1073\/pnas.78.3.1327","volume":"78","author":"T Ratiu","year":"1981","unstructured":"Ratiu T (1981) Euler\u2013Poisson equations on Lie algebras and theN-dimensional heavy rigid body. Proc Natl Acad Sci USA 78:1327\u20131328","journal-title":"Proc Natl Acad Sci USA"},{"key":"326_CR158_326","doi-asserted-by":"crossref","first-page":"409","DOI":"10.2307\/2374165","volume":"104","author":"T Ratiu","year":"1982","unstructured":"Ratiu T (1982) Euler\u2013Poisson equations on Lie algebras and theN-dimensional heavy rigid body. Am J Math 104:409\u2013448, 1337","journal-title":"Am J Math"},{"key":"326_CR159_326","doi-asserted-by":"crossref","first-page":"1719","DOI":"10.1088\/0951-7715\/10\/6\/015","volume":"10","author":"M Roberts","year":"1997","unstructured":"Roberts M, de Sousa Dias M (1997) Bifurcations from relative equilibria ofHamiltonian systems. Nonlinearity 10:1719\u20131738","journal-title":"Nonlinearity"},{"key":"326_CR160_326","doi-asserted-by":"crossref","first-page":"562","DOI":"10.1006\/jdeq.2001.4045","volume":"179","author":"M Roberts","year":"2002","unstructured":"Roberts M, Wulff C, Lamb J (2002) Hamiltonian systems near relativeequilibria. J\u00a0Diff Eq 179:562\u2013604","journal-title":"J Diff Eq"},{"key":"326_CR161_326","volume-title":"Treatise on the Dynamics of a System of Rigid Bodies","author":"EJ Routh","year":"1860","unstructured":"Routh EJ (1860) Treatise on the Dynamics of a\u00a0System of Rigid Bodies.MacMillan, London"},{"key":"326_CR162_326","volume-title":"Stability of a given state of motion","author":"EJ Routh","year":"1877","unstructured":"Routh EJ (1877) Stability of a\u00a0given state of motion. Halsted Press,New York. Reprinted (1975) In: Fuller AT (ed) Stability of Motion"},{"key":"326_CR163_326","volume-title":"Advanced Rigid Dynamics","author":"EJ Routh","year":"1884","unstructured":"Routh EJ (1884) Advanced Rigid Dynamics. MacMillian,London"},{"key":"326_CR164_326","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1073\/pnas.42.6.359","volume":"42","author":"I Satake","year":"1956","unstructured":"Satake I (1956) On a\u00a0generalization of the notion of manifold. ProcNat Acad Sci USA 42:359\u2013363","journal-title":"Proc Nat Acad Sci USA"},{"key":"326_CR165_326","doi-asserted-by":"crossref","first-page":"951","DOI":"10.1512\/iumj.1977.26.26077","volume":"26","author":"WJ Satzer","year":"1977","unstructured":"Satzer WJ (1977) Canonical reduction of mechanical systems invariant underAbelian group actions with an application to celestial mechanics. Ind Univ Math J 26:951\u2013976","journal-title":"Ind Univ Math J"},{"key":"326_CR166_326","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1007\/BF01881678","volume":"115","author":"JC Simo","year":"1991","unstructured":"Simo JC, Lewis DR, Marsden JE (1991) Stability of relative equilibria I: Thereduced energy momentum method. Arch Ration Mech Anal 115:15\u201359","journal-title":"Arch Ration Mech Anal"},{"key":"326_CR167_326","doi-asserted-by":"crossref","first-page":"375","DOI":"10.2307\/2944350","volume":"134","author":"R Sjamaar","year":"1991","unstructured":"Sjamaar R, Lerman E (1991) Stratified symplectic spaces and reduction. AnnMath 134:375\u2013422","journal-title":"Ann Math"},{"key":"326_CR168_326","doi-asserted-by":"crossref","unstructured":"Smale S (1970) Topology and Mechanics. Inv Math 10:305\u2013331,11:45\u201364","DOI":"10.1007\/BF01418778"},{"key":"326_CR169_326","volume-title":"Structure des Systemes Dynamiques","author":"JM Souriau","year":"1970","unstructured":"Souriau JM (1970) Structure des Systemes Dynamiques. Dunod,Paris"},{"key":"326_CR170_326","first-page":"374","volume":"1","author":"J Souriau","year":"1966","unstructured":"Souriau J (1966) Quantification g\u00e9om\u00e9trique. Comm Math Phys1:374\u2013398","journal-title":"Comm Math Phys"},{"key":"326_CR171_326","doi-asserted-by":"crossref","first-page":"5253","DOI":"10.1073\/pnas.74.12.5253","volume":"74","author":"S Sternberg","year":"1977","unstructured":"Sternberg S (1977) Minimal coupling and the symplectic mechanics ofa\u00a0classical particle in the presence of a\u00a0Yang\u2013Mills field. Proc Nat Acad Sci 74:5253\u20135254","journal-title":"Proc Nat Acad Sci"},{"key":"326_CR172_326","volume-title":"Classical Mechanics: A Modern Perspective. Wiley, New York. (1983)","author":"ECG Sudarshan","year":"1974","unstructured":"Sudarshan ECG, Mukunda N (1974) Classical Mechanics: A\u00a0ModernPerspective. Wiley, New York. (1983) 2nd edn. Krieber, Melbourne, Fl","edition":"2"},{"key":"326_CR173_326","unstructured":"Tulczyjew WM, Urbanski P (1999) A\u00a0slow and careful Legendretransformation for singular Lagrangians. Acta Phys Polon B 30:2909\u20132978. The Infeld Centennial Meeting, Warsaw, 1998"},{"key":"326_CR174_326","volume-title":"the Realm of Algebraic Geometry. Lecture Notes in Mathematics","author":"P Vanhaecke","year":"1996","unstructured":"Vanhaecke P (1996) Integrable Systems in the Realm of AlgebraicGeometry. Lecture Notes in Mathematics, vol\u00a01638. Springer, New York"},{"key":"326_CR175_326","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1007\/BF00400169","volume":"2","author":"A Weinstein","year":"1978","unstructured":"Weinstein A (1978) A\u00a0universal phase space for particles inYang\u2013Mills fields. Lett Math Phys 2:417\u2013420","journal-title":"Lett Math Phys"},{"key":"326_CR176_326","first-page":"95","volume":"1","author":"A Weinstein","year":"1983","unstructured":"Weinstein A (1983) Sophus Lie and symplectic geometry. Expo Math1:95\u201396","journal-title":"Expo Math"},{"key":"326_CR177_326","first-page":"207","volume":"7","author":"A Weinstein","year":"1996","unstructured":"Weinstein A (1996) Lagrangian mechanics and groupoids. Fields Inst Commun7:207\u2013231","journal-title":"Fields Inst Commun"},{"key":"326_CR178_326","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1016\/S0167-2789(97)00051-1","volume":"106","author":"JM Wendlandt","year":"1997","unstructured":"Wendlandt JM, Marsden JE (1997) Mechanical integrators derived froma\u00a0discrete variational principle. Physica D 106:223\u2013246","journal-title":"Physica D"},{"key":"326_CR179_326","volume-title":"A Treatise on the Analytical Dynamics of Particles and Rigid Bodies","author":"E Whittaker","year":"1937","unstructured":"Whittaker E (1937) A\u00a0Treatise on the Analytical Dynamics of Particlesand Rigid Bodies, 4th edn. Cambridge University Press, Cambridge. (1904) 1st edn. (1937) 5th edn. (1944) Reprinted by Dover and (1988)4th edn, Cambridge University Press","edition":"4"},{"key":"326_CR180_326","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1088\/0951-7715\/16\/1\/305","volume":"16","author":"C Wulff","year":"2003","unstructured":"Wulff C (2003) Persistence of relative equilibria in Hamiltonian systemswith non-compact symmetry. Nonlinearity 16:67\u201391","journal-title":"Nonlinearity"},{"key":"326_CR181_326","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1137\/S1111111101387760","volume":"1","author":"C Wulff","year":"2002","unstructured":"Wulff C, Roberts M (2002) Hamiltonian systems near relative periodic orbits.SIAM J Appl Dyn Syst 1:1\u201343","journal-title":"SIAM J Appl Dyn Syst"},{"key":"326_CR182_326","doi-asserted-by":"crossref","first-page":"3431","DOI":"10.1063\/1.532897","volume":"40","author":"N Zaalani","year":"1999","unstructured":"Zaalani N (1999) Phase space reduction and Poisson structure. J\u00a0Math Phys40:3431\u20133438","journal-title":"J Math Phys"}],"container-title":["Encyclopedia of Complexity and Systems Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-0-387-30440-3_326","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,20]],"date-time":"2024-08-20T16:52:08Z","timestamp":1724172728000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-0-387-30440-3_326"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009]]},"ISBN":["9780387758886","9780387304403"],"references-count":182,"URL":"https:\/\/doi.org\/10.1007\/978-0-387-30440-3_326","relation":{},"subject":[],"published":{"date-parts":[[2009]]}}}