Geometrija realnih form kompleksnega Neumannovega sistema : doktorska disertacija

Novak, Tina (author), Saksida, Pavle (mentor)

C. Neumannov sistem opisuje gibanje delca na ▫$n$▫-dimenzionalni sferi ▫$S^n$▫ v polju sil s kvadratnim potencialom ▫$U(q_1, \ldots, q_{n+1}) = \sum a_jq_j^2$▫. Znano je, da je Neumannov sistem popolnoma Liouvilleovo integrabilen. Prvi integrali Neumannovega sistema so integrali Uhlenbeckove. Poleg tega je kompleksen Neumannov sistem algebraično popolnoma integrabilen, nivojske množice kompleksne momentne preslikave pa so afini deli kompleksnih torusov. Nivojske množice realne momentne preslikave so potemtakem njihovi realni deli. V disertaciji natančno definiramo realne forme kompleksnega Neumannovega sistema. Realne forme so Hamiltonovi sistemi na kotangentih svežnjih nad hiperboloidi. Pokažemo, da so tudi novi sistemi popolnoma Liouvilleovo integrabilni in eksplicitno zapišemo njihove prve integrale (ohranitvene količine). Kompleksen Neumannov sistem je poseben primer splošnejšega Mumfordovega sistema. Mumfordov sistem je karakteriziran z Laxovo enačbo ▫$\frac{d}{dt}L^{\mathbb{C}}(\lambda) = [M^\mathbb{C}(\lambda), L^\mathbb{C}(\lambda)]$▫ v zančni algebri ▫$\mathfrak{sl}(2, \mathbb{C})[\lambda, \lambda^{-1}]$▫, pri čemer so koeficienti ▫$U^\mathbb{C}$▫, ▫$V^\mathbb{C}$▫, ▫$W^\mathbb{C}$▫ matrike ▫$L^\mathbb{C}(\lambda)$▫ polinomi določene oblike. Če so ▫$u_1, \ldots, u_n$▫ ničle ustrezne realne forme polinoma ▫$U^\mathbb{C}$▫, je topologija regularne nivojske množice momentne preslikave realne forme kompleksnega generičnega Neumannovega sistema določena z lego ničel ▫$u_1, \ldots, u_n$▫ glede na konstante ▫$a_1, \ldots, a_{n+1}$▫ in ostalih določenih parametrov sistema. Za dve družini realnih form je topologija nivojskih množic neodvisna od lege regularnih vrednosti momentne preslikave. Za eno od njiju so nivojske množice nekompaktne. Opazimo, da so v posebnih primerih ničle realne forme polinoma ▫$U^\mathbb{C}$▫ koordinate na enakoosnem hiperboloidu, ki je ustrezna realna forma kompleksne kvadrike ▫$(S^n)^\mathbb{C}$▫. Definiramo konično-hiperboloidne koordinate na enakoosnih hiperboloidih, ki so posplošitev Jacobijevih eliptično-sferičnih koordinat na sferi ▫$S^n$▫. Ker ima Neumannov sistem Laxovo enačbo tudi v zančni algebri ▫$\mathfrak{sl}(n+1, \mathbb{R})[\lambda, \lambda^{-1}]$▫, nam ta porodi še eno družino prvih integralov sistema. V disertaciji je podana in dokazana zveza med omenjeno družino integralov in družino integralov Uhlenbeckove.

• Language: Slovenian
• Keywords: matematika, integrabilni sistemi, Neumannov sistem, Arnold-Liouvilleove nivojske množice, spektralna krivulja, realne strukture, realne forme
• Work type: Doctoral dissertation
• Typology: 2.08 - Doctoral Dissertation
• Organization: FMF - Faculty of Mathematics and Physics
• Place of publishing: Ljubljana
• Publisher: [T. Novak]
• Year: 2015
• Number of pages: XI, 72 str.
• PID: 20.500.12556/RUL-95863
• UDC: 517.9(043.3)
• COBISS.SI-ID: 17567065
• Publication date in RUL: 24.10.2017

Secondary language

• Language: English

Abstract

C. Neumann system describes the motion of a particle on the sphere ▫$S^n$▫ under the influence of a quadratic potential ▫$U(q_1, \ldots, q_{n+1}) = \sum a_jq_j^2$▫. The Neumann system is completely Liouville integrable. First integrals in involution are well known Uhlenbeck's integrals. In addition, the complex Neumann system is completely algebraically integrable and the regular level sets of the complex momentum map are affine parts of complex tori. In the dissertation, we precisely define real forms of the complex Neumann system. We obtain new Hamiltonian systems on the cotangent bundles of hyperboloids. We prove that the real forms are completely integrable Hamiltonian systems and write down their first integrals (conserved quantities). The complex Neumann system is an example of the more general Mumford system. The Mumford system is characterized by the Lax equation ▫$\frac{d}{dt}L^{\mathbb{C}}(\lambda) = [M^\mathbb{C}(\lambda), L^\mathbb{C}(\lambda)]$▫ in the loop algebra ▫$\mathfrak{sl}(2, \mathbb{C})[\lambda, \lambda^{-1}]$▫. Coefficients ▫$U^\mathbb{C}$▫, ▫$V^\mathbb{C}$▫, ▫$W^\mathbb{C}$▫ of the matrix ▫$L^\mathbb{C}(\lambda)$▫ are suitable polynomials. If ▫$u_1, \ldots, u_n$▫ are roots of the appropriate real form of the polynomial ▫$U^\mathbb{C}$▫, the topology of a regular level set of the moment map of the real form is determined by the positions of the roots ▫$u_1, \ldots, u_n$▫ with respect to the constants ▫$a_1, \ldots, a_{n+1}$▫ and to the suitable parameters of the system. For two families of the real forms of the complex Neumann system, we describe the topology of the regular level set of the moment map. For one of these two families the level sets are noncompact. We observe that in some special cases the roots of a real form of the polynomial ▫$U^\mathbb{C}$▫ determine coordinates on a suitable hyperboloid. We define conical hyperboloidal coordinates on equiaxed hyperboloids and they can be interpreted as a generalization of the Jacobian elliptic spherical coordinates on ▫$S^n$▫. Since the Neumann system has another Lax equation in the loop algebra ▫$\mathfrak{sl}(n+1, \mathbb{R})[\lambda, \lambda^{-1}]$▫, there exists another family of the first integrals in involution. In the dissertation, we also give the formula which provides the relation between this family and the family of Uhlenbeck's integrals.

• Keywords: mathematics, integrable systems, Neumann system, Arnold-Liouville level sets, spectral curve, real structures, real forms