Theorie der algebraischen Funktionen einer Variabeln und ihre Anwendung auf algebraische Kurven und Abelsche Integrale. By K. Hensel and G. Landsberg. Pp. xvi, 708. (Leipzig: Teubner.)

Mathews, G. B.

doi:10.2307/3605115

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New Superintegrable Systems On the Plane2

Arithmetic Of Elliptic Curves1

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Cited by 2 publications

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“…and, therefore, algebra of the rst integrals coincides with well-know relations between Weierstrass ℘-function and its derivatives, see [2,5,8].…”

Section: Superintegrable System With Two Degrees Of Freedommentioning

confidence: 77%

Discretization and superintegrability all rolled into one

Tsiganov

2020

Nonlinearity

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Abelian integrals appear in mathematical descriptions of various physical processes. According to Abel's theorem these integrals are related to motion of a set of points along a plane curve around xed points, which are rarely used in physical applications. We propose to interpret coordinates of the xed points either as parameters of exact discretization or as additional rst integrals for equations of motion reduced to Abelian quadratures on a symmetric product of algebraic curve.

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“…and, therefore, algebra of the rst integrals coincides with well-know relations between Weierstrass ℘-function and its derivatives, see [2,5,8].…”

Section: Superintegrable System With Two Degrees Of Freedommentioning

confidence: 77%

Discretization and superintegrability all rolled into one

Tsiganov

2020

Nonlinearity

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“…In 1863 Clebsch proposed geometric approach to construction of algebraic constraints, closely interwoven with the intersection theory, which was continued by Brill and Noether in 1857 and formalized by Poincaré in 1901 and Severy in 1914, see classical textbooks [17,19], review [20]. and modern discussion in [21].…”

Section: Arithmetic Of Elliptic Curvesmentioning

confidence: 99%

“…In our opinion, it is practically impossible to use such sophisticated expressions in the direct search of additional integrals of motion or for investigations of algebras of integrals of motion, see also the discussion in [26]. we can derive the algebra of integrals using the well-known syzygies on an elliptic curve [17,19].…”

Section: New Superintegrable Systems On the Planementioning

confidence: 99%

Elliptic curve arithmetic and superintegrable systems

Tsiganov¹

2019

Phys. Scr.

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Harmonic oscillator and the Kepler problem are superintegrable systems which admit more integrals of motion than degrees of freedom and all these integrals are polynomials in momenta. We present superintegrable deformations of the oscillator and the Kepler problem with algebraic and rational first integrals. Also, we discuss a family of superintegrable metrics on the two-dimensional sphere, which have similar first integrals.The prime divisors are points on X, denoted P i = (x i , y i ), including point at infinity P ∞ , which plays the role of neutral element 0 in arithmetic of elliptic curves.

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Theorie der algebraischen Funktionen einer Variabeln und ihre Anwendung auf algebraische Kurven und Abelsche Integrale. By K. Hensel and G. Landsberg. Pp. xvi, 708. (Leipzig: Teubner.)

Cited by 2 publications

References 0 publications

Discretization and superintegrability all rolled into one

Discretization and superintegrability all rolled into one

Elliptic curve arithmetic and superintegrable systems

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