Lagrange-Schwarzian derivative and symplectic Sturm theory. (English) Zbl 0780.34004

Ovsienko, Valentin

Ann. Fac. Sci. Toulouse, VI. Sér., Math. 2, No. 1, 73-96 (1993).

The scalar Schwarzian derivative \(S\{f\}=f'''/f'-(3/2)(f''/f')^ 2\) of \(C^ 3\)-function \(f\), \(f'\neq 0\) is very important for example in the theory of linear differential equations [see e.g., F. Neuman, Global properties of linear ordinary differential equations, Kluwer Academic Publishers (1991)]. The main property of this derivative is its projective invariance, that is, \(S\{f\}=S\{g\}\) if and only if \(g=(af+b)/(cf+d)\), where \(a\), \(b\), \(c\), \(d\) are constants. This interesting paper is connected with a recent author’s paper [Mosc. Univ. Mech. Bull. 44, No. 6, 8-13 (1989); translation from Vestn. Mosk. Univ., Ser. I 1989, No. 6, 42-45 (1989; Zbl 0707.70005)] and gives more details and discusses relations of multidimensional Schwarzian derivative with loop groups. The author presents two constructions in the linear symplectic space which can be considered as analogues of the classical cross-ratio and Schwarzian derivative. The symplectic cross-ratio is the unique invariant of four Lagrangian subspaces in the linear symplectic space. The (multidimensional) Lagrange-Schwarzian derivative is defined by \(\text{LS} \{{\mathcal F}\}= (F'{}^{-1})^{1\over 2} (F'''-(3/2) F'' F'{}^{-1} F'')((F'{}^{-1})^*)^{1\over 2}\) for any 1-parameter smooth family of symmetric matrices \(F(t)\) such that \(F'(t)\) is positive definite for every \(t\). The derivative recovers the system of linear Newton equations \(y''=a(t)y\), \(y\in\mathbb{R}^ n\), \(A^*=A\) from the evolution of a Lagrangian subspace in a linear symplectic space. It is also proved a theorem which gives a nonoscillation condition for the system of Newton equations.

Reviewer: S.Staněk (Olomouc)

MSC:

34A30	Linear ordinary differential equations and systems
37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
30C10	Polynomials and rational functions of one complex variable
34A26	Geometric methods in ordinary differential equations

Keywords:

multidimensional Lagrange-Schwarzian derivative; linear differential equations; loop groups; linear symplectic space; Newton equations; nonoscillation condition

Citations:

Zbl 0707.70005

References:

[1]	Arnold, V.I.) . - Ordinary differential equations, MIT Press (1973). · Zbl 0296.34001
[2]	Arnold, V.I.) .- Sturm theorems and symplectic geometry, Function Anal. and its Appl.19, n° 1 (1985), pp. 1-10. · Zbl 0606.58017
[3]	Arnold, V.I.) . - On a characteristic class intervening in quantization conditions, Funct. Anal. and its Appl.1, n° 1 (1967), pp. 1-14. · Zbl 0175.20303
[4]	Bott, R.) . - On the iterations of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math.9, n° 2 (1956), pp. 171-206. · Zbl 0074.17202
[5]	Brown, K.S.) . - Buildings, Springer-Verlag (1988).
[6]	Carne, K.) . - The Schwarzian derivative for conformal maps, J. Reine Angew Math.408 (1990), pp. 10-33. · Zbl 0705.30010
[7]	Coppel, J.) . - Disconjugacy, 220 (1970). · Zbl 0224.34003
[8]	Flanders, H.) .- The Schwarzian as a curvature, J. Diff. Geometry4, n° 4, pp. 515-519. · Zbl 0232.53005
[9]	Fuchs, D.B.) . - Cohomologies of infinite-dimensional Lie algebras, Consultants Bureau, New York (1986). · Zbl 0667.17005
[10]	Kirillov, A.A.) . - Infinite-dimensional Lie groups: their orbits, invariants and representations. Geometry of moments, 970 (1982), pp. 101-123. · Zbl 0498.22017
[11]	Klein, F.) . - Vorlesungen uber das ikosaeder und die auflosing der gleichungen vom funfen grade, Leipzig (1884). · JFM 16.0061.01
[12]	Lagrange, J.-L.) . - Sur la construction des cartes géographiques, Nouveaux Mémoires de l’Académie de Berlin (1779).
[13]	Morse, M.) . - A generalization of the Sturm theorems in n-space, Math. Ann.103 (1930), pp. 52-69. · JFM 56.1078.03
[14]	Nehari, Z.) . - The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc.55, n° 6 (1949). · Zbl 0035.05104
[15]	Ovsienko, V. Yu.) .- Hook Law and Denogardus Great Number, Kvant8 (1989) pp. 8-16 (Russian).
[16]	Ovsienko, V. Yu.). - Lagrange Schwarzian derivative, Vestnik Moscow State University6 (1889), pp. 42-45 (Russian). · Zbl 0707.70005
[17]	Royan, J.) . - Generalised Schwarzian derivatives for generalised fractional linear transformations, Annales Polinici Math. (to appear). · Zbl 0762.15013
[18]	Retakh, V.) and Shander, V.) .- Noncommutative analogues of the Schwarzian derivative, Preprint. · Zbl 0841.17003
[19]	Tabachnikov, S.) .- Projective structures and group Vey cocycle, Preprint E.N.S. de Lyons (1992).