Journal of Nonlinear Mathematical Physics

Volume 12, Issue 2, May 2005
Research Article

2. A Lie Symmetry Connection between Jacobi's Modular Differential Equation and Schwarzian Differential Equation

L. Rosati, M.C. Nucci
Pages: 144 - 161
In [18] Jacobi introduced a third-order nonlinear ordinary differential equation which links two different moduli of an elliptic integral. In the present paper Lie group analysis is applied to that equation named Jacobi's modular differential equation. A six-dimensional Lie symmetry algebra is obtained...
Research Article

3. On Properties of Elliptic Jacobi Functions and Applications

A. Raouf Chouikha
Pages: 162 - 169
In this paper we are interested in developments of the elliptic functions of Jacobi. In particular a trigonometric expansion of the classical theta functions introduced by the author (Algebraic methods and q-special functions, C.R.M. Proceedings and Lectures Notes, A.M.S., vol 22, Providence, 1999, 53-57)...
Research Article

4. Separabilty in Hamilton­Jacobi Sense in Two Degrees of Freedom and the Appell Hypergeometric Functions

Vladimir Dragovi
Pages: 170 - 177
A huge family of separable potential perturbations of integrable billiard systems and the Jacobi problem for geodesics on an ellipsoid is given through the Appell hypegeometric functions F4 of two variables, leading to an interesting connection between two classical theories: separability in Hamilton­Jacobi...
Research Article

5. Subgroup Type Coordinates and the Separationof Variables in Hamilton-Jacobi and Schrödinger Equations

E.G. Kalnins, Z. Thomova, P. Winternitz
Pages: 178 - 208
Separable coordinate systems are introduced in complex and real four-dimensional flat spaces. We use maximal Abelian subgroups to generate coordinate systems with a maximal number of ignorable variables. The results are presented (also graphically) in terms of subgroup chains. Finally the explicit solutions...
Research Article

6. Jacobi, Ellipsoidal Coordinates and Superintegrable Systems

E.G. Kalnins, J.M. Kress, W. Miller
Pages: 209 - 229
We describe Jacobi's method for integrating the Hamilton-Jacobi equation and his discovery of elliptic coordinates, the generic separable coordinate systems for real and complex constant curvature spaces. This work was an essential precursor for the modern theory of second-order superintegrable systems...
Research Article

7. Separable Potentials and a Triality in Two-Dimensional Spaces of Constant Curvature

José F. Cariñena, Manuel F. Rañada, Mariano Santander
Pages: 230 - 252
We characterize and completely describe some types of separable potentials in twdimensional spaces, S2 [1]2 , of any (positive, zero or negative) constant curvature and either definite or indefinite signature type. The results are formulated in a way which applies at once for the two-dimensional sphere...
Research Article

8. Triangular Newton Equations with Maximal Number of Integrals of Motion

Fredrik Persson, Stefan Rauch-Wojciechowski
Pages: 253 - 267
We study two-dimensional triangular systems of Newton equations (acceleration = velocity-independent force) admitting three functionally independent quadratic intgrals of motion. The main idea is to exploit the fact that the first component M1(q1) of a triangular force depends on one variable only. By...
Research Article

9. Symmetry Reductions of a Hamilton-Jacobi-Bellman Equation Arising in Financial Mathematics

V. Naicker, K. Andriopoulos, P.G.L. Leach
Pages: 268 - 283
We determine the solutions of a nonlinear Hamilton-Jacobi-Bellman equation which arises in the modelling of mean-variance hedging subject to a terminal condition. Firstly we establish those forms of the equation which admit the maximal number of Lie point symmetries and then examine each in turn. We...
Research Article

10. Jacobi Last Multiplier and Lie Symmetries: A Novel Application of an Old Relationship

M.C. Nucci
Pages: 284 - 304
After giving a brief account of the Jacobi last multiplier for ordinary differential equtions and its known relationship with Lie symmetries, we present a novel application which exploits the Jacobi last multiplier to the purpose of finding Lie symmetries of first-order systems. Several illustrative...
Research Article

11. Jacobi's Last Multiplier and the Complete Symmetry Group of the Ermakov-Pinney Equation

M.C. Nucci, P.G.L. Leach
Pages: 305 - 320
The Ermakov-Pinney equation possesses three Lie point symmetries with the algebra sl(2, R). This algebra does not provide a representation of the complete symmetry group of the Ermakov-Pinney equation. We show how the representation of the group can be obtained with the use of the method described in...