Pages: 144 - 161
In  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...
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,...
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...
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...
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...
Pages: 230 - 252
We characterize and completely describe some types of separable potentials in twdimensional spaces, S2
, 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...
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....
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....
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...
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...