Publication: Symmetries of differential equations. IV
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1983
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American Institute of Physics
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Abstract
By an application of the geometrical techniques of Lie, Cohen, and Dickson it is shown that a system of differential equations of the form [x^(r_i)]_i = F_i(where r_i > 1 for every i = 1 , ... ,n) cannot admit an infinite number of pointlike symmetry vectors. When r_i = r for every i = 1, ... ,n, upper bounds have been computed for the maximum number of independent symmetry vectors that these systems can possess: The upper bounds are given by 2n_ 2 + nr + 2 (when r> 2), and by 2n_2 + 4n + 2 (when r = 2). The group of symmetries of ͞x^r = ͞0 (r> 1) has also been computed, and the result obtained shows that when n > 1 and r> 2 the number of independent symmetries of these equations does not attain the upper bound 2n _2 + nr + 2, which is a common bound for all systems of differential equations of the form ͞x^r = F[t, ͞x, ... , ͞x^(r - 1 )] when r> 2. On the other hand, when r = 2 the first upper bound obtained has been reduced to the value n^2 + 4n + 3; this number is equal to the number of independent symmetry vectors of the system ͞x= ͞0, and is also a common bound for all systems of the form ͞x = ͞F (t ,͞x, ‾̇x).
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©1983 American Institute of Physics.
It is a pleasure to express our gratitude to Dr. C. Ruiz and Dr. M. Amores for useful discussions with them and for providing some bibliography. It is also a pleasure to acknowledge the constant encouragement given by M. C. Hidalgo-Brinquis.
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1. F. G. Gascón, 1. Math. Phys. 18, 1763 (1977); 21, 2046 (1980); Hadronic J. 3,1457 (1980). F. G. Gascón and A. G. López, Lett. Nuovo Cimento 32, 353 (1981). For a global formulation of certain relations arising between symmetries and first integrals see the following papers: F. G. Gascón, Lett. Nuovo Cimento 20,54 (1977); F. G. Gascón, F. Moreno, and E. Rodríguez, Lett. Nuovo Cimento 21,595 (1978); F. G. Gascón, Lett. Nuovo Cimento 26,385 (1979); F. G. Gascón and E. Rodriguez, Lett. Nuovo Cimento 27,363 (1980); Hadronic 1.3, 1059 (1980); F. A. Gascón, Phys. Lett. A 76, 205 (1980); Lett. Nuovo Cimento 29, 73 (1980) and the references therein. See also the following papers: F. G. Gascón, F. Moreno, and E. Rodríguez, Hadronic 1. 3, 1491 (1980); F. G. Gascón and E. Rodríguez, Phys. Lett. A 80,133 (1980); F. G. Gascón, Phys. Lett. A 87, 385 (1982); Lett. Nuovo Cimento 33,97 (1982); Lett. Nuovo Cimento 34, 35 (1982). See also the paper "The inverse problem concerning the Poincare symmetry for second order differential equations" by F. G. Gascón and A. G. López (Hadronic I., accepted for publication).
2. S. Lie and G. Scheffers, Vorlesungen uber Continuierliche Groppen (Chelsea, Bronx, NY, 1971). A. Cohen, An Introduction to the Lie Theory of One-Parameter Groups (Heath, Boston, 1911). L. E. Dickson, Ann. Math. 25,287 (1924).
3. W. Fleming, Functions of Several Variables (Springer-Yerlag, New York, 1977), p. 148.
4. See the Addendum, following the Appendix.
5. I. E. Campbell, Introductory Treatise on Lie's Theory of Finite Continuous Transformation Groups (Chelsea, Bronx, NY, 1966), pp. 23-27.
6. See the first two papers quoted in Ref. 1. 7. See Cohen, Ref. 2, p. 211.
8. K. Gruenberger and A. Weis, Linear Geometry, University Series Undergraduate Mathematics (Yan Nostrand, Princeton, NI, 1967).
9. V. Arnold: Equations Differentielles Ordinaires (Ed. MIR, Moscow, 1974), p. 222.
10. See the first four papers quoted in Ref. 1
11. I. G. Petrovskii, Partial Differential Equations (Iliffe, London, 1967), p. 19.
12. S. Sternberg, Lectures on Differential Geometry (Prentice-Hall, Englewood Cliffs, NI, 1964). S. Helgason, Differential Geometry and Symmetric Spaces (Academic, New York, 1962). M. Spivak, Calculus on Manifolds (Benjamin, New York, 1965).
13. G. Chilov, Analyse Mathematique (Fonctionsd 'une Variable, Tome 1 ) (Ed. MIR, Moscow, 1974), p. 290.