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SL(3,R) as the group of symmetry transformations for all one‐dimensional linear systems. III. Equivalent Lagrangian formalisms
1.It is well known indeed that the full point symmetry group of all such equations corresponds to This fact figures in the current literature, [cf. L. V. Ovsiannikov, Group Analysis of Differential Equations (Academic, New York, 1982)], and moreover, it was well known to Lie himself [S. Lie; Vorlesungen über Differential-Gleichungen Mit Bekannten Infinitesimal Transformationen (B. G. Teubner, Leipzig, 1891;
1.reprinted by Chelsea, New York, 1967);
1.however, it seems to be not so well known to most physicists. For instance, the rediscovery 14 years ago [i.e., C. E. Wulfman and B. G. Wyborne, J. Phys. A: Math. Gen. 9, 507 (1976)] that the Newtonian equation for the simple harmonic oscillator has point symmetry was a surprise to physicists. Later on it was found that is also the complete symmetry group of the one-dimensional time-dependent harmonic oscillator;
1.cf., P. G. L. Leach, J. Math. Phys. 21, 300 (1980). Furthermore, it has been shown that the full symmetry group of the time-dependent n-dimensional harmonic oscillator is
1.cf., G. E. Prince and C. J. Eliezer, J. Phys. A: Math. Gen. 13, 815 (1980).
1.Also see M. Aguirre and K. Krause, J. Phys. A: Math. Gen. 20, 3553 (1987),
1.concerning the finite point realizations of for the simple harmonic oscillator. By the way, the present paper belongs to a line of research that was initiated by us some few years ago; cf. M. Aguirre and J. Krause, J. Math. Phys. 25, 210 (1984);
1.and M. Aguirre and J. Krause, J. Math. Phys. 26, 593 (1985).
2.M. Aguirre and J. Krause, J. Math. Phys. 29, 9 (1988).
3.M. Aguirre and J. Krause, J. Math. Phys. 29, 1746 (1988).
4.A brief survey of recent work can be seen in P. Rudra, Pramana 23, 445 (1984).
5.A lucid discussion of this subject can be found in G. E. Prince, Bull. Austr. Math. Soc. 25, 309 (1982);
5.G. E. Prince, Bull. Austr. Math. Soc. 27, 53 (1983), , Bull. Aust. Math. Soc.
5.G. E. Prince, J. Phys. A: Math. Gen. 16, L105 (1983),
5.and G. E. Prince, Bull. Austr. Math. Soc. 32, 299 (1985).
6.Concerning quantum kinematics, see J. Krause, J. Phys. A: Math. Gen. 18, 1309 (1985);
6.J. Krause, J. Math. Phys. 27, 2922 (1986);
6.J. Krause, J. Math. Phys. 29, 393 (1988); , J. Math. Phys.
6.J. Krause, J. Math. Phys. 32, 348 (1991)., J. Math. Phys.
7.Also see M. Aguirre and J. Krause, Int. J. Theor. Phys. 30, 495, 1461 (1991), and references quoted therein.
8.E. L. Hill, Rev. Mod. Phys. 23, 253 (1951).
9.S. Hojman and H. Harleston, J. Math. Phys. 22, 1414 (1981).
10.D. G. Currie and E. J. Saletan, J. Math. Phys. 7, 967 (1966). This gives rise to the inverse problem of the calculus of variations, which consists in trying to find all Lagrangians that yield Euler-Lagrange equations that are equivalent to a given system of equations of motion. This problem was first solved for by Darboux. The case was treated much later by Douglas. A large amount of noteworthy work devoted to this fundamental problem has been published lately;
10.see, for instance, Sarlet. For a recent approach, see Cariñena and Martínez (1989) and references therein. See G. Darboux, Lecons Sur la Theorie Generale des Surfaces (Gauthier-Villars, Paris, 1891);
10.J. Douglas, Trans. Am. Math. Soc. 50, 71 (1941);
10.W. Sarlet, J. Phys. A: Math. Gen. 15, 1503 (1982);
10.and J. F. Cariñena, and E. Martínez, J. Phys. A: Math. Gen. 22, 2659 (1989).
11.M. Lutzky, Phys. Lett. A 72, 86 (1979);
11.M. Lutzky, Phys. Lett. A 75, 8 (1979); , Trans. Am. Math. Soc.
11.and M. Lutzky, J. Math. Phys. 22, 1626 (1981).
12.See M. Trümper, Ann. Phys. N.Y. 149, 203 (1983).
13.G. E. Prince and C. J. Eliezer, J. Phys. A: Math. Gen. 14, 587 (1981).
14.J. F. Cariñena and L. A. Ibort, J. Phys. A: Math. Gen. 16, 1 (1983).
15.V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations (Springer-Verlag, New York, 1988), p. 44.
16.See, i.e., M. Aguirre and J. Krause, in Ref. 1.
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