Canonical and Hamiltonian formalism applied to the Sturm-Liouville equation
Authors:
M. A. Biot and I. Tolstoy
Journal:
Quart. Appl. Math. 18 (1960), 163-172
MSC:
Primary 34.00
DOI:
https://doi.org/10.1090/qam/111889
MathSciNet review:
111889
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Abstract: The Sturm-Liouville equation is expressed in Hamiltonian form. A simple generating function is derived which defines a large class of canonical transformations and reduces the Sturm-Liouville equation to the solution of a first order equation with a single unknown. The method is developed with particular reference to the wave equation. The procedure unifies many apparently diverse treatments and leads to new insights and procedures. Some new transformations are obtained, useful in the turning point region and for the improvement of accuracy in the region of validity of W.K.B. solutions. In addition a new power series expansion near the turning point is obtained.
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S. C. Miller and R. H. Good, Phys. Rev. 91, 174 (1953)
H. Bremmer, Communs. Pure and Appl. Math. IV, 1, 103 (1951)
R. E. Langer, Phys. Rev. 51, 669 (1937)
T. L. Eckersley, Proc. Roy. Soc. A132, 83 (1931)
D. R. Hartee, Proc. Roy. Soc. A131, 428 (1931)
H. Bremmer, Terrestrial radio waves, Elsevier, N. Y., 1949
L. M. Brekhovskih, Zhur. Tekh. Fiz. XVIII 4, 455 (1948); Izvest. Akad. Nauk., Ser. Fiz XIII 5, 505 (1949)
L. R. Walker and N. Wax, J. Appl. Phys. 17, 1043 (1946)
I. Tolstoy, J. Acoust. Soc. Am. 27, 274 (1955); 27, 897 (1955)
M. A. Biot, J. Appl. Phys. 11, 530 (1940)
H. Prüfer, Math. Ann. 94, 498 (1928)
J. H. Barrett, Proc. Am. Math. Soc. 6, 247 (1955)
F. V. Atkinson, Univ. Nac. Tucumah, Revista A, 8, 71 (1951)
S. A. Schelkunoff, Communs. Pure and Appl. Math. 4, 1, 117 (1951)
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© Copyright 1960
American Mathematical Society
