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On Integrals developable about a Singular Point of a Hamiltonian System of Differential Equations

Published online by Cambridge University Press:  24 October 2008

T. M. Cherry
T. M. Cherry
Affiliation:
Trinity College
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Extract

Let

be a system of differential equations of Hamiltonian form, the characteristic function H being independent of t and expansible in a convergent series of powers of x1, … xn, y1, … yn in which the terms of lowest degree are

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 22 , Issue 3 , 20 September 1924 , pp. 325 - 349
Copyright
Copyright © Cambridge Philosophical Society 1924

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References

* The coefficients in this transformation are not necessarily real.

† If the origin had been an ordinary point of the equations there would have been 2n − 1 integrals developable about it. See p. 273, above, “Integrals of Systems of Ordinary Differential Equations.”

* ”On the Adelphic Integral of the Differential Equations of Dynamics” (Proc. Roy. Soc. Edin., November, 1916).Google Scholar

* Whittaker, , Analytical Dynamics (2nd ed.), § 146.Google Scholar

* Here, and in (c) below, we consider only the general case; this is sufficient for illustrative purposes, since we are only verifying a result which we know from § 3, that the coefficients of all critical terms vanish.

† It is supposed that the terms ø1, ø2, ø3, ø4 are all different and that none of D 1, D 2, D 12, D 13, … vanishes.

* This determinant has the value and is non-zero because and, from (59),

† Whittaker, , Analytical Dynamics (2nd ed.), § 130.Google Scholar

* If n = 2 and the ratio λ1: λ2 is not real the modulus of the expression (63) has a definite non-zero lower bound for all possible integral choices of A 1, A 2 except A 1 = A 2=0; if n=2 and λ1: λ2 is real, or if n > 2, the lower bound is zero and the statement in the text is valid.

* I.e. products of powers of x 1y 1, x 2y 2, … x ny n.

† Thus

but though the two sserie on the right are divergent for we cannot conclude the same range of divergence for the series on the left, which is in fact convergent for | x | <1.

* Arising in the problem of the motion of a rigid body about a fixed point under no forces; q is the constant in an angular momentum integral by means of which the number of degrees of freedom has been reduced from 3 to 2.

† Since H enters the equations only through its derivatives, the presence of the constant term q 2/2A is immaterial.