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Integrals developable about a singular point of a Hamiltonian system of differential equations. Part II

Published online by Cambridge University Press:  24 October 2008

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

This paper completes an investigation, of which the first part has already been published, into the integrals of a Hamiltonian system which are formally developable about a singular point of the system. Let

be a system of differential equations of which the origin is a singular point of the first type, i.e. a point at which H is developable in a convergent Taylor series, but at which its first derivatives all vanish. We suppose that H does not involve t, and we consider only integrals not involving t. Let the exponents of this singular point be ± λ1, ± λ2,…±λn. In Part I, I considered the case in which the constants λ1,…λn are connected by no relation of commensurability, i.e. a relation of the form

where A1…An are integers (positive, negative or zero) not all zero, and showed that the equations (1) possess n, and only n, integrals not involving t which are formally developable as power series in the xk, yk. In this paper I consider the case in which λ1 … λn are connected by one or more relations of commensur-ability. Suppose that there are p, and only p, such relations linearly independent (p > 0): it will be shown that the equations (1) possess (n − p) independent integrals not involving t, formally developable about the origin and independent of H.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 22 , Issue 4 , March 1925 , pp. 510 - 533
Copyright
Copyright © Cambridge Philosophical Society 1925

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References

* Proc. Comb. Phil. Soc. 22, 325 (1924)CrossRefGoogle Scholar: “On integrals developable about a singular point of a Hamiltonian system of differential equations.” This paper will be referred to as Part I.

† So called to distinguish it from a point (singular point of the second type) at which H is not developable.

* In fact, from (12), no term α1a1β1b 1… αnanβnb n for which a 2 = b 2 = 0 is of a type which can occur in [β2, α 3], for this Lagrange bracket has α2 as a factor.

* It must be remembered that these equations do not give the dependence of αk, βk upon t, the equations which do this being since the αk, βk are constants of integration.

* It arises out of their Hamiltonian form; for equations there is no integral developable about a singular point unless some restriction is placed upon the form of x 1, … xn.

* I.e. necessary when λ1, … λn are connected by relations of commensurability; when there are no such relations it is unnecessary, as shown in Part I, § 3.

* Whittaker, (Proc. Boy. Soc. Edin. 37, 95)Google Scholar uses this principle in a problem very similar to the present one, but does not give a general proof that by its application all the critical terms can be annulled.

* From an integral we may derive by combination with H the integral Z – cH 2, which is of the same form as Z with c = 0.

* Since the product χα χβ is critical we have Dαβ = 0, i.e. Dα + Dβ = 0.

* Compare § 2 (ii) above.

* The argument which follows is clearly of a general type.

* Since α2β1 = α1β1.α2β2/α1β2, the fifth series is a function of the first, second and fourth.