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

1924 ◽  
Vol 22 (3) ◽  
pp. 325-349 ◽  
Author(s):  
T. M. Cherry
Keyword(s):  
Singular Point ◽  
Hamiltonian System ◽  
Characteristic Function ◽  
Differential Equations ◽  
Convergent Series ◽  
Hamiltonian Form ◽  
System Of Differential Equations ◽  
Image Position

Letbe 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

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

1925 ◽  
Vol 22 (4) ◽  
pp. 510-533 ◽  
Author(s):  
T. M. Cherry
Keyword(s):  
Singular Point ◽  
Hamiltonian System ◽  
Power Series ◽  
Differential Equations ◽  
Taylor Series ◽  
System Of Differential Equations ◽  
Linearly Independent ◽  
Image Position

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. Letbe 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 formwhere 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.

Integrals of Systems of Ordinary Differential Equations

1924 ◽  
Vol 22 (3) ◽  
pp. 273-281 ◽  
Author(s):  
T. M. Cherry
Keyword(s):  
Singular Point ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Analytic Functions ◽  
Convergent Series ◽  
System Of Differential Equations ◽  
Image Position

Letbe a system of differential equations in which X1, … Xn are analytic functions independent of t, expansible in convergent series of powers of x1, … xn. Suppose further that not all the functions X1, … Xn vanish when x1 = … = xn = 0. It will be shown in §§ 1–3 that these equations possess (n — 1) independent integrals independent of t, expansible in convergent series of powers of x1, … xn; i.e. functionssuch thatidentically. In § 4 it is shown how the series for ϕ1,…ϕn−1 may be most directly constructed, and in § 5 is briefly considered the corresponding problem when the origin is a singular point of the equations, i.e. when the expansions of X1,…Xn all begin with terms of the first or higher degrees.

Some General Instability Criteria for Ordinary Differential Equations

1965 ◽  
Vol 8 (4) ◽  
pp. 453-457
Author(s):  
T. A. Burton
Keyword(s):  
Singular Point ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
System Of Differential Equations ◽  
Instability Criteria ◽  
Image Position

We consider a system of differential equations1where 0 = (o, o) is an isolated singular point. Thus, there exists B > o such that S(0, B) contains only one singular point. Here, S(0, B) denotes a sphere centered at 0 with radius B. We shall denote the boundary of S(0, B) by ∂ S(0, B).

scholarly journals Stability on the basis of Orthogonal Trajectories

1965 ◽  
Vol 8 (5) ◽  
pp. 647-658
Author(s):  
T. A. Burton
Keyword(s):  
Singular Point ◽  
Differential Equations ◽  
Second Order ◽  
System Of Differential Equations ◽  
Partial Derivatives ◽  
Connected Set ◽  
Image Position ◽  
Simply Connected

We consider a system of differential equations of second order given by1(' = d/dt) where P and Q have continuous first partial derivatives with respect to x and y in some open and simply connected set R containing O = (0, 0) which we assume to be the only singular point in R. In fact, let R be the whole plane; for if not then the following discussion can be modified.

XXVI.—The Universal Integral Invariants of Hamiltonian Systems and Application to the Theory of Canonical Transformations

1948 ◽  
Vol 62 (3) ◽  
pp. 237-246 ◽  
Author(s):  
Hwa-Chung Lee
Keyword(s):  
Hamiltonian System ◽  
Differential Equations ◽  
Hamiltonian Systems ◽  
Hamiltonian Function ◽  
Integral Invariant ◽  
Canonical Transformations ◽  
System Of Differential Equations ◽  
Integral Invariants ◽  
Image Position ◽  
Universal Integral

I. Introduction.—Consider a Hamiltonian system of differential equationswhere H is a function of the 2n variables qi and pi involving in general also the time t. For each given Hamiltonian function H the system (1.1) possesses infinitely many absolute and relative integral invariants of every order r = 1,…, 2n, which can all be written out when (1.1) is integrated. Our interest now is not in these integral invariants, which are possessed by one Hamiltonian system, but in those which are possessed by all Hamiltonian systems. Such an integral invariant, which is independent of the Hamiltonian H, is said to be universal.

scholarly journals Solutions of first level of meromorphic differential equations

1982 ◽  
Vol 25 (2) ◽  
pp. 183-207 ◽  
Author(s):  
W. Balser
Keyword(s):  
Differential Equation ◽  
Power Series ◽  
Differential Equations ◽  
Formal Power Series ◽  
Convergent Series ◽  
Constant Matrix ◽  
Image Position ◽  
Formal Power ◽  
Meromorphic Differential Equation ◽  
Meromorphic Differential

Let a meromorphic differential equationbe given, where r is an integer, and the series converges for |z| sufficiently large. Then it is well known that (0.1) is formally satisfied by an expressionwhere F( z) is a formal power series in z–1 times an integer power of z, and F( z) has an inverse of the same kind, L is a constant matrix, andis a diagonal matrix of polynomials qj( z) in a root of z, 1≦ j≦ n. If, for example, all the polynomials in Q( z) are equal, then F( z) can be seen to be a convergent series (see Section 1), whereas if not, then generally the coefficients in F( z) grow so rapidly that F( z) diverges for every (finite) z.

Bounds for the characteristic exponents of linear systems

1965 ◽  
Vol 61 (4) ◽  
pp. 889-894 ◽  
Author(s):  
R. A. Smith
Keyword(s):  
Differential Equations ◽  
Linear Systems ◽  
Characteristic Exponent ◽  
Zero Solution ◽  
System Of Differential Equations ◽  
Characteristic Exponents ◽  
Integrable Functions ◽  
Image Position ◽  
Complex Valued

For an n-vector x = (xi) and n × n matrix A = (aij) with complex elements, let |x|2 = Σi|xi|2,|A|2 = ΣiΣj|aij|2. Also, M(A), m(A) denoteℜλ1,ℜλn, respectively, where λ1,…,λA are the eigenvalues of A arranged so that ℜλ1 ≥ … ≥ ℜλn. Throughout this paper A(t) denotes a matrix whose elements aij(t) are complex valued Lebesgue integrable functions of t in (0, T) for all T > 0. Then M(A(t)), m(A(t)) are also Lebesgue integrable in (0, T) for all T > 0. The characteristic exponent μ of a non-zero solution x(t) of the n × n system of differential equationscan be defined, following Perron ((12)), aswhere ℒ denotes lim sup as t → + ∞. When |A(t)| is bounded in (0,∞), μ is finite; in other cases it could be ± ∞.

scholarly journals V. On periodic solutions of Hamiltonian systems differential equations

1928 ◽  
Vol 227 (647-658) ◽  
pp. 137-221 ◽  
Keyword(s):  
Hamiltonian System ◽  
Differential Equations ◽  
Ab Initio ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Celestial Mechanics ◽  
Divergent Series ◽  
Hamiltonian Form ◽  
Systems Of Differential Equations ◽  
Independent Variable

In the following pages it is proposed to develop ab initio a theory of periodic solutions of Hamiltonian systems of differential equations. Such solutions are of theoretical importance for the following reason: that whereas the attempt to obtain, for a real Hamiltonian system, solutions valid for all real values of the independent variable leads in general to divergent series, for certain solutions which are formally periodic the series can be proved convergent. In the words of Poincare, “ce qui nous rend ces solutions périodiques si précieuses, c’est qu’elles sont, pour ainsi dire, la seule breche paroù nous puissions essayer de pénétrer dans une place jusqu'ici reputee inabordable. The existing theory of periodic solutions of differential equations was developed by Poincare mainly with reference to the equations of Celestial Mechanics. With a suitable choice of co-ordinates these are of the Hamiltonian form.

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