...is a special case of the $ n $-body problem, which may be described by a system of ordinary differential equations of order $ 6n $,
...rals (one of which is independent of the preceding ones) and is completely integrable [[#References|[2]]].
4 KB (624 words) - 16:44, 1 November 2023
...they generate (known as higher Korteweg–de Vries equations) are completely integrable.
...ry solutions of the higher KdV-equations; the latter constitute completely-integrable finite-dimensional Hamiltonian systems. Any periodic potential can be appro
14 KB (2,037 words) - 20:24, 16 January 2024
...quation (in case (1), for example) is the [[Hamiltonian system|Hamiltonian system]] with Hamiltonian
This system is completely integrable, and replacing the variables $ u $
11 KB (1,465 words) - 20:18, 16 January 2024
and form a completely-integrable subsystem of forms in the system $ ( \theta ^ {k} , \theta ^ \alpha ) $.
...ffian equation]]; [[Completely-integrable differential equation|Completely-integrable differential equation]]).
15 KB (2,363 words) - 08:15, 18 August 2022
...s on the line [[#References|[a1]]]. A convenient container is the $2$-Toda system, first introduced and studied comprehensively in [[#References|[a2]]]; see
The $1$-Toda system (which can always be imbedded in the $2$-Toda system) is just the $x$-flow for $L_1$, i.e. it just involves ignoring $L_{2}$ and
8 KB (1,147 words) - 17:45, 1 July 2020
...etely-integrable Hamiltonian systems (cf. [[Hamiltonian system|Hamiltonian system]]; [[Soliton|Soliton]]), cf. [[#References|[a12]]], and the results of [[#R
...n to the Yang–Baxter equation" M. Jimbo (ed.) , ''Yang–Baxter equation in integrable systems'' , World Sci. (1990) pp. 111–134</TD></TR><TR><TD valign="top"
10 KB (1,364 words) - 08:29, 6 June 2020
...otion are integrable in the sense of Kowalewski. The latter means that the system admits solutions, expressible as [[Laurent series|Laurent series]] in time
...eger spectrum. This strong condition leads to the following three cases of integrable rotating bodies: i) the Euler top, for which the fixed point and the centre
13 KB (1,808 words) - 22:15, 5 June 2020
==Fuchsian singularity of a system==
A singular point $t=t_*$ of a system linear of first order ordinary differential equations with meromorphic coef
7 KB (1,237 words) - 11:48, 23 November 2023
A class of dynamical systems (cf. [[Dynamical system|Dynamical system]]). An example is the flow generated by all translations of a torus (consid
...e also frequently encountered in Hamiltonian systems sufficiently close to integrable ones (this problem is closely connected with [[Small denominators|small den
12 KB (1,841 words) - 18:33, 5 June 2020
...es/data/L059/L.0509270 Linear hyperbolic partial differential equation and system
A partial differential equation (or system) of the form
20 KB (2,792 words) - 08:50, 7 January 2022
[[Absolutely integrable function]] |
[[Borel system of sets]] |
11 KB (1,076 words) - 07:40, 12 July 2014
$#C+1 = 128 : ~/encyclopedia/old_files/data/A014/A.0104190 Autonomous system
A system of ordinary differential equations which does not explicitly contain the in
13 KB (1,960 words) - 07:35, 26 March 2023
...e integral manifolds of maximal dimension for a [[Pfaffian system|Pfaffian system]] of Pfaffian equations
is called an integral manifold of the system (*) if the restrictions of the forms $ \theta ^ \alpha $
17 KB (2,624 words) - 19:27, 9 January 2024
is almost-everywhere finite, measurable, integrable, etc., if the function $ f ( t ( s)) $
is integrable on $ \Gamma $.
27 KB (3,955 words) - 10:05, 8 May 2022
...the study of unitary representations. Thus, any unitary representation is completely reducible; for a unitary representation the conditions of complete irreduci
of groups containing a fundamental system of neighbourhoods of the unit element that are invariant under inner automo
24 KB (3,516 words) - 08:27, 6 June 2020
...K$ are matrices and $f$, $\phi$ are vector functions, then (1) is called a system of linear integral equations. If $f=0$, then the integral equation is said
...dholm kernel]], that is, if the integral operator in equations (2), (3) is completely continuous (also called compact, see
14 KB (2,157 words) - 17:38, 3 September 2013
of square-integrable functions on $ D $
...oblem (1) is then treated as a problem of finding the eigen values of some completely-continuous operator $ A $
11 KB (1,669 words) - 19:37, 5 June 2020
...onstruction can be considered as a quantization operation of a Hamiltonian system for which $ \Omega $
...ic hypothesis can therefore be reformulated thus: Every elementary quantum system with time (or group of symmetries) $ G $
12 KB (1,682 words) - 08:04, 6 June 2020
Consider a system of $n$ equations in matrix notation:
...sumed that $A(t)$ is meromorphic in $G$, and one considers the homogeneous system
16 KB (2,410 words) - 11:15, 28 January 2020
...ominus \theta H ^ { 2 }$, each of which leads to a model for an arbitrary, completely non-unitary [[Contraction operator|contraction operator]] on a Hilbert spac
...fer function $\theta ( z ) = d + c z ( I - z A ) ^ { - 1 } b$ of a unitary system
12 KB (1,802 words) - 17:01, 1 July 2020