...lled totally integrable if for each point $a\in M^n$ there is a coordinate system $(U,\phi)$, $x\in U$, $\phi(x)=\phi(x_1,\dots,x_n)$, such that for any cons
...involution|integrals in involution]], cf. [[Hamiltonian system|Hamiltonian system]].
2 KB (254 words) - 23:02, 22 December 2018
$#C+1 = 50 : ~/encyclopedia/old_files/data/C023/C.0203970 Completely\AAhintegrable differential equation
Instead of equation (*) the following system of equations is sometimes considered [[#References|[2]]]:
5 KB (702 words) - 17:45, 4 June 2020
The geometric interpretation of a completely-integrable differential system on an $ n $-dimensional differentiable manifold $ M ^ {n} $
A $ p $-dimensional distribution (or a differential system of dimension $ p $)
3 KB (378 words) - 10:11, 21 March 2022
Usually a Pfaffian structure is given by a system of Pfaffian equations (cf. [[Pfaffian equation|Pfaffian equation]]) $ \th
A Pfaffian structure is said to be completely integrable if through each point $ x \in M $
5 KB (750 words) - 08:06, 6 June 2020
...the spectral curve is an $n$-fold covering of the parameter space and the system lives on a co-adjoint orbit in a loop algebra, by the Adler–Kostant–Sym
...valign="top">[a9]</TD> <TD valign="top"> N.J. Hitchin, "Stable bundles and integrable systems" ''Duke Math. J.'' , '''54''' (1987) pp. 91–114 {{MR|0885778}} {{
5 KB (752 words) - 15:33, 4 October 2014
...occur explicitly in the expression of the characteristic function of this system. When one uses the corresponding equations of motion, one may obtain at onc
...ates in the theory of completely-integrable Hamiltonian systems. Each such system (with finite degrees of freedom) can be transformed into one with coordinat
1 KB (239 words) - 14:39, 28 August 2014
...on the conditions for a system of [[Pfaffian equation]]s to be completely integrable, or (in geometrical terms) on conditions under which a given field of $n$-d
1 KB (157 words) - 19:10, 24 September 2017
Consider a system of ordinary differential equations of first order in the unknowns $x: \math
A first integral of the system is a (non-constant) continuously-differentiable function $\Psi: \mathbb R \
3 KB (563 words) - 18:44, 15 January 2015
The Pfaffian equation is said to be completely integrable if there is one and only one integral manifold of maximum possible dimensio
...ry and sufficient condition for the Pfaffian equation (1) to be completely integrable is
9 KB (1,342 words) - 19:08, 9 January 2024
...equations defining the geometric object $\Phi$ is called the stationarity system of equations of $F$.
...where $N_i$ $(i = 1, 2)$ is the rank of $F_i$, and $N$ is the rank of the system of forms $\Omega^{J_1}, \Omega^{J_2}$, $J_i = 1, \dots, N_i$, that are the
3 KB (538 words) - 00:43, 27 November 2013
.../encyclopedia/old_files/data/C110/C.1100030 Calogero\ANDMoser\ANDKrichever system,
''Calogero–Moser–Sutherland–Krichever system''
13 KB (1,901 words) - 09:09, 26 March 2023
...rable system (cf. [[Completely-integrable differential equation|Completely-integrable differential equation]]). Then there are new symplectic coordinates, called
Now consider a small perturbation of an integrable system satisfying (a2):
8 KB (1,212 words) - 17:33, 7 June 2020
...adiabatic invariant — undergo significant changes.) Thus, for the simplest system
the system (*) describes an ordinary harmonic oscillator with frequency $ \omega $.
7 KB (1,019 words) - 16:09, 1 April 2020
in addition, the usual laws for a hypercomplex system (i.e. an associative algebra) are required to hold. The algebra of dimensio
is called the associated system of linear forms of $ \Omega _ {p} $.
21 KB (3,193 words) - 11:01, 4 June 2020
...sformation effected by the product of these elements, and let a coordinate system be suitably introduced into the space. One then says that $ G $
An intransitivity system is called a space of geometric objects in the proper sense. A representatio
21 KB (3,190 words) - 19:41, 5 June 2020
...x_1,\dots,x_n)$ are functionally independent first integrals in $G$ of the system of ordinary differential equations in symmetric form
which is completely integrable in some domain $G$ of three-dimensional space and does not have any singula
2 KB (314 words) - 17:39, 14 February 2020
$#C+1 = 55 : ~/encyclopedia/old_files/data/H046/H.0406270 Hamiltonian system
A system of ordinary differential equations in $ 2n $
13 KB (1,837 words) - 07:56, 21 January 2024
A system $ w ^ \prime = A ( z) w $
are singular for the equation (1) and the system (2). Fuchs' identity holds for (1):
12 KB (1,732 words) - 17:50, 5 May 2024
If the linear system $ | M | $
and the linear system $ | L | $
18 KB (2,511 words) - 06:25, 26 March 2023
...es equation|Korteweg–de Vries equation]]; [[Hamiltonian system|Hamiltonian system]]; [[Painlevé-type equations|Painlevé-type equations]]). Non-linear parti
...is widely regarded as the prototypical example of a completely-integrable system.
18 KB (2,677 words) - 19:56, 4 February 2024
...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
$#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
...heory of completely-integrable systems. Indeed, consider an overdetermined system of linear partial differential equations (cf. [[#References|[a6]]] for more
a so-called Zakharov–Shabat system. Many integrable systems can be put in this form. Now let $ \phi $
17 KB (2,509 words) - 03:12, 13 January 2022
...olutions with an admissible infinity at the end of order less than one (an integrable infinity) is equal to one plus the index in the class of solutions bounded
considered above. The investigation is carried out by reduction to a system of integral equations. The number of linearly independent solutions or solv
10 KB (1,509 words) - 08:11, 6 June 2020
...sidered as stationary stochastic processes if the chain is in a stationary system; the pulsations of velocity or pressure at a point of a turbulent flow are
...nto the theory of stationary stochastic processes, the properties that are completely defined by the characteristics $ m $
24 KB (3,614 words) - 08:52, 21 January 2024
...6/h046420/h0464203.png" /> be its Fourier coefficients with respect to the
system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
...yclopediaofmath.org/legacyimages/h/h046/h046420/h046420168.png" /> will be
integrable with respect to the measure <img align="absmiddle" border="0" src="https://
66 KB (9,085 words) - 17:28, 31 March 2020
...f the variables") transforms a local dynamical system to an ''equivalent'' system. The local classification problem is to describe the equivalence classes of
The advantage of considering the local system rests in the hope that the classification will be determined by (semi)algeb
37 KB (5,881 words) - 19:10, 24 November 2023
...tended in both directions up to the boundary of any closed subregion lying completely in $ D $
...ivatives. A natural generalization of equation (2) is the normal form of a system of differential equations of order $ n $:
33 KB (4,933 words) - 01:50, 23 January 2022
...s of finite-dimensional spaces over an algebraically closed field has been
completely solved, and the similarity classes have been described in terms of the inva
...Compact operator|compact operator]], or a [[
Completely-continuous operator|
completely-continuous operator]], if it maps any bounded set in <img align="absmiddle"
67 KB (9,247 words) - 17:12, 29 October 2017
...hinsky equation is rigorously equivalent to a finite-dimensional dynamical system (for an overview of analytical results in an appropriate functional setting
...y dynamics is its dissipativity (cf. also [[Dissipative system|Dissipative system]]): solutions are attracted to an absorbing ball, with $L$-dependent radius
21 KB (3,050 words) - 17:43, 1 July 2020
...st natural normal form are of substantial importance in linear control and system theory [[#References|[a1]]], [[#References|[a2]]]. Here one studies systems
is called completely controllable if the rank of the block matrix
51 KB (7,267 words) - 07:39, 14 January 2024
in the commonly-used curvilinear coordinate system) has the meaning of the gravitational field of a spherically-symmetric poin
...diation have been verified indirectly (by the loss of energy of a two-body system revolving around a common centre of mass). Not a single fact contradicting
18 KB (2,719 words) - 19:42, 5 June 2020
...number of parameters, and to find then the actual space implicitly using a system of equations ( "state equations" ), or to parametrize the space locally ( "
are homeomorphisms in a system $ \Gamma $
30 KB (4,462 words) - 07:59, 6 June 2020