...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 (1,730 words) - 08:25, 14 January 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