Chetaev theorems

Chetaev's theorems on stability are general theorems about the instability of motion, established by N.G. Chetaev for the equations of a perturbed motion of the form

$$ \tag{1 } \frac{dx _ {s} }{dt} = X _ {s} ( t , x _ {1} \dots x _ {n} ) ,\ \ s = 1 \dots n , $$

in which the right-hand sides $ X _ {s} $ are holomorphic functions of the real variables $ x _ {s} $ with coefficients that are continuous bounded functions of a real variable — the time $ t $— defined on some domain

$$ \tag{2 } t \geq t _ {0} ,\ \ \sum _ { s= } 1 ^ { n } x _ {s} ^ {2} < A , $$

where $ X _ {s} ( t , 0 \dots 0 ) = 0 $.

Chetaev's theorem on instability of motion. The functions $ V $ which turn up in the theory below are supposed to be real-valued functions of the variables $ x _ {s} $ and $ t $, single valued and continuous in the domain

and with single-valued and continuous total time derivatives in the domain $ G $ occurring below for $ h $ small enough,

$$ \dot{V} = \ \frac{\partial V }{\partial t } + \sum _ {s = 1 } ^ { n } \frac{\partial V }{\partial x _ {s} } X _ {s} . $$

It is also supposed that $ V ( 0, t) = 0 $. Suppose that for the reduction of the system

to a domain $ Z = \{ t _ {0} \leq t < \infty, \| x \| \leq h < H \} $ there exists a function $ V ( t, x) $ such that its domain of positivity $ G = \{ {( t, x) \in Z } : {V ( t, x) > 0 } \} $ has a non-empty open section $ D _ {t} $ contiguous with the coordinate origin 0 for all $ t \in [ t _ {0} , \infty ] $, while on the part of the boundary of the domain $ G $ lying inside the cylinder $ Z $ and including the axis $ 0 t $ there holds the equality $ V ( t, x) = 0 $. Now if 1) the function $ V ( x, t) $ is bounded on the domain $ G $; 2) the total derivative $ \dot{V} ( t, x) $ is positive on this domain; and 3) for each subdomain $ \{ {( x, t) } : {V ( x, t) \geq \alpha > 0 } \} $ there holds an inequality $ \dot{V} ( t, x) \geq \beta > 0 $, where $ \beta = \beta ( \alpha ) $ is some positive number depending on $ \alpha $, then the trivial solution $ x = 0 $ of

is instable in the sense of Lyapunov for $ t \rightarrow \infty $.

Figure: c022080a

There are various other versions of this theorem. For example, a version involving two functions $ V, W $, cf. . Still another version can be found in , p. 103. These theorems imply the so-called first instability theorem of Lyapunov, which says that if there exists a function $ V ( x) $ with a negative-definite total derivative and with $ V ( x) $ itself either negative definite or indefinite, then the equilibrium is unstable.

Chetaev's theorem on perturbations of stable motions of Hamiltonian systems. This Chetaev theorem is a theorem on the properties of the Poincaré variational equations

$$ \tag{3 } \left . \begin{array}{c} \frac{d \xi _ {j} }{dt} = \sum _ { i= } 1 ^ { n } \left ( \frac{\partial ^ {2} H }{\partial p _ {j} \partial q _ {i} } \xi _ {i} + \frac{\partial ^ {2} H }{\partial p _ {j} \partial p _ {i} } \eta _ {i} \right ) , \\ \frac{d \eta _ {j} }{dt} = - \sum _ { i= } 1 ^ { n } \left ( \frac{\partial ^ {2} H }{\partial q _ {j} \partial q _ {i} } \xi _ {i} + \frac{\partial ^ {2} H }{\partial q _ {j} \partial p _ {i} } \eta _ {i} \right ) , \end{array} \right \} $$

$$ j = 1 \dots n , $$

for an unperturbed motion $ q _ {i} = q _ {i} ( t) $, $ p _ {i} = p _ {i} ( t) $, under the assumption that the coefficients of the equations (3) are all continuous bounded real functions of $ t $, $ H ( t , q _ {i} , p _ {i} ) $ is the Hamiltonian, and $ \xi _ {i} $ and $ \eta _ {i} $ are the deviations of the coordinates $ q _ {i} $ and the momenta $ p _ {i} $. The equations (3) are of great significance in studies on the stability of the motion of conservative holonomic systems.

Theorem) If the unperturbed motion of a holonomic potential system is stable, then the characteristic numbers of all solutions of the variational equations (3) are equal to zero, the equations (3) are regular in the sense of Lyapunov, reduce to a system of equations with constant coefficients and have a quadratic integral of definite sign.

Chetaev's theorem generalizes Lagrange's theorem on an equilibrium and the Poincaré–Lyapunov theorem on a periodic motion. According to the theorem, for a stable unperturbed motion of a potential system, an infinitely near perturbed motion has an oscillatory, wave-like, character. From this Chetaev drew the conclusion that if there is an analogy between dynamics and Cauchy's mathematical theory of light, then one should look for it in perturbed motions close to stable motions of potential systems. And Chetaev found such an analogy (cf. [1e]), showing that a necessary condition for the stability of a holonomic conservative system leads to the wave equation. The optic-mechanical analogy was completely investigated by Chetaev, in the light also of the theory of Lie groups, based on replacing the original idea of the existence of an analogy between the two phenomena by that of the identification of the group of transformations of one phenomenon (the oscillatory process of diffusion of light) with that of other phenomenon (perturbed motion of a conservative system close to its stable motion). Chetaev proved (cf. [1f]) that this latter group is a unimodular group of linear transformations and has a presentation in the full group of Lorentz transformations, which is fundamental to the theory of light of Cauchy and Maxwell.

References

[1a]	N.G. Chetaev, , Stability of motion. Papers on analytical mechanics , Moscow (1962) pp. 222–224 (In Russian)
[1b]	N.G. Chetaev, , Stability of motion. Papers on analytical mechanics , Moscow (1962) pp. 225–238 (In Russian)
[1c]	N.G. Chetaev, , Stability of motion. Papers on analytical mechanics , Moscow (1962) pp. 232 (In Russian)
[1d]	N.G. Chetaev, , Stability of motion. Papers on analytical mechanics , Moscow (1962) pp. 5–152 (In Russian)
[1e]	N.G. Chetaev, , Stability of motion. Papers on analytical mechanics , Moscow (1962) pp. 404–406 (In Russian)
[1f]	N.G. Chetaev, , Stability of motion. Papers on analytical mechanics , Moscow (1962) pp. 393–403 (In Russian)
[2]	N.N. Krasovskii, "Stability of motion. Applications of Lyapunov's second method to differential systems and equations with delay" , Stanford Univ. Press (1963) (Translated from Russian)
[3]	W. Hahn, "Stability of motion" , Springer (1967)