Instability of the equilibrium positions of restricted mechanical systems

Furta, Stanislav D.

doi:10.1007/bf00887812

Cited by 4 publications

(15 citation statements)

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“…Theorem 1 represents generalization of the theorem formulated in [6] to the case when the Pfaffian form B…”

Section: Resultsmentioning

confidence: 99%

“…Starting from the results of Furta [6] three theorems on the instability are formulated. Theorem 1 represents generalization of the theorem formulated in [6] to the case when the Pfaffian form B…”

Section: Resultsmentioning

confidence: 99%

“…Also, instead of a single constraint, in this paper, l constraints are imposed on the system (2). Theorems 2 and 3 proved without seeking the asymptotic solution for Lagrange's multipliers of constraints (as has been done in [6]) represent generalization of the results from [6] (the case of weakly nonholonomic systems), especially for the case when difficulties occur in determining the sign of the form Π (m+1) = Π (m+1) (q) with limitations imposed by equations of constraints. It is evident from the conditions of Theorem 3 that in the case of a singular nonholonomic constraint (the case considered in [6]), the condition that is imposed on the coefficients B ν(r) i in [6] is weakened in such a way that there is no requirement that at least one of the coefficients should be different from zero at any point of q = 0 but only at the point representing a non-trivial solution of Eqs.…”

Section: ν(R) Imentioning

confidence: 95%

“…Under these conditions, the equilibrium position q = 0,q = 0 is unstable. Further considerations, in this paper, lead to the generalization of the results obtained in [6]. The motion of a mechanical system, whose configuration is determined by a set of generalized coordinates q = (q 1 , .…”

Section: Introductionmentioning

confidence: 96%

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On the instability of equilibrium position of a mechanical system with singular constraints

et al. 2013

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Key words Lyapunov first method, nonholonomic system, singular constraints, instability of equilibrium.The Lyapunov first method generalized to the case of nonlinear differential equations is applied to the study of the instability of the equilibrium position of a mechanical system, whose motion is constrained by singular nonholonomic constraints. Starting from the results of S. D. Furta (On the instability of equilibrium position of constrained mechanical systems) three theorems on the instability are formulated. The first theorem considers the case of nonholonomic constraints that do not satisfy the condition of weak nonholonomity. The other two theorems are related to the case of weakly nonholonomic systems. In each of the formulated theorems it is shown that the minimum form of Maclaurin series for the potential energy has not a local minimum. Thus, a contribution has been made to the inversion of Lagrange's theorem.

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“…Theorem 1 represents generalization of the theorem formulated in [6] to the case when the Pfaffian form B…”

Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: ν(R) Imentioning

confidence: 95%

Section: Introductionmentioning

confidence: 96%

See 2 more Smart Citations

On the instability of equilibrium position of a mechanical system with singular constraints

et al. 2013

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“…Such a problem, in the case of a mechanical system with a single nonholonomic constraint, was solved in [13] by applying Kozlov's generalization of Lyapunov's first method.…”

Section: Introductionmentioning

confidence: 99%

Lyapunov-Kozlov method for singular cases

Čović

Djuric

Vesković

et al. 2011

Appl. Math. Mech.-Engl. Ed.

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Lyapunov's first method, extended by Kozlov to nonlinear mechanical systems, is applied to study the instability of the equilibrium position of a mechanical system moving in the field of conservative and dissipative forces. The cases with a tensor of inertia or a matrix of coefficients of the Rayleigh dissipative function are analyzed singularly in the equilibrium position. This fact renders the impossible application of Lyapunov's approach in the analysis of the stability because, in the equilibrium position, the conditions of the existence and uniqueness of the solutions to the differential equations of motion are not fulfilled. It is shown that Kozlov's generalization of Lyapunov's first method can also be applied in the mentioned cases on the conditions that, besides the known algebraic expression, more are fulfilled. Three theorems on the instability of the equilibrium position are formulated. The results are illustrated by an example.

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Inversion Problem for the Lagrange Theorem on the Stability of Equilibrium and Related Problems

Kozlov

Furta

2012

Springer Monographs in Mathematics

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Instability of the equilibrium positions of restricted mechanical systems

Cited by 4 publications

References 0 publications

On the instability of equilibrium position of a mechanical system with singular constraints

On the instability of equilibrium position of a mechanical system with singular constraints

Lyapunov-Kozlov method for singular cases

Inversion Problem for the Lagrange Theorem on the Stability of Equilibrium and Related Problems

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