Differential and Integral Equations

Influence of dissipative forces on the stability behavior of the steady motions of Lagrangian mechanical systems with cyclic coordinates

Stephen R. Bernfeld, Luigi Salvadori, and Francesca Visentin

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Abstract

We are concerned with stability problems for a holonomic system $\mathcal {S}$ with $n$ degrees of freedom having $n-m$ cyclic coordinates, $m<n$. Let $x$ be the set of the acyclic coordinates and let $v$ be the set of the generalized velocities corresponding to the cyclic coordinates. For $\mathcal {S}$ conservative or subject to a dissipation restricted to the acyclic coordinates, we revisit classical stability results concerning the steady motions of the system and give some new contribution. When $\mathcal {S}$ is strictly dissipative with respect to all the coordinates, the integrals of momenta disappear and so do the steady motions. In this case, under suitable conditions there exist motions for which $x$ is constant and, consequently, $v \rightarrow 0$ as $t \rightarrow +\infty$ (pseudosteady motions). We analyze the stability properties with respect to $(x, \dot x)$ of these motions. Such properties define a stable or unstable behavior with respect to $(x, \dot x)$ of corresponding steady motions of the conservative system under the influence of strictly dissipative perturbing

Article information

Source
Differential Integral Equations, Volume 11, Number 6 (1998), 807-822.

Dates
First available in Project Euclid: 30 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1367329477

Mathematical Reviews number (MathSciNet)
MR1659272

Zentralblatt MATH identifier
1015.34045

Subjects
Primary: 34D20: Stability
Secondary: 37C75: Stability theory 70F20: Holonomic systems 70K20: Stability

Citation

Bernfeld, Stephen R.; Salvadori, Luigi; Visentin, Francesca. Influence of dissipative forces on the stability behavior of the steady motions of Lagrangian mechanical systems with cyclic coordinates. Differential Integral Equations 11 (1998), no. 6, 807--822. https://projecteuclid.org/euclid.die/1367329477


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