Abstract
In this chapter we introduce the basics in the theory of Lyapunov stability, which was first formulated at the end of the nineteenth century.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The function W is real-valued even when we work in \({\mathbb K}^n={\mathbb C}^n\).
- 2.
If not, outside a certain ball B 1∕n∗ there would be a sequence x 1, x 2, … with W(x k) → 0 as k → +∞. Since the x k would belong to the compact set \(K:= \overline {B}\setminus B_{1/n*}\), we could find a subsequence \(\{{\mathbf {x}}_{k_m}\}_{m}\) with \({\mathbf {x}}_{k_m} \to {{\mathbf {x}}^*} \in K\) as m → +∞. By continuity W(x ∗) = 0. But since x ∗≠ x 0, W could not be strictly minimised by x 0.
- 3.
We cannot have \(||\mathbf {u}||{ }^2+ |\mathbf {u}\cdot \mathbf {u}|\cos \phi = 0\), i.e. ||u||2 + Re(u ⋅u) = 0, since the computation would force u ∗ = −u, which we have excluded a priori.
- 4.
That (x 0,1, …, x 0,N) gives a strict local minimum for the potential energy is not enough to ensure it is an isolated stationary point: consider, for N = 1, the map \(\mathscr {U}(x) := x^4(2+ \sin {}(1/x))\) if x ≠ 0 and \(\mathscr {U}(0):=0\). \(\mathscr {U}\) is twice differentiable, x = 0 is a strict local minimum but not an isolated stationary point.
References
Malkin, I.G.: Theory of stability of motion, in ACE-tr-3352 Physics and Mathematics. US Atomic Energy Commission, Washington (1952)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Moretti, V. (2023). Introduction to Stability Theory with Applications to Mechanics. In: Analytical Mechanics. UNITEXT(), vol 150. Springer, Cham. https://doi.org/10.1007/978-3-031-27612-5_6
Download citation
DOI: https://doi.org/10.1007/978-3-031-27612-5_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-27611-8
Online ISBN: 978-3-031-27612-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)