View source for Lagrange stability

A property of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057240/l0572401.png" /> (a trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057240/l0572402.png" />) of a [[Dynamical system|dynamical system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057240/l0572403.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057240/l0572404.png" />, cf. [[#References|[2]]]) given on a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057240/l0572405.png" />, requiring that the trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057240/l0572406.png" /> is contained in a pre-compact set (cf. [[Pre-compact space|Pre-compact space]]).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057240/l0572407.png" />, then Lagrange stability is the same as boundedness of the trajectory. If for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057240/l0572408.png" /> (respectively, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057240/l0572409.png" />) the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057240/l05724010.png" /> is contained in a pre-compact set, then the trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057240/l05724011.png" /> (the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057240/l05724012.png" />) is called positively (respectively, negatively) Lagrange stable. The concept of Lagrange stability was introduced by H. Poincaré in connection with analyzing the results of J.L. Lagrange on the stability of planetary orbits.

Birkhoff's theorem: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057240/l05724013.png" /> is complete, then the closure of a positively or negatively Lagrange-stable trajectory contains at least one compact [[Minimal set|minimal set]]. Every point of a compact minimal set is a [[Recurrent point|recurrent point]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,   "Les méthodes nouvelles de la mécanique céleste" , '''3''' , Blanchard, reprint  (1987)  pp. Chapt. 26</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.V. Nemytskii,   V.V. Stepanov,   "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR></table>



====Comments====
In [[#References|[1]]], Poincaré explicitly introduces the term  "stabilité à la Poisson" , but merely implicitly suggests the notion  "stabilité à la Lagrange"  by mentioning that boundedness of the planetary orbits was proved by Lagrange.

The above definitions can be given for any dynamical system, not necessarily defined on a metric space. In particular, for the first part of Birkhoff's theorem as formulated above, it is not necessary to require that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057240/l05724014.png" /> is metrizable, let alone complete. Metrizability and completeness are needed to prove that every point of a minimal set is a [[Recurrent point|recurrent point]]. In the general case, every point of a compact minimal set is an almost-periodic point.