---
tag: MPE4AWEP6K
kind: concept
name: Conservative system
msc: [37A40, 37A05, 28D05]
sources:
  - url: https://en.wikipedia.org/wiki/Conservative_system
---

## Summary

A conservative system is a measurable dynamical system in which every set of positive measure returns to itself with positive measure.
It is the opposite of a dissipative system, in which some mass escapes and never comes back.
On a finite measure space every [[measure-preserving transformation|556WBDAHNP]] is conservative, by the [[Poincaré recurrence theorem|TVYJPBHD0S]].
The notion matters most for infinite measures and for maps that only respect null sets.
In physics the word refers to systems without friction, such as Hamiltonian systems, whose flows preserve phase-space volume.

## Explanations

### L1 human

Picture a system that moves mass around without creating or destroying it, and without letting it run away for good.
Any blob of positive size that one follows will, at some later step, overlap its starting position again.
A rotation of a circle is conservative: an arc keeps coming back to overlap itself.
The translation $x \mapsto x + 1$ on the real line is not: the interval $[0, 1)$ moves off to the right and never overlaps itself again.