Conserved quantity

In dynamics, a conserved quantity is any function of state $𝑔 (𝐱)$ that remains constant for all time, #m/def/dynamics i.e. $˙ 𝑔 = 0$ for a differential system or $𝑔 𝑛 + 1 = 𝑔 𝑛$ for an iterated map.

Properties

One is guaranteed that the time evolution of any trajectory will remain on the manifold defined by $𝑔 (𝐱) = 𝑔 (𝐱 0)$ , i.e. such manifolds are invariant sets.
Nonlinear centres with conserved quantities

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