I am reading ana cannas' lecture notes to try understanding a bit about symplectic geometry.
At page 147, there is this definition :
Definition 24.2 A G-invariant function f : M → R is called an integral of motion of (M, ω, G, μ). If μ is constant on the trajectories of a hamiltonian vector field $v_f$ , then the corresponding one-parameter group of diffeomorphisms $\{exp(tv_f) | t ∈ \mathbb{R}\}$ is called a symmetry of (M, ω, G, μ).
Until now, I actually thought that an integral of motion is a function $f$ which is constant on the trajectories. So how is this definition motivated?