First integral

Definition

Let $U$ be an open subset of $\mathbb{R}^n$ and let $f : U \to \mathbb{R}^n$ be a function. Let $H : U \to \mathbb{R}$ be a continuously differentiable function. It is a first integral of the autonomous system $x' = f(x)$ if $$\nabla H(x) \cdot f(x) = 0 \quad \text{for each } x \in U,$$ and if $H$ is not constant on any non-empty open subset of $U$.

Remarks

The condition means that $\frac{d}{dt} H(\gamma(t)) = 0$ for each solution $\gamma$. So $H$ is constant along each solution. Each trajectory lies in a level set $\{x : H(x) = c\}$. In the plane, the level curves of $H$ give the phase portrait. For the harmonic oscillator $x' = y$, $y' = -x$, the function $H(x, y) = x^2 + y^2$ is a first integral. The Hamiltonian of a time-independent Hamiltonian system is a first integral.

Explanations

L1 human

A first integral is a quantity that does not change while the system moves, such as the energy of a system without friction. Knowing it tells us that each motion stays on one level set of this quantity.