First integral - Imported Knowlpedia Content

Definition

Let $(M,\omega,H)$ be a Hamiltonian system with Hamiltonian vector field $X_H$ . A smooth function $F:M\to\mathbb R$ is a first integral if

dF(X_H)=0

at every point of $M$ . Equivalently, $F$ is constant along each integral curve of $X_H$ , or $F\circ\varphi_t=F$ wherever the Hamiltonian flow $\varphi_t$ is defined. With the Poisson-bracket convention $\dot F=\{F,H\}$ , the defining condition is $\{F,H\}=0$ . The zero condition is independent of the opposite sign convention for the bracket.

Equivalent formulations and use

The flow formulation follows from

\frac{d}{dt}F(\varphi_t(x))=dF_{\varphi_t(x)}(X_H)=\{F,H\}(\varphi_t(x)).

Thus a first integral confines every trajectory to a level set of $F$ . Several independent first integrals can reduce the effective dimension of the motion; when half the phase-space dimension is supplied by pairwise commuting integrals, one obtains a completely integrable Hamiltonian system. This criterion is developed in Arnol'd, Chapters 9–10.

Examples and non-examples

The Hamiltonian $H$ itself is a first integral because $\{H,H\}=0$ . If a Lie-group symmetry preserves $H$ , the corresponding component of a moment map is another standard example Abraham–Marsden, Chapters 3–4. A function that is constant on one selected trajectory but not on all trajectories is not a first integral on $M$ ; the defining identity must hold everywhere in its stated domain.

Conventions and scope

For a general vector field $X$ , the same term means a function satisfying $dF(X)=0$ . For a time-dependent Hamiltonian $H_t$ , a time-dependent conserved quantity $F_t$ instead satisfies $\partial_tF_t+\{F_t,H_t\}=0$ . “First integral” does not mean an antiderivative, despite that usage of “integral” in elementary calculus.

References

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, 2nd ed., Springer, 1989. DOI record. Relevant: Chapters 9–10, first integrals, Poisson brackets, and complete integrability.
Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, 2nd ed., AMS Chelsea Publishing 364, 2008 reprint. DOI record. Relevant: Chapters 3–4, Hamiltonian systems, conserved quantities, and symmetry.