Definition:First Integral of System of Differential Equations
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Establish the domain and codomain of these functions
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Definition
Let $S$ be a system of differential equations.
Let $g$ be a function which satisfies $S$.
Let $f$ be a function.
Establish the domain and codomain of these functions
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Let $f$ depend on variables (denoted by ordered tuples here) of $S$ independently as well as through $g$ and its derivatives:
- $f = \map f {\sequence {x_i} _{1 \mathop \le i \mathop \le n}, \sequence {g^{\paren j} \paren {\sequence {x_i}_{1 \mathop \le i \mathop \le n} }_{0 \mathop \le j \mathop \le k} } }, \quad {n, k} \in \N$
Suppose there exists $g$ such that $f$ is a constant.
Then $f$ is the first integral of $S$.
Sources
- 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 4.17$: The Canonical Form of the Euler's Equations