On the existence of the Lagrangian for a system of non-autonomous ordinary differential equations. (Russian)

In this paper the results of the author [ibid. 13, 34-38 (1981; Zbl 0478.34033)] are generalized to the case of a time-dependent Lagrangian. A criterion for the existence of the Lagrangian for a system of differential equations $λ_{i} (t, t^{j}, x_{(1)}^{j}, \cdot \cdot \cdot, x_{(r)}^{j}) = 0 (i, j = 1, \cdot \cdot \cdot, n)$ of arbitrary order r is given. The following theorem holds: The system $λ_{i} = 0$ is an Euler-Poisson equations system, if and only if the identities

\partial λ_{i} / \partial x^{j} - \partial λ_{j} / \partial x^{i} - \sum_{s = o}^{r} {(- 1)}^{s} (d^{s} / d t^{s}) (\partial λ_{j} / \partial x_{(s)}^{i} / \partial λ_{i} / \partial x_{(s)}^{j}) = 0,

\partial λ_{i} / \partial x_{(v)}^{j} - \sum_{s = v}^{r} {(- 1)}^{s} (s! / (s - v)! v!) (d^{s - v} / d t^{s - v}) \partial λ_{j} / \partial x_{(s)}^{i} = 0 (i \leq v \leq r)

are fulfilled. In detail, a system of fourth order is studied. Additionally, systems of first up to third order are discussed.

Reviewer: G.Stiller

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