On the existence of the Lagrangian for a system of non-autonomous ordinary differential equations. (Russian) Zbl 0536.34007

Matsyuk, R. Ya

Mat. Metody Fiz.-Mekh. Polya 20, 16-19 (1984).

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 \(\lambda_ i(t,t^ j,x^ j_{(1)},...,x^ j_{(r)})=0 (i,j=1,...,n)\) of arbitrary order r is given. The following theorem holds: The system \(\lambda_ i=0\) is an Euler-Poisson equations system, if and only if the identities \[ \partial \lambda_ i/\partial x^ j-\partial \lambda_ j/\partial x^ i-\sum^{r}_{s=o}(-1)^ s(d^ s/dt^ s)(\partial \lambda_ j/\partial x^ i_{(s)}/\partial \lambda_ i/\partial x^ j_{(s)})=0, \]
\[ \partial \lambda_ i/\partial x^ j_{(v)}-\sum^{r}_{s=v}(-1)^ s(s!/(s- v)!v!)(d^{s-v}/dt^{s-v})\partial \lambda_ j/\partial x^ i_{(s)}=0\quad(i\leq v\leq r) \] are fulfilled. In detail, a system of fourth order is studied. Additionally, systems of first up to third order are discussed.

Reviewer: G.Stiller

MSC:

34A99

General theory for ordinary differential equations

Keywords:

fourth order differential equation; Lagrangian; Euler-Poisson equations system

Citations:

Zbl 0478.34033