Matsyuk, R.Ya
On the existence of the Lagrangian for a system of non-autonomous ordinary differential equations. (Russian) Zbl 0536.34007
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 λ i (t,t j ,x (1) j ,···,x (r) j )=0(i,j=1,···,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

λ i /x j -λ j /x i - s=o r (-1) s (d s /dt s )(λ j /x (s) i /λ i /x (s) j )=0,
λ i /x (v) j - s=v r (-1) s (s!/(s-v)!v!)(d s-v /dt s-v )λ j /x (s) i =0(ivr)

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 of ODE
Keywords:
fourth order differential equation; Lagrangian; Euler-Poisson equations system