  We study distributional solutions of pressureless Euler systems on the line.
In particular we show that Lagrangian solutions, introduced by Brenier, Gangbo,
Savar\'{e} and Westdickenberg, and entropy solutions, studied by Nguyen and
Tudorascu for the Euler--Poisson system, are equivalent. For the Euler--Poisson
system this can be seen as a generalization to second-order systems of the
equivalence between $L^2$-gradient flows and entropy solutions for a
first-order aggregation equation proved by Bonaschi, Carrillo, Di Francesco and
Peletier. The key observation is an equivalence between Ole\u{\i}nik's
E-condition for conservation laws and a characterization due to Natile and
Savar\'{e} of the normal cone for $L^2$-gradient flows. This new equivalence
allows us to define unique solutions after blow-up for classical solutions of
the Euler--Poisson system with quadratic confinement due to Carrillo, Choi and
Zatorska, as well as to describe their asymptotic behavior.
