This paper presents the local classification of Lagrange functions $L:TX\to\R$ satisfying i) $L$ is continuous, ii) $L$ is a generalized invariant under the group of all affine time-scalings $\R\to\R$, iii) $L$ generates a Poisson structure on $TX$.
Keywords. Time scaling, Lagrange mechanics, Poisson manifolds.
MS 2000 Classification. 53D17, 70G, 70H.