  We study the geometry of dynamic pairs $(X,\mathcal{V})$ on a manifold $M$,
where $X$ is a vector field and $\mathcal{V}$ is a distribution on $M$, both
satisfying a regularity condition. Special cases are pairs defined by systems
of second order ODEs, geodesic sprays in Riemannian, Finslerian and Lagranian
geometries, semi-Hamiltonian systems and control-affine systems. Analogs of
conjugate points from the calculus of variations are defined for the pair
$(X,\mathcal{V})$. The main results give estimates for the position of
conjugate points in terms of a curvature operator, analogously to the
Cartan--Hadamard and Bonet--Myers theorems. Contrary to classical cases, no
metric is given a priori, the distribution $\mathcal{V}$ may be nonintegrable
and the curvature operator is defined in terms of $(X,\mathcal{V})$.
