  We prove that on any closed Riemannian three-manifold $(M,g)$ the
time-dependent Euler equations are non-mixing on the space of smooth
volume-preserving vector fields endowed with the $C^1$-topology, for any fixed
helicity and large enough energy, solving a problem posed by Khesin, Misiolek,
and Shnirelman. To prove this, we introduce a new framework that assigns
contact/symplectic geometry invariants to large sets of time-dependent
solutions to the Euler equations on any 3-manifold with an arbitrary fixed
metric. This greatly broadens the scope of contact topological methods in
hydrodynamics, which so far have had applications only for stationary solutions
and without fixing the ambient metric. We further use this framework to prove
that spectral invariants obtained from Floer theory, concretely embedded
contact homology, define new non-trivial continuous first integrals of the
Euler equations in certain regions of the phase space endowed with the
$C^{1,s}$-topology, producing countably many disjoint invariant open sets.
