  We propose a new explicit pseudo-energy and momentum conserving scheme for
the time integration of Hamiltonian systems. The scheme, which is formally
second-order accurate, is based on two key ideas: the integration during the
time-steps of forces between free-flight particles and the use of momentum
jumps at the discrete time nodes leading to a two-step formulation for the
acceleration. The pseudo-energy conservation is established under exact force
integration, whereas it is valid to second-order accuracy in the presence of
quadrature errors. Moreover, we devise an asynchronous version of the scheme
that can be used in the framework of slow-fast time-stepping strategies. The
scheme is validated against classical benchmarks and on nonlinear or
inhomogeneous wave propagation problems.
