  In this paper, we investigate the long-time near-conservations of energy and
kinetic energy by the widely used exponential integrators to highly oscillatory
conservative systems. The modulated Fourier expansions of two kinds of
exponential integrators have been constructed and the long-time numerical
conservations of energy and kinetic energy are obtained by deriving two
almost-invariants of the expansions. Practical examples of the methods are
given and the theoretical results are confirmed and demonstrated by a numerical
experiment.
