  We explain from first principles why satisfying conservation laws in Bose
Einstein condensate dynamics requires many-body theory. For the
Gross-Pitaevskii mean-field we show analytically and numerically that
conservation laws are violated. We provide examples for angular momentum and
linear momentum conservation. Arbitrarily large violations occur despite
negligible depletion and interaction energy. For the case of angular momentum
we show through extensive many-body simulations how the conservation law can be
gradually restored on the many-body level. Implications are discussed.
