Changes in the circulation of the earth's atmosphere cause fluctuations in the length of day (LOD) by exchanging angular momentum with the surface of the earth. In this paper we examine the earth's polar angular momentum budget using earth rotation data from optical astrometry and lunar laser ranging together with simultaneous estimates of the atmospheric angular momentum Matm derived from meteorological data. The atmospheric excitation can explain most of the observed LOD fluctuations at periods between 700 and 40 days. The observed Matm spectral power is proportional to the inverse frequency squared for periods between 1 year and 3 days and flattens out at higher frequencies. The LOD and Matm spectra are very similar except at periods of <40 days, where the LOD measurement noise dominates. The Matm-LOD coherence is highly significant between periods of 700 and 40 days. There is no evidence for phase shifts between the Matm and LOD data at periods between 600 and 40 days. A broad peak is clearly evident at periods of 40--60 days in the spectra of all three time series. The LOD data have a long-term drift, no matched by the Matm data, that is probably the result of a core-mantle angular momentum exchange. There are significant imbalances in the angular momentum budget. Most ot the seasonal imbalance can be attributed to the effects of systematic errors in the meteorological data, but some of this discrepancy may be caused by exchanges of angular momentum with the oceans. The semiannual, monthly, and fortnightly tidal variations in the earth rotation were detected, and the observed amplitudes of these variations are consistent with theoretical estimates.