Semantic drift: the gradual change in word meaning over time, poses a critical challenge to natural language understanding in diachronic corpora. Existing methods detect drift via alignment or distance metrics but lack the ability to model its dynamics or forecast future semantic change. We present Koopman Spectral Dynamics, the first framework to model semantic drift as a nonlinear dynamical system in word embedding space, linearly lifted via the Koopman operator. This enables interpretable forecasting, spectral decomposition, and drift attribution using Koopman eigenfunctions and modes. Applied to the HistWords: All English dataset (1800s-1990s), our method outperforms strong baselines: PCA, LSTM, and Pro-crustes, with up to 24.5% lower RMSE and 19.2% lower MAE in word forecasting, and +7.8 % accuracy in semantic coherence classification. We analyze drift trajectories using spectral energy curves, degradation plots, and mode attributions, and introduce causal influence maps via Granger and mutual information matrices. Our model generalizes to multi-word phrases (e.g., gay marriage) and supports a real-time interactive forecasting tool. This work introduces Koopman operator theory to NLP and data mining, offering a principled, spectral foundation for modeling and interpreting meaning evolution over time.