Learning the underlying dynamics from data with deep neural networks has shown remarkable potential in modeling various complex physical dynamics. However, current approaches are constrained in their ability to make reliable predictions in a specific domain and struggle with generalizing to unseen systems that are governed by the same general dynamics but differ in environmental characteristics. In this work, we formulate a parameter-efficient method, Fourier Neural Simulator for Dynamical Adaptation (FNSDA), that can readily generalize to new dynamics via adaptation in the Fourier space. Specifically, FNSDA identifies the shareable dynamics based on the known environments using an automatic partition in Fourier modes and learns to adjust the modes specific for each new environment by conditioning on low-dimensional latent systematic parameters for efficient generalization. We evaluate our approach on four representative families of dynamic systems, and the results show that FNSDA can achieve superior or competitive generalization performance compared to existing methods with a significantly reduced parameter cost. Our code is available at https://github.com/WonderSeven/FNSDA.