State-Space Geometry and Geometric Phase in Static and Modulated Elastic Lattices

Geometric phases play a central role in the topology of elastic lattices, but they are often extracted indirectly from band eigenvectors rather than from the motion itself. We present a state-based formulation for one-dimensional mass-spring lattices in which intracell motion is written as a normalized superposition of orthogonal in-phase and out-of-phase modes, and the resulting complex coefficients are tracked as trajectories on a Bloch sphere. This construction turns amplitude hierarchies and phase evolution into directly interpretable geometry, allowing Berry and Zak phases to be identified from measurable amplitude-phase data. For diatomic lattices, the method shows how inversion symmetry locks the relative phase between modal components and enforces Zak-phase quantization to 0 or pi, with band-dependent jumps at high-symmetry points and the expected exchange under a shift of unit-cell origin. Extending the analysis to triatomic lattices reveals how restoring inversion preserves quantization, while symmetry breaking lifts it without changing the spectral origin. The same coefficient-based picture also supports a control interpretation: norm-preserving changes of the modal pair act as Bloch-sphere rotations, giving classical analogs of qubit-like state transfer and phase-flip operations. We then apply the framework to spatiotemporally modulated lattices, where carrier and Floquet sideband hybridization replaces piecewise-locked phases by continuous winding and generates open-path geometric phases accumulated along the actual trajectory. By moving from overlap-based phase extraction to explicit state reconstruction, this work provides a compact, gauge-robust, and experimentally accessible language for connecting symmetry, modulation, and topology in elastic media, and for designing programmable vibration and acoustic functionalities in classical mechanical structures.