I was reading a book about hamiltonian mechanics. After computing the divergence of the hamiltonian vector field to be identically zero, the author adds: "...thus the vector field is divergence-free and its flow preserves area in the phase plane." What does it mean to preserve area? I just couldn't see the area here..
2026-03-25 06:01:53.1774418513
What does it mean for a vector field to preserve area?
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It means that the flow of the vector field is an area-preserving map (for each $t$). That is, if you take any region in the phase plane, and let its points “go with the flow” for a certain time $t$, they will then form a new region whose area is the same as the area of the original region.