What does it mean for a vector field to preserve area?

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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..






vector-analysis hamilton-equations







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  • you didn't see the in phase space?
    – user10354138
    9 hours ago










  • This should be related.
    – Giuseppe Negro
    8 hours ago















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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..






vector-analysis hamilton-equations







share|cite|improve this question







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FreeLanding45 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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  • you didn't see the in phase space?
    – user10354138
    9 hours ago










  • This should be related.
    – Giuseppe Negro
    8 hours ago













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down vote

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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..






vector-analysis hamilton-equations







share|cite|improve this question







New contributor




FreeLanding45 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











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..






vector-analysis hamilton-equations




vector-analysis hamilton-equations






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  • you didn't see the in phase space?
    – user10354138
    9 hours ago










  • This should be related.
    – Giuseppe Negro
    8 hours ago


















  • you didn't see the in phase space?
    – user10354138
    9 hours ago










  • This should be related.
    – Giuseppe Negro
    8 hours ago
















you didn't see the in phase space?
– user10354138
9 hours ago




you didn't see the in phase space?
– user10354138
9 hours ago












This should be related.
– Giuseppe Negro
8 hours ago




This should be related.
– Giuseppe Negro
8 hours ago










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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.






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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.






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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.






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        up vote
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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.






        share|cite|improve this answer












        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.







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        answered 9 hours ago









        Hans Lundmark

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