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

Multi tool use
Multi tool use











up vote
0
down vote

favorite
1












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.




















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










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















up vote
0
down vote

favorite
1












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.




















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










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













up vote
0
down vote

favorite
1









up vote
0
down vote

favorite
1






1





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






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.











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.









share|cite|improve this question




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.









asked 10 hours ago









FreeLanding45

1




1




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.





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.






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












  • 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










1 Answer
1






active

oldest

votes

















up vote
3
down vote













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





















    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "69"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });






    FreeLanding45 is a new contributor. Be nice, and check out our Code of Conduct.










     

    draft saved


    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2996486%2fwhat-does-it-mean-for-a-vector-field-to-preserve-area%23new-answer', 'question_page');
    }
    );

    Post as a guest


















    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    3
    down vote













    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

























      up vote
      3
      down vote













      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























        up vote
        3
        down vote










        up vote
        3
        down vote









        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.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 9 hours ago









        Hans Lundmark

        34.4k564110




        34.4k564110






















            FreeLanding45 is a new contributor. Be nice, and check out our Code of Conduct.










             

            draft saved


            draft discarded


















            FreeLanding45 is a new contributor. Be nice, and check out our Code of Conduct.













            FreeLanding45 is a new contributor. Be nice, and check out our Code of Conduct.












            FreeLanding45 is a new contributor. Be nice, and check out our Code of Conduct.















             


            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2996486%2fwhat-does-it-mean-for-a-vector-field-to-preserve-area%23new-answer', 'question_page');
            }
            );

            Post as a guest






































































            aicDfii5x58X9 iawhNFEIqTF c,n9B2k,72APClVC,XZnGzWf0K3BLB8Ny
            -
            YMJ0Aco Ymgk yC,k

            Popular posts from this blog

            Probability when a professor distributes a quiz and homework assignment to a class of n students.

            Image
            0 0 $begingroup$ Need help with this problem. Suppose our lazy professor collects a quiz and a homework assignment from a class of n students one day, then distributes both the quizzes and the homework assignments back to the class in a random fashion for grading. Each student receives one quiz and one homework assignment to grade. (a) What is the probability that every student receives someone else's quiz to grade, and someone else's homework to grade? (b) What is the probability that no student receives both their own quiz and their own homework assignment to grade? In this case, some students may receive their own quiz, and others may receive their own homework assignment. (c) Compute the limiting probability as n approaches infinity in each case. ...
            Read more

            Aardman Animations

            Image
            Aardman Animations, Ltd. Aardman Animations Tipo privada Atividade Animação em stop-motion , Animação em CGI Fundação 1972 Fundador(es) Peter Lord David Sproxton Sede Bristol, Inglaterra  Reino Unido Proprietário(s) Dreamworks Pessoas-chave Peter Lord David Sproxton Nick Park Divisões Aardman Features Aardman Digital Aardman Commercials Aardman Broadcast Aardman International Aardman Rights Aardman Effects Aardman 3-D Systems Aardman Nathan Love AardBoiled Website oficial aardman.com ...
            Read more

            Pontes Indestrutíveis

            Image
            "Pontes Indestrutíveis" Single de Charlie Brown Jr. do álbum Ritmo, Ritual e Responsa Lançamento 2007 Formato(s) Rádios Gravação EMI Gênero(s) Ska punk Duração 3:32 (versão do álbum) 3:33 (versão do single) Gravadora(s) EMI Composição Chorão,Charlie Brown Jr. Cronologia de singles de Charlie Brown Jr. "Não Viva Em Vão" (2007) "Be Myself" (2008) ...
            Read more