The physical meaning of a symplectic form.












5












$begingroup$


So I've studied a bit about symplectic geometry, and I know that phase space is a symplectic manifold, and the symplectic form induces a poisson bracket. However, what is the physical meaning of the symplectic form? Perhaps it's simply the same as asking what the meaning is of the poisson bracket. What does the poisson bracket say about a system? And what is the physical meaning of observables?






differential-geometry classical-mechanics symplectic-geometry







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    Have you looked at the first chapter of McDuff and Salamon's book on symplectic topology? I don't know if that will answer your question but it should be useful.
    $endgroup$
    – Mike Miller
    4 hours ago












  • $begingroup$
    a little bit, just scanned through it
    $endgroup$
    – AkatsukiMaliki
    3 hours ago
















5












$begingroup$


So I've studied a bit about symplectic geometry, and I know that phase space is a symplectic manifold, and the symplectic form induces a poisson bracket. However, what is the physical meaning of the symplectic form? Perhaps it's simply the same as asking what the meaning is of the poisson bracket. What does the poisson bracket say about a system? And what is the physical meaning of observables?






differential-geometry classical-mechanics symplectic-geometry







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    Have you looked at the first chapter of McDuff and Salamon's book on symplectic topology? I don't know if that will answer your question but it should be useful.
    $endgroup$
    – Mike Miller
    4 hours ago












  • $begingroup$
    a little bit, just scanned through it
    $endgroup$
    – AkatsukiMaliki
    3 hours ago














5












5








5





$begingroup$


So I've studied a bit about symplectic geometry, and I know that phase space is a symplectic manifold, and the symplectic form induces a poisson bracket. However, what is the physical meaning of the symplectic form? Perhaps it's simply the same as asking what the meaning is of the poisson bracket. What does the poisson bracket say about a system? And what is the physical meaning of observables?






differential-geometry classical-mechanics symplectic-geometry







share|cite|improve this question









$endgroup$




So I've studied a bit about symplectic geometry, and I know that phase space is a symplectic manifold, and the symplectic form induces a poisson bracket. However, what is the physical meaning of the symplectic form? Perhaps it's simply the same as asking what the meaning is of the poisson bracket. What does the poisson bracket say about a system? And what is the physical meaning of observables?






differential-geometry classical-mechanics symplectic-geometry




differential-geometry classical-mechanics symplectic-geometry






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 4 hours ago









AkatsukiMalikiAkatsukiMaliki

338110




338110








  • 1




    $begingroup$
    Have you looked at the first chapter of McDuff and Salamon's book on symplectic topology? I don't know if that will answer your question but it should be useful.
    $endgroup$
    – Mike Miller
    4 hours ago












  • $begingroup$
    a little bit, just scanned through it
    $endgroup$
    – AkatsukiMaliki
    3 hours ago














  • 1




    $begingroup$
    Have you looked at the first chapter of McDuff and Salamon's book on symplectic topology? I don't know if that will answer your question but it should be useful.
    $endgroup$
    – Mike Miller
    4 hours ago












  • $begingroup$
    a little bit, just scanned through it
    $endgroup$
    – AkatsukiMaliki
    3 hours ago








1




1




$begingroup$
Have you looked at the first chapter of McDuff and Salamon's book on symplectic topology? I don't know if that will answer your question but it should be useful.
$endgroup$
– Mike Miller
4 hours ago






$begingroup$
Have you looked at the first chapter of McDuff and Salamon's book on symplectic topology? I don't know if that will answer your question but it should be useful.
$endgroup$
– Mike Miller
4 hours ago














$begingroup$
a little bit, just scanned through it
$endgroup$
– AkatsukiMaliki
3 hours ago




$begingroup$
a little bit, just scanned through it
$endgroup$
– AkatsukiMaliki
3 hours ago










1 Answer
1






active

oldest

votes


















5












$begingroup$

This topic is incredibly deep so my answer barely skims the surface, however I believe you're looking for an intuitive meaning of the symplectic form. Consider the standard symplectic form



$$
omega = sum_i d x_i wedge d lambda_i
$$



This can also be written as



$$
omega = begin{bmatrix}
mathbf{0} & mathbf{I} \
-mathbf{I} & mathbf{0}
end{bmatrix}
$$



Now consider a rotation matrix



$$
R_theta = begin{bmatrix}
cos(theta) & sin(theta) \
-sin(theta) & cos(theta)
end{bmatrix}
$$



By setting $theta = 90^o$, you find the data of these tensors are equivalent. Ergo the standard symplectic form is very similar to a rotation matrix that rotates vectors by 90 degrees. Consider the symplectic form of Hamiltonian mechanics



$$
dH = iota_{X_H} omega
$$



where $dH$ is the exterior derivative of the Hamiltonian and $iota$ is the interior derivative. This exterior derivative generates a covector field whose covectors point in the direction of steepest descent. Rotate these covectors by 90 degrees and they point tangential to the level sets of $H$. In Hamiltonian mechanics, motion almost always lies on level sets of $H$.



Ultimately, I like to think about the symplectic form as a rotation matrix that properly encodes the structure of mechanics, but be aware that due to its differential geometric nature, it's much deeper than I lead on.






share|cite|improve this answer









$endgroup$













    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',
    autoActivateHeartbeat: false,
    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
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3096840%2fthe-physical-meaning-of-a-symplectic-form%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    5












    $begingroup$

    This topic is incredibly deep so my answer barely skims the surface, however I believe you're looking for an intuitive meaning of the symplectic form. Consider the standard symplectic form



    $$
    omega = sum_i d x_i wedge d lambda_i
    $$



    This can also be written as



    $$
    omega = begin{bmatrix}
    mathbf{0} & mathbf{I} \
    -mathbf{I} & mathbf{0}
    end{bmatrix}
    $$



    Now consider a rotation matrix



    $$
    R_theta = begin{bmatrix}
    cos(theta) & sin(theta) \
    -sin(theta) & cos(theta)
    end{bmatrix}
    $$



    By setting $theta = 90^o$, you find the data of these tensors are equivalent. Ergo the standard symplectic form is very similar to a rotation matrix that rotates vectors by 90 degrees. Consider the symplectic form of Hamiltonian mechanics



    $$
    dH = iota_{X_H} omega
    $$



    where $dH$ is the exterior derivative of the Hamiltonian and $iota$ is the interior derivative. This exterior derivative generates a covector field whose covectors point in the direction of steepest descent. Rotate these covectors by 90 degrees and they point tangential to the level sets of $H$. In Hamiltonian mechanics, motion almost always lies on level sets of $H$.



    Ultimately, I like to think about the symplectic form as a rotation matrix that properly encodes the structure of mechanics, but be aware that due to its differential geometric nature, it's much deeper than I lead on.






    share|cite|improve this answer









    $endgroup$


















      5












      $begingroup$

      This topic is incredibly deep so my answer barely skims the surface, however I believe you're looking for an intuitive meaning of the symplectic form. Consider the standard symplectic form



      $$
      omega = sum_i d x_i wedge d lambda_i
      $$



      This can also be written as



      $$
      omega = begin{bmatrix}
      mathbf{0} & mathbf{I} \
      -mathbf{I} & mathbf{0}
      end{bmatrix}
      $$



      Now consider a rotation matrix



      $$
      R_theta = begin{bmatrix}
      cos(theta) & sin(theta) \
      -sin(theta) & cos(theta)
      end{bmatrix}
      $$



      By setting $theta = 90^o$, you find the data of these tensors are equivalent. Ergo the standard symplectic form is very similar to a rotation matrix that rotates vectors by 90 degrees. Consider the symplectic form of Hamiltonian mechanics



      $$
      dH = iota_{X_H} omega
      $$



      where $dH$ is the exterior derivative of the Hamiltonian and $iota$ is the interior derivative. This exterior derivative generates a covector field whose covectors point in the direction of steepest descent. Rotate these covectors by 90 degrees and they point tangential to the level sets of $H$. In Hamiltonian mechanics, motion almost always lies on level sets of $H$.



      Ultimately, I like to think about the symplectic form as a rotation matrix that properly encodes the structure of mechanics, but be aware that due to its differential geometric nature, it's much deeper than I lead on.






      share|cite|improve this answer









      $endgroup$
















        5












        5








        5





        $begingroup$

        This topic is incredibly deep so my answer barely skims the surface, however I believe you're looking for an intuitive meaning of the symplectic form. Consider the standard symplectic form



        $$
        omega = sum_i d x_i wedge d lambda_i
        $$



        This can also be written as



        $$
        omega = begin{bmatrix}
        mathbf{0} & mathbf{I} \
        -mathbf{I} & mathbf{0}
        end{bmatrix}
        $$



        Now consider a rotation matrix



        $$
        R_theta = begin{bmatrix}
        cos(theta) & sin(theta) \
        -sin(theta) & cos(theta)
        end{bmatrix}
        $$



        By setting $theta = 90^o$, you find the data of these tensors are equivalent. Ergo the standard symplectic form is very similar to a rotation matrix that rotates vectors by 90 degrees. Consider the symplectic form of Hamiltonian mechanics



        $$
        dH = iota_{X_H} omega
        $$



        where $dH$ is the exterior derivative of the Hamiltonian and $iota$ is the interior derivative. This exterior derivative generates a covector field whose covectors point in the direction of steepest descent. Rotate these covectors by 90 degrees and they point tangential to the level sets of $H$. In Hamiltonian mechanics, motion almost always lies on level sets of $H$.



        Ultimately, I like to think about the symplectic form as a rotation matrix that properly encodes the structure of mechanics, but be aware that due to its differential geometric nature, it's much deeper than I lead on.






        share|cite|improve this answer









        $endgroup$



        This topic is incredibly deep so my answer barely skims the surface, however I believe you're looking for an intuitive meaning of the symplectic form. Consider the standard symplectic form



        $$
        omega = sum_i d x_i wedge d lambda_i
        $$



        This can also be written as



        $$
        omega = begin{bmatrix}
        mathbf{0} & mathbf{I} \
        -mathbf{I} & mathbf{0}
        end{bmatrix}
        $$



        Now consider a rotation matrix



        $$
        R_theta = begin{bmatrix}
        cos(theta) & sin(theta) \
        -sin(theta) & cos(theta)
        end{bmatrix}
        $$



        By setting $theta = 90^o$, you find the data of these tensors are equivalent. Ergo the standard symplectic form is very similar to a rotation matrix that rotates vectors by 90 degrees. Consider the symplectic form of Hamiltonian mechanics



        $$
        dH = iota_{X_H} omega
        $$



        where $dH$ is the exterior derivative of the Hamiltonian and $iota$ is the interior derivative. This exterior derivative generates a covector field whose covectors point in the direction of steepest descent. Rotate these covectors by 90 degrees and they point tangential to the level sets of $H$. In Hamiltonian mechanics, motion almost always lies on level sets of $H$.



        Ultimately, I like to think about the symplectic form as a rotation matrix that properly encodes the structure of mechanics, but be aware that due to its differential geometric nature, it's much deeper than I lead on.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 3 hours ago









        Michael SparapanyMichael Sparapany

        715




        715






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3096840%2fthe-physical-meaning-of-a-symplectic-form%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Усть-Каменогорск

            Image
            У этого термина существуют и другие значения, см. Усть-Каменогорск. Город Усть-Каменогорск каз. Өскемен Флаг Герб 49°57′ с. ш. 82°37′ в. д. H G Я O L Страна Казахстан   Казахстан Статус областной центр Область Восточно-Казахстанская Аким Жаксылык Омар [1] История и география Основан 1720 Город с 1868 Площадь 303,15 [2] км² Высота центра 283 ± 1 м Тип климата резко-континентальный Часовой пояс UTC+6 Население Население 329 090 [3]  человек ( 2018 ) Национальности русские 53,56 % казахи 42,99 % немцы 0,82 % украинцы 0,57 % татары 0,79 % белорусы 0,14 % азербайджанцы 0,23 % корейцы 0,20 % прочие 1,08 % -1 шт [4] Этнохороним усть-каменогорцы, усть-каменогорец Цифровые идентификаторы Телефонный код +7 (7232) Почтовый индекс F0***** [5] Автомобильный код 16 (ранее F) oskemen.kz Усть-Каменогорск Усть-Каменогорск ...
            Read more

            Халкинская богословская школа

            Image
            Халкинская богословская школа тур. Rum Ortodoks Ruhban Okulu греч. Ιερά Θεολογική Σχολή της Χάλκης Главное здание Халкинской школы Международное название Theological School of Halki Год основания 1844 Конфессия православие Церковь Константинопольская Ректор Елпидифор (Ламбриниадис) Расположение о. Хейбелиада, Турция Юридический адрес Ümit tepesi No 5, 34973 Heybeliada, Istanbul, Turkey Сайт www.greece.org  [[commons:Category:Halki seminary|Халкинская богословская школа тур. Rum Ortodoks Ruhban Okulu греч. Ιερά Θεολογική Σχολή της Χάλκης на Викискладе]] Ха́лкинская богосло́вская шко́ла (или Духовная семинария на Халки , тур. Rum Ortodoks Ruhban Okulu , греч. Ιερά Θεολογική Σχολή της Χάλκης ) — недействующее среднее специальное учебное заведение Константинопольской православной церкви, расположенное на территории Свято-Троицкого мужского монастыря на вершине холма Надежды (тур. Ümit Tepesi ) острова Хейбелиада, в Турции....
            Read more

            Высокополье (Харьковская область)

            Image
            У этого термина существуют и другие значения, см. Высокополье. Село Высокополье укр. Високопілля Герб 49°55′07″ с. ш. 35°25′13″ в. д. H G Я O L Страна Украина   Украина Область Харьковская Район Валковский Сельский совет Высокопольский История и география Основан 1670 Прежние названия Хмелевое Площадь 55,56 км² Часовой пояс UTC+2, летом UTC+3 Население Население 1957 человек ( 2001 ) Плотность 35,220 чел./км² Цифровые идентификаторы Телефонный код +380 5753 Почтовый индекс 63020 Автомобильный код AX, КХ / 21 КОАТУУ 6321281501 Высокополье Высокополье Высокополье  Аудио, фото и видео на Викискладе Высокопо́лье (укр. Високопілля ) — село, Высокопольский сельский совет, Валковский район, Харьковская область, Украина. Код КОАТУУ — 6321281501. Население по переписи 2001 г. составляет 1957 (866/1091 м/ж) человек. Является административным центр...
            Read more