Nonlinear oscillator with velocity dependent frequency












5












$begingroup$


In a physical problem I need to investigate the following nonlinear differential equation
$$ddot x+omega^2left (1+frac{m^2dot x^2}{p^2}right)x=0,$$ where $p$ is some constant with a dimension of momentum. I will be grateful for references about such type of oscillators. So far I only found https://www.sciencedirect.com/science/article/pii/S0022460X00929894






mp.mathematical-physics differential-equations







share|cite|improve this question









$endgroup$

















    5












    $begingroup$


    In a physical problem I need to investigate the following nonlinear differential equation
    $$ddot x+omega^2left (1+frac{m^2dot x^2}{p^2}right)x=0,$$ where $p$ is some constant with a dimension of momentum. I will be grateful for references about such type of oscillators. So far I only found https://www.sciencedirect.com/science/article/pii/S0022460X00929894






    mp.mathematical-physics differential-equations







    share|cite|improve this question









    $endgroup$















      5












      5








      5


      1



      $begingroup$


      In a physical problem I need to investigate the following nonlinear differential equation
      $$ddot x+omega^2left (1+frac{m^2dot x^2}{p^2}right)x=0,$$ where $p$ is some constant with a dimension of momentum. I will be grateful for references about such type of oscillators. So far I only found https://www.sciencedirect.com/science/article/pii/S0022460X00929894






      mp.mathematical-physics differential-equations







      share|cite|improve this question









      $endgroup$




      In a physical problem I need to investigate the following nonlinear differential equation
      $$ddot x+omega^2left (1+frac{m^2dot x^2}{p^2}right)x=0,$$ where $p$ is some constant with a dimension of momentum. I will be grateful for references about such type of oscillators. So far I only found https://www.sciencedirect.com/science/article/pii/S0022460X00929894






      mp.mathematical-physics differential-equations




      mp.mathematical-physics differential-equations






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 4 hours ago









      Zurab SilagadzeZurab Silagadze

      10.8k2569




      10.8k2569






















          1 Answer
          1






          active

          oldest

          votes


















          6












          $begingroup$

          This type of ODE,
          $$ddot{x}+f(x)dot{x}^2+g(x)=0$$
          is known as a Liénard equation of the second kind. It has been studied for example in Monotonicity of the period function of the Liénard equation of second kind (2016).



          A particular case with nice properties is
          $$ddot{x}-frac{f'(x)}{f(x)}dot{x}^2+f(x)int_0^xfrac{1}{f(u)}du=0,$$
          see Design of nonlinear isochronous oscillators.






          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: "504"
            };
            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%2fmathoverflow.net%2fquestions%2f321379%2fnonlinear-oscillator-with-velocity-dependent-frequency%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









            6












            $begingroup$

            This type of ODE,
            $$ddot{x}+f(x)dot{x}^2+g(x)=0$$
            is known as a Liénard equation of the second kind. It has been studied for example in Monotonicity of the period function of the Liénard equation of second kind (2016).



            A particular case with nice properties is
            $$ddot{x}-frac{f'(x)}{f(x)}dot{x}^2+f(x)int_0^xfrac{1}{f(u)}du=0,$$
            see Design of nonlinear isochronous oscillators.






            share|cite|improve this answer









            $endgroup$


















              6












              $begingroup$

              This type of ODE,
              $$ddot{x}+f(x)dot{x}^2+g(x)=0$$
              is known as a Liénard equation of the second kind. It has been studied for example in Monotonicity of the period function of the Liénard equation of second kind (2016).



              A particular case with nice properties is
              $$ddot{x}-frac{f'(x)}{f(x)}dot{x}^2+f(x)int_0^xfrac{1}{f(u)}du=0,$$
              see Design of nonlinear isochronous oscillators.






              share|cite|improve this answer









              $endgroup$
















                6












                6








                6





                $begingroup$

                This type of ODE,
                $$ddot{x}+f(x)dot{x}^2+g(x)=0$$
                is known as a Liénard equation of the second kind. It has been studied for example in Monotonicity of the period function of the Liénard equation of second kind (2016).



                A particular case with nice properties is
                $$ddot{x}-frac{f'(x)}{f(x)}dot{x}^2+f(x)int_0^xfrac{1}{f(u)}du=0,$$
                see Design of nonlinear isochronous oscillators.






                share|cite|improve this answer









                $endgroup$



                This type of ODE,
                $$ddot{x}+f(x)dot{x}^2+g(x)=0$$
                is known as a Liénard equation of the second kind. It has been studied for example in Monotonicity of the period function of the Liénard equation of second kind (2016).



                A particular case with nice properties is
                $$ddot{x}-frac{f'(x)}{f(x)}dot{x}^2+f(x)int_0^xfrac{1}{f(u)}du=0,$$
                see Design of nonlinear isochronous oscillators.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 3 hours ago









                Carlo BeenakkerCarlo Beenakker

                74.9k9170277




                74.9k9170277






























                    draft saved

                    draft discarded




















































                    Thanks for contributing an answer to MathOverflow!


                    • 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%2fmathoverflow.net%2fquestions%2f321379%2fnonlinear-oscillator-with-velocity-dependent-frequency%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

                    SQL Server 17 - Attemping to backup to remote NAS but Access is denied

                    Image
                    .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ margin-bottom:0; } 0 I was attemping to use Ola Hallengrens scripts to backup to a remote NAS, but it was failing and not giving any direct errors, so I tried doing it manually using Server Management Studio and got the following error: TITLE: Microsoft SQL Server Management Studio Failed to retrieve data for this request. (Microsoft.SqlServer.Management.Sdk.Sfc) For help, click: http://go.microsoft.com/fwlink?ProdName=Microsoft%20SQL%20Server&LinkId=20476 ADDITIONAL INFORMATION: An exception occurred while executing a Transact-SQL statement or batch. (Microsoft.SqlServer.ConnectionInfo) The operating system returned the error '0x80070005(Access is denied.)&#
                    Read more

                    Always On Availability groups resolving state after failover - Remote harden of transaction...

                    Image
                    0 My explanation will be opinion based as it is based on my findings. If more information is needed, I am always happy to add it. The setup We have 4 SQL Server 2016 Instances, and all 4 underlying servers are in a WSFC (In my eyes a strange setup since Always On AG's are 2 by 2. But that is a different question.) 4 nodes + Fileshare witness Version: 13.0.5026.0 (all nodes) The instances that are having the failover issues are instances 1 & 2. They have 3 AG's between them. AG1 is primary on instance 1 and is having the issue. AG2 and AG3 are primary on instance 2. Instances 3 and 4 have their own separate AG's (2). AG1's settings The week before Last week, after checking for long running queries (none) and the synchronization state (green accross the board) we initiated a
                    Read more

                    Restoring from pg_dump with foreign key constraints

                    Image
                    5 2 In restoring a database from a pg_dump , a number of errors are being generated and the whole table is subsequently being ignored. An example: ERROR: insert or update on table "channelproducts" violates foreign key constraint "fk_rails_dfaae373a5" DETAIL: Key (channel_id)=(1) is not present in table "channels". Interestingly enough, I've noted that all these instances are popping up because of the sequence of loading. channels is after channelproducts both alphabetically and in the file and thus I can understand why postgres complains about having to create a child without a parent. caveat: the foreign key is being generated automatically by rails 4.2: I could remove the problem at the source but that still does not really solve the problem... version: PostgreSQL
                    Read more