How to apply d/dt to solve for equations of motion (Applying Lagrangian to Vibrations)












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In my engineering vibrations course, I am encountering derivatives that are partial as well as total. I am a little rusty and am having a really hard time grasping why the differentiation with respect to time is done differently in some scenarios.



Here is an example from my book:



enter image description here



I circled the two confusing terms in red and green. Why is the red term using chain rule while the green term simply takes a derivative with respect to x_dot?






derivatives euler-lagrange-equation







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    1












    $begingroup$


    In my engineering vibrations course, I am encountering derivatives that are partial as well as total. I am a little rusty and am having a really hard time grasping why the differentiation with respect to time is done differently in some scenarios.



    Here is an example from my book:



    enter image description here



    I circled the two confusing terms in red and green. Why is the red term using chain rule while the green term simply takes a derivative with respect to x_dot?






    derivatives euler-lagrange-equation







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      In my engineering vibrations course, I am encountering derivatives that are partial as well as total. I am a little rusty and am having a really hard time grasping why the differentiation with respect to time is done differently in some scenarios.



      Here is an example from my book:



      enter image description here



      I circled the two confusing terms in red and green. Why is the red term using chain rule while the green term simply takes a derivative with respect to x_dot?






      derivatives euler-lagrange-equation







      share|cite|improve this question









      $endgroup$




      In my engineering vibrations course, I am encountering derivatives that are partial as well as total. I am a little rusty and am having a really hard time grasping why the differentiation with respect to time is done differently in some scenarios.



      Here is an example from my book:



      enter image description here



      I circled the two confusing terms in red and green. Why is the red term using chain rule while the green term simply takes a derivative with respect to x_dot?






      derivatives euler-lagrange-equation




      derivatives euler-lagrange-equation






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 29 at 3:23









      LearnITLearnIT

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          $begingroup$

          It is
          $$ddot{x}/dt=ddot{x}$$
          but
          $$d(dot{x}^2)/dt=2dot{x},ddot{x}/dt=2dot{x}ddot{x}$$






          share|cite|improve this answer









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            $begingroup$

            It is
            $$ddot{x}/dt=ddot{x}$$
            but
            $$d(dot{x}^2)/dt=2dot{x},ddot{x}/dt=2dot{x}ddot{x}$$






            share|cite|improve this answer









            $endgroup$


















              1












              $begingroup$

              It is
              $$ddot{x}/dt=ddot{x}$$
              but
              $$d(dot{x}^2)/dt=2dot{x},ddot{x}/dt=2dot{x}ddot{x}$$






              share|cite|improve this answer









              $endgroup$
















                1












                1








                1





                $begingroup$

                It is
                $$ddot{x}/dt=ddot{x}$$
                but
                $$d(dot{x}^2)/dt=2dot{x},ddot{x}/dt=2dot{x}ddot{x}$$






                share|cite|improve this answer









                $endgroup$



                It is
                $$ddot{x}/dt=ddot{x}$$
                but
                $$d(dot{x}^2)/dt=2dot{x},ddot{x}/dt=2dot{x}ddot{x}$$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 29 at 5:41









                MASLMASL

                708313




                708313






























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