Getting Canonical Coordinates in differential equations from Lie Group












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$begingroup$


I've been trying to understand how Lie Groups can help solve differential equations (this 12-page pdf is the most straightforward explanation I've seen).



My understanding is that when a first-order differential equation $frac{dy}{dx}=h(x,y)$ has a continuous "translation symmetry" $(x,y)mapsto(x,y+lambda)$, it becomes very easy to solve the equation using basic integration of one variable.



The problem is that most differential equations in standard $(x,y)$ coordinates don't have this nice translation symmetry, and so we need to change to a set of "canonical coordinates" $(X,Y)$ which do have the desired translation symmetry $(X,Y)mapsto(X,Y+lambda)$.



The main condition that these new canonical coordinates need to satisfy, is that the $Y$ coordinate should be aligned with the $lambda$ variable ($frac{dY}{dlambda}=1,frac{dX}{dlambda}=0$), so that we get the nice translation symmetry we want:



$$frac{dY}{dlambda}=frac{dY}{dx}frac{dx}{dlambda}+frac{dY}{dy}frac{dy}{dlambda}=Y_xxi+Y_yeta=1$$



$$frac{dX}{dlambda}=frac{dX}{dx}frac{dx}{dlambda}+frac{dX}{dy}frac{dy}{dlambda}=X_xxi+X_yeta=0$$



We also require that the symmetry condition be satisfied:



$$frac{dY}{dX}=frac{D_xY}{D_xX}=frac{Y_x+Y_y y'}{X_x+X_y y'}=h(X,Y)$$



Here's where I'm stuck. Basically all the terms in the above equations are unknowns, and it seems incredibly difficult to solve for the new coordinates $(X,Y)$. I've also seen the "linearized symmetry condition" below, but it still results in a difficult-to-solve equation:



$$eta_x-xi_y h^2 +(eta_y -xi_x)h - (xi h_x + eta h_y) = 0$$



Is there a nice, algorithmic approach for deducing a set of canonical coordinates? Or does it ultimately come down to "guess and check" with the above equation?










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    0












    $begingroup$


    I've been trying to understand how Lie Groups can help solve differential equations (this 12-page pdf is the most straightforward explanation I've seen).



    My understanding is that when a first-order differential equation $frac{dy}{dx}=h(x,y)$ has a continuous "translation symmetry" $(x,y)mapsto(x,y+lambda)$, it becomes very easy to solve the equation using basic integration of one variable.



    The problem is that most differential equations in standard $(x,y)$ coordinates don't have this nice translation symmetry, and so we need to change to a set of "canonical coordinates" $(X,Y)$ which do have the desired translation symmetry $(X,Y)mapsto(X,Y+lambda)$.



    The main condition that these new canonical coordinates need to satisfy, is that the $Y$ coordinate should be aligned with the $lambda$ variable ($frac{dY}{dlambda}=1,frac{dX}{dlambda}=0$), so that we get the nice translation symmetry we want:



    $$frac{dY}{dlambda}=frac{dY}{dx}frac{dx}{dlambda}+frac{dY}{dy}frac{dy}{dlambda}=Y_xxi+Y_yeta=1$$



    $$frac{dX}{dlambda}=frac{dX}{dx}frac{dx}{dlambda}+frac{dX}{dy}frac{dy}{dlambda}=X_xxi+X_yeta=0$$



    We also require that the symmetry condition be satisfied:



    $$frac{dY}{dX}=frac{D_xY}{D_xX}=frac{Y_x+Y_y y'}{X_x+X_y y'}=h(X,Y)$$



    Here's where I'm stuck. Basically all the terms in the above equations are unknowns, and it seems incredibly difficult to solve for the new coordinates $(X,Y)$. I've also seen the "linearized symmetry condition" below, but it still results in a difficult-to-solve equation:



    $$eta_x-xi_y h^2 +(eta_y -xi_x)h - (xi h_x + eta h_y) = 0$$



    Is there a nice, algorithmic approach for deducing a set of canonical coordinates? Or does it ultimately come down to "guess and check" with the above equation?










    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      I've been trying to understand how Lie Groups can help solve differential equations (this 12-page pdf is the most straightforward explanation I've seen).



      My understanding is that when a first-order differential equation $frac{dy}{dx}=h(x,y)$ has a continuous "translation symmetry" $(x,y)mapsto(x,y+lambda)$, it becomes very easy to solve the equation using basic integration of one variable.



      The problem is that most differential equations in standard $(x,y)$ coordinates don't have this nice translation symmetry, and so we need to change to a set of "canonical coordinates" $(X,Y)$ which do have the desired translation symmetry $(X,Y)mapsto(X,Y+lambda)$.



      The main condition that these new canonical coordinates need to satisfy, is that the $Y$ coordinate should be aligned with the $lambda$ variable ($frac{dY}{dlambda}=1,frac{dX}{dlambda}=0$), so that we get the nice translation symmetry we want:



      $$frac{dY}{dlambda}=frac{dY}{dx}frac{dx}{dlambda}+frac{dY}{dy}frac{dy}{dlambda}=Y_xxi+Y_yeta=1$$



      $$frac{dX}{dlambda}=frac{dX}{dx}frac{dx}{dlambda}+frac{dX}{dy}frac{dy}{dlambda}=X_xxi+X_yeta=0$$



      We also require that the symmetry condition be satisfied:



      $$frac{dY}{dX}=frac{D_xY}{D_xX}=frac{Y_x+Y_y y'}{X_x+X_y y'}=h(X,Y)$$



      Here's where I'm stuck. Basically all the terms in the above equations are unknowns, and it seems incredibly difficult to solve for the new coordinates $(X,Y)$. I've also seen the "linearized symmetry condition" below, but it still results in a difficult-to-solve equation:



      $$eta_x-xi_y h^2 +(eta_y -xi_x)h - (xi h_x + eta h_y) = 0$$



      Is there a nice, algorithmic approach for deducing a set of canonical coordinates? Or does it ultimately come down to "guess and check" with the above equation?










      share|cite|improve this question









      $endgroup$




      I've been trying to understand how Lie Groups can help solve differential equations (this 12-page pdf is the most straightforward explanation I've seen).



      My understanding is that when a first-order differential equation $frac{dy}{dx}=h(x,y)$ has a continuous "translation symmetry" $(x,y)mapsto(x,y+lambda)$, it becomes very easy to solve the equation using basic integration of one variable.



      The problem is that most differential equations in standard $(x,y)$ coordinates don't have this nice translation symmetry, and so we need to change to a set of "canonical coordinates" $(X,Y)$ which do have the desired translation symmetry $(X,Y)mapsto(X,Y+lambda)$.



      The main condition that these new canonical coordinates need to satisfy, is that the $Y$ coordinate should be aligned with the $lambda$ variable ($frac{dY}{dlambda}=1,frac{dX}{dlambda}=0$), so that we get the nice translation symmetry we want:



      $$frac{dY}{dlambda}=frac{dY}{dx}frac{dx}{dlambda}+frac{dY}{dy}frac{dy}{dlambda}=Y_xxi+Y_yeta=1$$



      $$frac{dX}{dlambda}=frac{dX}{dx}frac{dx}{dlambda}+frac{dX}{dy}frac{dy}{dlambda}=X_xxi+X_yeta=0$$



      We also require that the symmetry condition be satisfied:



      $$frac{dY}{dX}=frac{D_xY}{D_xX}=frac{Y_x+Y_y y'}{X_x+X_y y'}=h(X,Y)$$



      Here's where I'm stuck. Basically all the terms in the above equations are unknowns, and it seems incredibly difficult to solve for the new coordinates $(X,Y)$. I've also seen the "linearized symmetry condition" below, but it still results in a difficult-to-solve equation:



      $$eta_x-xi_y h^2 +(eta_y -xi_x)h - (xi h_x + eta h_y) = 0$$



      Is there a nice, algorithmic approach for deducing a set of canonical coordinates? Or does it ultimately come down to "guess and check" with the above equation?







      ordinary-differential-equations lie-groups






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      asked Nov 24 '18 at 22:52









      eigenchriseigenchris

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