- #1
guest1234
- 41
- 1
I have to derive equations of motion from Lagrangian and stumbled upon the following system of equations (constants are simplified, that information is unneeded)
[itex]
\begin{cases}
\ddot{x}-A\dot{y}+Bx=0 \\
\ddot{y}+A\dot{x}+Dy=0
\end{cases}
[/itex]
This is an extension of a simpler problem where B=D. There I just multiplied the second equation by i, added equations together, substituted z=x+iy and solved for z.
But doing the same with current system doesn't help much (can't substitute z=x+iy, b/c [itex]B\ne{D}[/itex]).
When choosing E=(B+D)/2 and F=E-B then
[itex]
\begin{cases}
\ddot{x}-A\dot{y}+Ex=Fx\\
\ddot{y}+A\dot{x}+Ey=-Fy
\end{cases}
[/itex]
isn't much help either.
I know there are some techniques for solving coupled differential equations like writing a reciprocal matrix for the system but it seems that it applies only to 1st order ODEs.
Is there any analytic solutions to this?
When punching it into Maple, it throws a huge block of square roots, although simpler problem gave me a combination of exponentials (for z). I know this problem must give similar answer with little modifications, but don't know how to tackle it.
Help and pointers much appreciated
[itex]
\begin{cases}
\ddot{x}-A\dot{y}+Bx=0 \\
\ddot{y}+A\dot{x}+Dy=0
\end{cases}
[/itex]
This is an extension of a simpler problem where B=D. There I just multiplied the second equation by i, added equations together, substituted z=x+iy and solved for z.
But doing the same with current system doesn't help much (can't substitute z=x+iy, b/c [itex]B\ne{D}[/itex]).
When choosing E=(B+D)/2 and F=E-B then
[itex]
\begin{cases}
\ddot{x}-A\dot{y}+Ex=Fx\\
\ddot{y}+A\dot{x}+Ey=-Fy
\end{cases}
[/itex]
isn't much help either.
I know there are some techniques for solving coupled differential equations like writing a reciprocal matrix for the system but it seems that it applies only to 1st order ODEs.
Is there any analytic solutions to this?
When punching it into Maple, it throws a huge block of square roots, although simpler problem gave me a combination of exponentials (for z). I know this problem must give similar answer with little modifications, but don't know how to tackle it.
Help and pointers much appreciated