Finding the constants of the general approximate solution to a double pendulum

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The discussion revolves around solving a system of equations related to the general approximate solution of a double pendulum. The user is attempting to find constants A1, A2, δ1, and δ2 by applying initial conditions to their derived equations. They present a general solution involving trigonometric functions and differentiate to obtain velocity equations. The user expresses confusion about whether to solve the equations simultaneously or individually. The thread highlights the importance of properly formatting LaTeX for clarity in mathematical discussions.
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Homework Statement
Given the (approximate) general solution to the double pendulum, find the constant $A_1$, $A_2$, $\delta_1$ and $\delta_2$ by stating some initial conditions for the double pendulum.
Relevant Equations
$\phi(t) = A_1 \begin{pmatrix} 1 \\ \sqrt{2} \end{pmatrix} \cos(\omega_1 t - \delta_1) + A_2 \begin{pmatrix} 1 \\ -\sqrt{2} \end{pmatrix} \cos(\omega_2 t - \delta_2)
$
EDIT: My Latex is not showing... Sorry. I attached a file with my "solution".

I though this would be quite easy, but I can't seem to solve this system of equations. Should I solve for each mode, or both of them together? I tried to solve them together, here's how far I get:

double.png

$\text{General solution:} \\

\phi(t)=A_1\binom{1}{\sqrt{2}}\cos\left(\omega_1t-\delta_1\right)+A_2\binom{1}{-\sqrt{2}}\cos\left(\omega_2t-\delta_2\right)

\text{Expanding the components, we get} \\

\phi_1(t)=A_1\cos\left(\omega_1t-\delta_1\right)+A_2\cos\left(\omega_2t-\delta_2\right) \\
\phi_2(t)=\sqrt{2}A_1\cos\left(\omega_1t-\delta_1\right)-\sqrt{2}A_2\cos\left(\omega_2t-\delta_2\right)

\text{Neglecting the amplitudes, this can be written as} \\

\phi_1(t)=A_1\cos\left(\omega_1t-\delta_1\right)+A_2\cos\left(\omega_2t-\delta_2\right) \\
\phi_2(t)=A_1\cos\left(\omega_1t-\delta_1\right)-A_2\cos\left(\omega_2t-\delta_2\right)

\text{Differentiating, we obtain} \\

\dot{\phi}_1(t)=-\omega_1A_1\sin\left(\omega_1t-\delta_1\right)-\omega_2A_2\sin\left(\omega_2t-\delta_2\right) \\
\dot{\phi}_2(t)=-\omega_1A_1\sin\left(\omega_1t-\delta_1\right)+\omega_2A_2\sin\left(\omega_2t-\delta_2\right)

\text{To find the particular solution, we need to determine } A_1, A_2, \delta_1, \text{ and } \delta_2. \text{ Choosing the initial conditions} \\

\phi_1(0)=\phi_2(0)=\frac{\pi}{6} \\
\dot{\phi}_1(0)=\dot{\phi}_2(0)=0

\text{We obtain a system of equations with four equations and four unknowns:} \\

A_1\cos(\delta_1)+A_2\cos(\delta_2)=\frac{\pi}{6} \\
A_1\cos(\delta_1)-A_2\cos(\delta_2)=\frac{\pi}{6} \\
-\omega_1A_1\sin(\delta_1)-\omega_2A_2\sin(\delta_2)=0 \\
-\omega_1A_1\sin(\delta_1)+\omega_2A_2\sin(\delta_2)=0

\text{Since } \cos(-x)=\cos(x) \text{ and } \sin(-x)=-\sin(x), \text{ we can rewrite this as} \\

A_1\cos(\delta_1)+A_2\cos(\delta_2)=\frac{\pi}{6}\ (1) \\
A_1\cos(\delta_1)-A_2\cos(\delta_2)=\frac{\pi}{6}\ (2) \\
\omega_1A_1\sin(\delta_1)+\omega_2A_2\sin(\delta_2)=0\ (3) \\
\omega_1A_1\sin(\delta_1)-\omega_2A_2\sin(\delta_2)=0\ (4)$
 
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