In order to decouple the equation of a chain of coupled harmonic oscillator I need to considerate the normal mode basis, here my Hamiltonian: $$H = \sum_{i=0}^{N-1}\frac{p_i^2}{2} + \frac{1}{2}\sum_{i=0}^{N-1}(q_{i+1} - q_i)^2$$ In my problem I have $N$ particles with periodic boundary conditions, so $q_0=q_N$.
First method:
Now I can rewrite $H$ as $H = \frac{1}{2}z^TCz$ where $z = (p_0,p_1,\dots,p_{N-1},q_0,q_1,\dots,q_{N-1})$ so $C$ is
$$C = \begin{bmatrix} 0 & A\\ 1 & 0 \end{bmatrix}$$
where $1$ is the identity and $A$ is the following one:
$$A = \begin{bmatrix} 2 & -1 & 0 & \dots & \dots & 0 & -1 \\ -1 & 2 & -1 & 0 & \dots & \dots & 0\\ 0 & -1 & 2 & -1 & 0 & \dots & 0\\ \vdots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots\\ 0 & \dots & 0 & -1 & 2 & -1 & 0\\ 0 & \dots & \dots & 0 & -1 & 2 & -1\\ -1 & 0 & \dots & \dots & 0 & -1 & 2 \\ \end{bmatrix}$$
Now in order to solve the problem I write down Hamilton equations
$$\dot{p_i}=-\frac{\partial{H}}{\partial{q_i}}=-2q_i+q_{i+1}+q_{i-1}$$ $$\dot{q_i}=\frac{\partial{H}}{\partial{p_i}}=p_i$$
So the problem to solve is given by
$$\dot{z}=JCz$$
where $J$ is a matrix given ($1$ is the identity matrix) by
$$J = \begin{bmatrix} 0 & 1\\ -1 & 0 \end{bmatrix}$$
To solve the problem I need to dyagonalize the $JC$ matrix:
$$M^{-1}(JC)M = \Lambda$$
And can be shown that the matrix $M$ is symplectic such that $M^TJM=J$.
Now how can I connect this with the normal modes? I think I understood that the normal modes are eigenvectors of the $JC$ matrix, am I supposed to know how to easily diagonalize this matrix with generic dimension? Where can I find infomation about how to diagonalize this matrix in generic dimension in order to obtain the eigenvectors?
On the other hand...
Secons method:
I only can find information online about how to obtain the normal modes with this method I'll show:
Given the Hamiltonian I can write the equation of motion:
$$\ddot{q}_j = -2q_j + q_{j+1} + q_{j-1}$$
Now I can make this ansatz: $q_l = A_le^{il\Omega}$, and if I insert this ansatz in the equation I obtain
$$-\Omega A_j = A_{i-1} - 2A_j + A_{j+1}$$
so if I suppose for $A_l = ae^{i\phi l}$ then $\Omega(\phi) = 2sin(\frac{\phi}{2})$
So $q_l = \mathfrak Re \left( a(\phi)e^{i(l\phi+\Omega(\phi)t)} \right)$
Now if I impose that $q_0 = q_{N}$ I obtain that $\phi = \frac{\pi k}{N}$ and $\Omega = 2\sin(\frac{\pi k}{2N})$, so I obtain:
$$q_l = \mathfrak Re \left(\sum_{k=0}^{N-1}ae^{i(\frac{\pi kl}{N} + \Omega t)} \right)$$
And this is the normal mode representation. I am totally confused.
How can I connect this method with the "algebric vision" that I mentioned before? I can't find anything that satisfies me online. The second method is somehow disconnected from the first for me, because I don't understand how my ansatz gave me the eigenvectors of the matrix. I don't know if I am totally wrong about this desire of link this two methods, If you know a book where I can follow a step by step indication to write my problem in normal modes with periodic boundary condition can you please indicate me the title? Thanks a lot.