N Oscillators and Normal Coordinates

The Idea of Decoupling

Imagine a row of balls connected by springs. If you push one ball, the motion propagates down the line, but the motion is messy; every ball moves at different times and amplitudes. However, there are special patterns of motion where every ball moves in unison, just with different amplitudes. In these special patterns, the internal forces between balls do no net work to change the shape of the motion; they only provide the restoring force. These special patterns are the normal modes. The key insight is that any arbitrary motion of the system can be viewed as a superposition of these independent modes. Instead of tracking the position of every single ball, we track the "amount" of each mode present. These amounts are the normal coordinates. They are independent variables, meaning the equation for one normal coordinate does not contain any other normal coordinate. This is the essence of diagonalization: we are rotating our coordinate system to align with the natural axes of the system's energy landscape.

Quadratic Forms and Diagonalization

For small oscillations about a stable equilibrium, the kinetic energy \(T\) and potential energy \(V\) can be expressed as quadratic forms in the generalized coordinates \(\mathbf q\). Let \(\mathbf q\) be a column vector of \(N\) coordinates. The kinetic energy is given by \(T = \frac{1}{2} \dot{\mathbf q}^{\mathsf T} \mathbf M \dot{\mathbf q}\), where \(\mathbf M\) is the symmetric, positive-definite mass matrix. The potential energy is \(V = \frac{1}{2} \mathbf q^{\mathsf T} \mathbf K \mathbf q\), where \(\mathbf K\) is the symmetric, positive-semidefinite stiffness matrix. The equations of motion are derived from the Lagrangian \(L = T - V\), leading to the matrix equation \(\mathbf M \ddot{\mathbf q} + \mathbf K \mathbf q = \mathbf 0\).

To decouple this system, we seek a linear transformation \(\mathbf q = \mathbf A \boldsymbol\eta\), where \(\boldsymbol\eta\) is the vector of normal coordinates and \(\mathbf A\) is a constant transformation matrix. We choose \(\mathbf A\) such that it simultaneously diagonalizes both \(\mathbf M\) and \(\mathbf K\). Specifically, we require \(\mathbf A^{\mathsf T} \mathbf M \mathbf A = \mathbf I\) (the identity matrix) and \(\mathbf A^{\mathsf T} \mathbf K \mathbf A = \boldsymbol\Omega^2\), where \(\boldsymbol\Omega^2\) is a diagonal matrix containing the squared normal frequencies \(\omega_r^2\). This is possible because \(\mathbf M\) and \(\mathbf K\) are real symmetric matrices, and \(\mathbf M\) is positive-definite, allowing us to define an inner product in which \(\mathbf K\) is symmetric and thus diagonalizable by an orthogonal transformation.

Substituting \(\mathbf q = \mathbf A \boldsymbol\eta\) into the equations of motion and premultiplying by \(\mathbf A^{\mathsf T}\), we obtain \(\ddot{\boldsymbol\eta} + \boldsymbol\Omega^2 \boldsymbol\eta = \mathbf 0\). Since \(\boldsymbol\Omega^2\) is diagonal, this single matrix equation splits into \(N\) independent scalar equations: \(\ddot{\eta}_r + \omega_r^2 \eta_r = 0\) for \(r = 1, \dots, N\).

The Eigenvalue Problem

The normal frequencies \(\omega_r\) and the mode shapes (columns of \(\mathbf A\)) are determined by the generalized eigenvalue problem. Assuming a solution of the form \(\mathbf q(t) = \mathbf a e^{i\omega t}\), we substitute this into the equation of motion \(\mathbf M \ddot{\mathbf q} + \mathbf K \mathbf q = \mathbf 0\) to get \((\mathbf K - \omega^2 \mathbf M) \mathbf a = \mathbf 0\). For a non-trivial solution \(\mathbf a \neq \mathbf 0\) to exist, the determinant of the coefficient matrix must vanish:

This is a polynomial equation of degree \(N\) in \(\omega^2\). The roots \(\omega_1^2, \dots, \omega_N^2\) are the squared normal frequencies. For each root \(\omega_r^2\), we solve for the corresponding eigenvector \(\mathbf a_r\). These eigenvectors define the relative displacements of the masses in the \(r\)-th normal mode. A crucial property of this system is that eigenvectors corresponding to distinct eigenvalues are orthogonal with respect to the mass matrix \(\mathbf M\). That is, \(\mathbf a_r^{\mathsf T} \mathbf M \mathbf a_s = 0\) for \(r \neq s\). We can normalize them such that \(\mathbf a_r^{\mathsf T} \mathbf M \mathbf a_s = \delta_{rs}\), where \(\delta_{rs}\) is the Kronecker delta. This orthogonality ensures that the normal coordinates are independent and that energy does not transfer between modes in the absence of damping or driving.

General Solution and Superposition

Once the normal coordinates are identified, the general solution for the displacement of the system is a superposition of the normal modes. The displacement vector \(\mathbf q(t)\) is given by:

where each normal coordinate \(\eta_r(t)\) evolves independently as a simple harmonic oscillator: \(\eta_r(t) = C_r \cos(\omega_r t + \phi_r)\). The constants \(C_r\) and \(\phi_r\) are determined by the initial conditions of the system. The vector \(\mathbf a_r\) is the \(r\)-th eigenvector, normalized such that \(\mathbf a_r^{\mathsf T} \mathbf M \mathbf a_r = 1\). This decomposition is unique. It allows us to analyze complex motions by breaking them down into their fundamental building blocks. For example, if the system is initially displaced in a way that matches a specific eigenvector, only that mode will be excited. If the initial displacement is arbitrary, multiple modes will be excited, and the resulting motion will be a complex combination of sinusoidal oscillations at different frequencies.