process then reverses. The energy oscillates between the two pendulums repeatedly until it is lost to friction. The behavior of this system can be understood by looking at its normal modes of oscillation. If the two pendulums are identical then one normal mode consists of both pendulums swinging in the same direction with a constant distance between them, while the other consists of the pendulums swinging in opposite (mirror image) directions. These normal modes have (slightly) different frequencies because the second involves the (weak) spring while the first does not. The initial state of the two-pendulum system is a combination of both normal modes. Over time, these normal modes drift out of phase, and this is seen as a transfer of motion from the first pendulum to the second. The description of the system in terms of the two pendulums is analogous to the flavor basis of neutrinos. These are the parameters that are most easily produced and detected (in the case of neutrinos, by weak interactions involving the W boson). The description in terms of normal modes is analogous to the mass basis of neutrinos. These modes do not interact with each other when the system is free of outside influence. When the pendulums are not identical the analysis is slightly more complicated. In the small-angle approximation, the potential energy of a single pendulum system is , where g is the standard gravity, L is the length of the pendulum, m is the mass of the pendulum, and x is the horizontal displacement of the pendulum. As an isolated system the pendulum is a harmonic oscillator with a frequency of . The potential energy of a spring is where k is the spring constant and x is the displacement. With a mass attached it oscillates with a period of . With two pendulums (labeled a and b) of equal mass but possibly unequal lengths and connected by a spring, the total potential energy is This is a quadratic form in xa and xb, which can also be written as a matrix product: The 2×2 matrix is real symmetric and so (by the spectral theorem) it is orthogonally diagonalizable. That is, there is an angle θ such that if we define then where λ1 and λ2 are the eigenvalues of the matrix. The variables x1 and x2 describe normal modes which oscillate with frequencies of and . When the two