# 6.2: N Coupled Oscillators

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The calculations of the previous section may be readily generalized to the case of an arbitrary number (say, $$N$$ ) coupled harmonic oscillators, with an arbitrary type of coupling. It is evident that in this case Eq. (4) should be replaced with $L=\sum_{j=1}^{N} L_{j}+\sum_{j, j^{\prime}=1}^{N} L_{j j^{\prime}}$ Moreover, we can generalize the above expression for the mixed terms $$L_{j j}$$, taking into account their possible dependence not only on the generalized coordinates but also on the generalized velocities, in a bilinear form similar to Eq. (4). The resulting Lagrangian may be represented in a compact form, $L=\sum_{j, j^{\prime}=1}^{N}\left(\frac{m_{i j^{\prime}}}{2} \dot{q}_{j} \dot{q}_{j^{\prime}}-\frac{\kappa_{i j^{\prime}}}{2} q_{j} q_{j^{\prime}}\right),$ where the off-diagonal terms are index-symmetric: $$m_{j j^{\prime}}=m_{j^{\prime} j}, \kappa_{j j^{\prime}}=\kappa_{j j}$$, and the factors $$1 / 2$$ compensate the double counting of each term with $$j \neq j$$ ’, taking place at the summation over two independently running indices. One may argue that Eq. (16) is quite general if we still want to keep the equations of motion linear - as they always are if the oscillations are small enough.

Plugging Eq. (16) into the general form (2.19) of the Lagrange equation, we get $$N$$ equations of motion of the system, one for each value of the index $$j^{\prime}=1,2, \ldots, N:$$ $\sum_{j=1}^{N}\left(m_{i j j} \ddot{q}_{j}+\kappa_{j j^{\prime}} q_{j}\right)=0 .$ Just as in the previous section, let us look for a particular solution to this system in the form $q_{j}=c_{j} e^{\lambda t} .$ As a result, we are getting a system of $$N$$ linear, homogeneous algebraic equations, $\sum_{j=1}^{N}\left(m_{i j^{\prime}} \lambda^{2}+\kappa_{j j^{\prime}}\right) c_{j}=0,$ for the set of $$N$$ distribution coefficients $$c_{j}$$. The condition that this system is self-consistent is that the determinant of its matrix equals zero: $\operatorname{Det}\left(m_{i j^{\prime}} \lambda^{2}+\kappa_{i j^{\prime}}\right)=0 .$ This characteristic equation is an algebraic equation of degree $$N$$ for $$\lambda^{2}$$, and so has $$N$$ roots $$\left(\lambda^{2}\right)_{n}$$. For any Hamiltonian system with stable equilibrium, the matrices $$m_{i j}$$ ’ and $$\kappa_{i j}$$ ’ ensure that all these roots are real and negative. As a result, the general solution to Eq. (17) is the sum of $$2 N$$ terms proportional to exp $$\left\{\pm i \omega_{n} t\right\}, n=1,2, \ldots, N$$, where all $$N$$ eigenfrequencies $$\omega_{n}$$ are real.

Plugging each of these $$2 N$$ values of $$\lambda=\pm i \omega_{n}$$ back into a particular set of linear equations (17), one can find the corresponding set of distribution coefficients $$c_{j \pm}$$. Generally, the coefficients are complex, but to keep $$q_{j}(t)$$ real, the coefficients $$c_{j+}$$ corresponding to $$\lambda=+i \omega_{n}$$, and $$c_{j}$$ - corresponding to $$\lambda$$ $$=-i \omega_{n}$$ have to be complex-conjugate of each other. Since the sets of the distribution coefficients may be different for each $$\lambda_{n}$$, they should be marked with two indices, $$j$$ and $$n$$. Thus, at general initial conditions, the time evolution of the $$j^{\text {th }}$$ coordinate may be represented as $q_{j}=\frac{1}{2} \sum_{n=1}^{N}\left(c_{j n} \exp \left\{+i \omega_{n} t\right\}+c_{j n}^{*} \exp \left\{-i \omega_{n} t\right\}\right) \equiv \operatorname{Re} \sum_{n=1}^{N} c_{j n} \exp \left\{i \omega_{n} t\right\}$ This formula shows very clearly again the physical sense of the distribution coefficients $$c_{j n}$$ : a set of these coefficients, with different values of index $$j$$ but the same $$n$$, gives the complex amplitudes of oscillations of the coordinates for the special initial conditions that ensure purely sinusoidal motion of all the system, with frequency $$\omega_{n}$$.

The calculation of the eigenfrequencies and distribution coefficients of a particular coupled system with many degrees of freedom from Eqs. (19)-(20) is a task that frequently may be only done numerically. $${ }^{4}$$ Let us discuss just two particular but very important cases. First, let all the coupling coefficients be small in the following sense: $$\left|m_{j j^{\prime}}\right|<<m_{j} \equiv m_{j j}$$ and $$\left|\kappa_{i j^{\prime}}\right|<<\kappa_{j} \equiv \kappa_{j j}$$, for all $$j \neq j$$, and all partial frequencies $$\Omega_{j} \equiv\left(\kappa_{j} / m_{j}\right)^{1 / 2}$$ be not too close to each other:

$\ \frac{\left|\Omega_{j}^{2}-\Omega_{j^{\prime}}^{2}\right|}{\Omega_{j}^{2}}>>\frac{\left|\kappa_{j j^{\prime}}\right|}{\kappa_{j}}, \frac{\left|m_{j j^{\prime}}\right|}{m_{j}}, \quad \text { for all } j \neq j^{\prime}.$

(Such situation frequently happens if parameters of the system are "random" in the sense that they do not follow any special, simple rule - for example, resulting from some simple symmetry of the system.) Results of the previous section imply that in this case, the coupling does not produce a noticeable change of oscillation frequencies: $$\left\{\omega_{n}\right\} \approx\left\{\Omega_{j}\right\}$$. In this situation, oscillations at each eigenfrequency are heavily concentrated in one degree of freedom, i.e. in each set of the distribution coefficients $$c_{j n}$$ (for a given $$n$$ ), one coefficient’s magnitude is much larger than all others.

Now let the conditions (22) be valid for all but one pair of partial frequencies, say $$\Omega_{1}$$ and $$\Omega_{2}$$, while these two frequencies are so close that coupling of the corresponding partial oscillators becomes essential. In this case, the approximation $$\left\{\omega_{n}\right\} \approx\left\{\Omega_{j}\right\}$$ is still valid for all other degrees of freedom, and the corresponding terms may be neglected in Eqs. (19) for $$j=1$$ and 2 . As a result, we return to Eqs. (7) (perhaps generalized for velocity coupling) and hence to the anticrossing diagram (Figure 2) discussed in the previous section. As a result, an extended change of only one partial frequency (say, $$\Omega_{1}$$ ) of a weakly coupled system produces a sequence of eigenfrequency anticrossings - see Figure 3 .

$${ }^{4}$$ Fortunately, very effective algorithms have been developed for this matrix diagonalization task - see, e.g., references in MA Sec. 16(iii)-(iv). For example, the popular MATLAB software package was initially created exactly for this purpose. ("MAT" in its name stands for "matrix" rather than "mathematics".)

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