Normal Modes

It follows from Equation (783) and (784), plus the mathematical results contained in the previous section, that the most general solution to Equation (782) can be written

$$\delta {\bf q}(t) = \sum_{k=1,{\cal F}}\delta {\bf q}_k(t),$$ (800)

where

$$\delta {\bf q}_k(t) = \left(\alpha_k \,{\rm e}^{+\sqrt{\lamb... ...,t}+\beta_k\,{\rm e}^{-\sqrt{\lambda_k}\,t}\right) {\bf x}_k.$$ (801)

Here, the $\lambda_k$ and the ${\bf x}_k$ are the eigenvalues and eigenvectors obtained by solving Equation (786). Moreover, the $\alpha_k$ and $\beta_k$ are arbitrary constants. Finally, we have made use of the fact that the two roots of $\gamma^2=\lambda_k$ are $\gamma=\pm \sqrt{\lambda}_k$.

According to Equation (800), the most general perturbed motion of the system consists of a linear combination of ${\cal F}$ different modes. These modes are generally termed normal modes, since they are mutually orthogonal (because the ${\bf x}_k$ are mutually orthogonal). Furthermore, it follows from the independence of the ${\bf x}_k$ that the normal modes are also independent (i.e., one mode cannot be expressed as a linear combination of the others). The $k$th normal mode has a specific pattern of motion which is specified by the $k$th eigenvector, ${\bf x}_k$. Moreover, the $k$th mode has a specific time variation which is determined by the associated eigenvalue, $\lambda_k$. Recall that $\lambda_k$ is real. Hence, there are only two possibilities. Either $\lambda_k$ is positive, in which case we can write

$$\delta {\bf q}_k(t) = \left(\alpha_k \,{\rm e}^{+\gamma_k\,t}+\beta_k\,{\rm e}^{-\gamma_k\,t}\right) {\bf x}_k,$$ (802)

where $\lambda_k = \gamma_k^{\,2}$, or $\lambda_k$ is negative, in which case we can write

$$\delta {\bf q}_k(t) = \left(\alpha_k \,{\rm e}^{+{\rm i}\,\o... ...,t}+\beta_k\,{\rm e}^{-{\rm i}\,\omega_k\,t}\right) {\bf x}_k,$$ (803)

where $\lambda_k = -\omega_k^{\,2}$. In other words, if $\lambda_k$ is positive then the $k$th normal mode grows or decays secularly in time, whereas if $\lambda_k$ is negative then the $k$th normal mode oscillates in time. Obviously, if the system possesses one or more normal modes which grow secularly in time then the equilibrium about which we originally expanded the equations of motion must be an unstable equilibrium. On the other hand, if all of the normal modes oscillate in time then the equilibrium is stable. Thus, we conclude that whilst Equation (773) is the condition for the existence of an equilibrium state in a many degree of freedom system, the condition for the equilibrium to be stable is that all of the eigenvalues of the stability equation (786) must be negative.

The arbitrary constants $\alpha_k$ and $\beta_k$ appearing in expression (801) are determined from the initial conditions. Thus, if $\delta {\bf q}^{(0)} = \delta {\bf q}(t=0)$ and $\delta \dot{\bf q}^{(0)} = \delta \dot{\bf q}(t=0)$ then it is easily demonstrated from Equations (799)-(801) that

$${\bf x}_k^{T}\,{\bf M}\,\delta {\bf q}^{(0)} = \alpha_k + \beta_k,$$ (804)

and

$${\bf x}_k^{T}\,{\bf M}\,\delta \dot{\bf q}^{(0)} =\sqrt{\lambda}_k\,(\alpha_k - \beta_k).$$ (805)

Hence,

$$\alpha_k = \frac{{\bf x}_k^{T}\,{\bf M}\,\delta{\bf q}^{(0)} +{\bf x}_k^{T}\,{\bf M}\,\delta \dot{\bf q}^{(0)}/\sqrt{\lambda}_k}{2},$$ (806)

$$\beta_k = \frac{{\bf x}_k^{T}\,{\bf M}\,\delta {\bf q}^{(0)} -{\bf x}_k^{T}\,{\bf M}\,\delta \dot{\bf q}^{(0)}/\sqrt{\lambda}_k}{2}.$$ (807)

Note, finally, that since there are $2{\cal F}$ arbitrary constants (two for each of the ${\cal F}$ normal modes), we can be sure that Equation (800) represents the most general solution to Equation (782).