Normal Coordinates

Since the eigenvectors ${\bf x}_k$, for $k=1,{\cal F}$, span ${\cal F}$-dimensional vector space, we can always write the displacement vector $\delta {\bf q}$ as some linear combination of the ${\bf x}_k$: i.e.,

$$\delta {\bf q}(t) = \sum_{k=1,{\cal F}}\eta_k(t)\,{\bf x}_k.$$ (808)

We can regard the $\eta_k(t)$ as a new set of generalized coordinates, since specifying the $\eta_k$ is equivalent to specifying the $\delta q_k$ (and, hence, the $q_k$). The $\eta_k$ are usually termed normal coordinates. According to Equations (799) and (808), the normal coordinates can be written in terms of the $\delta q_k$ as

$$\eta_k = {\bf x}_k^T\,{\bf M}\,\delta{\bf q}.$$ (809)

Let us now try to express $K$, $U$, and the equations of motion in terms of the $\eta_k$.

The kinetic energy can be written

$$K = \frac{1}{2}\,\delta \dot{\bf q}^T\,{\bf M}\,\delta\dot{\bf q},$$ (810)

where use has been made of Equation (775). It follows from (808) that

$$K = \frac{1}{2}\sum_{k,l=1,{\cal F}} \dot{\eta}_k\,\dot{\eta}_l\,{\bf x}_k^T\,{\bf M}\,{\bf x}_l.$$ (811)

Finally, making use of the orthonormality condition (799), we obtain

$$K = \frac{1}{2}\sum_{k=1,{\cal F}}\dot{\eta}_k^{\,2}.$$ (812)

Hence, the kinetic energy $K$ takes the form of a diagonal quadratic form when expressed in terms of the normal coordinates.

The potential energy can be written

$$U=- \frac{1}{2}\,\delta {\bf q}^T\,{\bf G}\,\delta{\bf q},$$ (813)

where use has been made of Equations (780). It follows from (808) that

$$U =- \frac{1}{2}\sum_{k,l=1,{\cal F}} \eta_k\,\eta_l\,{\bf x}_k^T\,{\bf G}\,{\bf x}_l.$$ (814)

Finally, making use of Equation (788) and the orthonormality condition (799), we obtain

$$U = -\frac{1}{2}\sum_{k=1,{\cal F}}\lambda_k\, \eta_k^{\,2}.$$ (815)

Hence, the potential energy $U$ also takes the form of a diagonal quadratic form when expressed in terms of the normal coordinates.

Writing Lagrange's equations of motion in terms of the normal coordinates, we obtain [cf., Equation (772)]

$$\frac{d}{dt}\!\left(\frac{\partial K}{\partial \dot{\eta}_k}... ...l K}{\partial \eta_k}+ \frac{\partial U}{\partial \eta_k} = 0,$$ (816)

for $k=1,{\cal F}$. Thus, it follows from Equations (812) and (815) that

$$\ddot{\eta}_k = \lambda_k\,\eta_k,$$ (817)

for $k=1,{\cal F}$. In other words, Lagrange's equations reduce to a set of ${\cal F}$ uncoupled simple harmonic equations when expressed in terms of the normal coordinates. The solutions to the above equations are obvious: i.e.,

$$\eta_k(t) = \alpha_k \,{\rm e}^{+\sqrt{\lambda_k}\,t}+\beta_k\,{\rm e}^{-\sqrt{\lambda_k}\,t},$$ (818)

where $\alpha_k$ and $\beta_k$ are arbitrary constants. Hence, it is clear from Equations (808) and (818) that the most general solution to the perturbed equations of motion is indeed given by Equations (800) and (801).

In conclusion, the equations of motion of a many degree of freedom dynamical system which is slightly perturbed from an equilibrium state take a particularly simple form when expressed in terms of the normal coordinates. Each normal coordinate specifies the instantaneous displacement of an independent mode of oscillation (or secular growth) of the system. Moreover, each normal coordinate oscillates at a characteristic frequency (or grows at a characteristic rate), and is completely unaffected by the other coordinates.