We examine the nature of the stationary character of the Hamilton action $S$ for a space-time trajectory (worldline) $x(t)$ of a single particle moving in one dimension with a general time-dependent potential energy function $U(x,t)$. We show that the action is a local minimum for sufficiently short worldlines for all potentials and for worldlines of any length in some potentials. For long enough worldlines in most time-independent potentials $U(x)$, the action is a saddle point, that is, a minimum with respect to some nearby alternative curves and a maximum with respect to others. The action is never a true maximum, that is, it is never greater along the actual worldline than along every nearby alternative curve. We illustrate these results for the harmonic oscillator, two different nonlinear oscillators, and a scattering system. We also briefly discuss two-dimensional examples, the Maupertuis action, and newer action principles.

## REFERENCES

*ibid*, p.

Systems with subsequent kinetic foci are discussed in Secs. VIII and IX. For examples with only a single kinetic focus, see Figs. 6 and 7.

Better estimates can be found using the Sturm and Sturm-Liouville theories; see Papastavridis, Refs. 26 and 66.

There are other systems for which $\u22023U\u2215\u2202x3$, etc., vanish, for example, $U(x)=C$, $U(x)=Cx$, $U(x)=\u2212Cx2$. For these systems the worldlines cannot have $\delta 2S=0$ [see Eq. (19)], so that kinetic foci do not exist.

The fact that $\delta 2S0\u21920$ for $\alpha \u21920$ is not surprising because $\delta 2S0$ is proportional to $\alpha 2$. The surprising fact is that here $\delta 2S0$ vanishes as $\alpha 3$ as $\alpha \u21920$ because the integral involved in the definition, Eq. (36a), of $\delta 2S0$ is itself $O(\alpha )$. We can see directly that $\delta 2S0$ becomes $O(\alpha 3)$ for $R$ near $Q$ for this special variation $\alpha \varphi $ by integrating the $\varphi \u0307$ term by parts in Eq. (36a) and using $\varphi =0$ at the end-points. The result is $\delta 2S0=\u2212(\alpha 2\u22152)\u222bPR\varphi [m\varphi \u0308+U\u2033(x0)\varphi ]dt$. Because $x0$ and $x1=x0+\alpha \varphi $ are both true worldlines, we can apply the equation of motion $mx\u0308+U\u2032(x)=0$ to both. We then subtract these two equations of motion and expand $U\u2032(x0+\alpha \varphi )$ as $U\u2032(x0)+U\u2033(x0)\alpha \varphi +(12)U\u2034(x0)(\alpha \varphi )2+O(\alpha 3)$, giving $m\varphi \u0308+U\u2033(x0)\varphi =\u2212(12)U\u2034(x0)\alpha \varphi 2+O(\alpha 2)$. (If the $\varphi 2$, etc., nonlinear terms on the right-hand side are neglected in the last equation, it becomes the Jacobi-Poincaré linear variation equation used in stability studies.) If we use this result in the previous expression for $\delta 2S0$, we find to lowest nonvanishing order $\delta 2S0=(\alpha 3\u22154)\u222bPRdtU\u2034(x0)\varphi 3$, which is $O(\alpha 3)$. This result and Eq. (40a) for $\delta 3S0$ give the desired result (42) for $S1\u2212S0$.

If we use arguments similar to those of this section and Sec. VI, we can show that $\delta 2S$ vanishes again at the second kinetic focus $Q2$, and that for $R$ beyond $Q2$ the wordline $PR$ has a second, independent variation leading to $\delta 2S<0$, in agreement with Morse’s general theory (Ref. 34).

If we use $L(x,x\u0307)=px\u0307\u2212H(x,p)$, we can rewrite the Hamilton action as a phase-space integral, that is, $S=\u222bPR[px\u0307\u2212H(x,p)]dt$. We set $\delta S=0$ and vary $x(t)$ and $p(t)$ independently and find (Ref. 63) the Hamilton equations of motion $x\u0307=\u2202H\u2215\u2202p$, $p\u0307=\u2212\u2202H\u2215\u2202x$. We can then show (Ref. 64) that in phase space, the trajectories $x(t)$, $p(t)$ that satisfy the Hamilton equations are always saddle points of $S$, that is, never a true maximum or a true minimum. In the proof it is assumed that $H$ has the normal form $H(x,p)=p2\u22152m+U(x)$.

Nearly pure quartic potentials have been found in molecular physics for ring-puckering vibrational modes (Ref. 75) and for the caged motion of the potassium ion $K+$ in the endohedral fullerene complex $K+@C60$ (Ref. 76), where the quadratic terms in the potential are small. Ferroelectric soft modes in solids are also sometimes approximately represented by quartic potentials (Refs. 77 and 78).

The converse effect cannot occur: a time-dependent potential $U(x,t)$ with $U\u2033<0$ at all times always has $\delta 2S>0$ as seen from Eq. (19). If $U(x,t)$ is such that $U\u2033$ alternates in sign with time, kinetic foci (and hence trajectory stability) may occur. An example is a pendulum with a rapidly vertically oscillating support point. In effect the gravitational field is oscillating. The pendulum can oscillate stably about the (normally unstable) upward vertical direction (Ref. 85). Two- and three-dimensional examples of this type are Paul traps (Ref. 86) and quadrupole mass filters (Ref. 85), which use oscillating quadrupole electric fields to trap ions. The equilibrium trajectory $x(t)=0$ at the center of the trap is unstable for purely electrostatic fields but is stabilized by using time-dependent electric fields. Focusing by alternating-gradients (also known as strong focusing) in particle accelerators and storage rings is based on the same idea (Ref. 42).

Note that for the actual 2D trajectories in the potential $U(x,y)=mgy$, kinetic foci exist for the spatial paths of the Maupertuis action $W$, but do not exist for the space-time trajectories of the Hamilton action $S$. This result illustrates the general result stated in Appendix A that the kinetic foci for $W$ and $S$ differ in general.

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