^{1,a)}and Edwin F. Taylor

^{2,b)}

### Abstract

We examine the nature of the stationary character of the Hamilton action for a space-time trajectory (worldline) of a single particle moving in one dimension with a general time-dependent potential energy function . 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 , 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.

CGG thanks NSERC for financial support. EFT acknowledges James Reid-Cunningham for providing opportunities for extended thought, and Alar Toomre for tutoring in mathematics. Both authors thank Bernie Nickel for many suggestions, including pointing out the insufficiency of the Culverwell-Whittaker argument as shown in Sec. VI and providing the original analysis elaborated in Sec. VII. We also thank Adam Pound, Slavomir Tuleja, Don Lemons, Donald Sprung, and John Taylor for a careful reading of the manuscript and many useful suggestions. Brenda Law, Igor Tolokh, and Will M. Farr are thanked for their assistance in the production of this paper.

I. INTRODUCTION

II. KINETIC FOCUS

III. WHY WORLDLINES CROSS

IV. VARIATION OF ACTION FOR AN ADJACENT CURVE

V. WHEN THE ACTION IS A MINIMUM

VI. MINIMUM ACTION WHEN WORLDLINE TERMINATES BEFORE KINETIC FOCUS

VII. SADDLE POINT IN ACTION WHEN WORLDLINE TERMINATES BEYOND KINETIC FOCUS

VIII. HARMONIC OSCILLATOR

IX. NONLINEAR OSCILLATORS

A. Piecewise-linear oscillator

B. Quartic oscillator

X. REPULSIVE INVERSE SQUARE POTENTIAL

XI. GENERALIZATIONS

XII. SUMMARY

### Key Topics

- Oscillators
- 83.0
- Lagrangian mechanics
- 15.0
- Milky Way Galaxy
- 6.0
- Equations of motion
- 4.0
- Potential energy surfaces
- 4.0

## Figures

(a) On a sphere the great circle line starting from the north pole at is the shortest distance between two points as long as it does not reach the south pole at . On a slippery sphere a rubber band stretched between and will snap back if displaced either locally, as at , or by pulling the entire line aside, as along . The point is called the *antipode* of or in general the *kinetic focus* of . We say that if a great circle path terminates before the kinetic focus of its initial point, the length of the great circle path is a minimum. (b) If the great circle passes through antipode of the initial point , then the resulting line has a minimum length only when compared with some alternative lines. For example on a slippery sphere a rubber band stretched along this path will still snap back from local distortion, as at . However, if the entire rubber band is pulled to one side, as along , then it will not snap back, but rather slide over to the portion of a great circle down the backside of the sphere. With respect to paths like , the length of the great circle line is a maximum. With respect to all possible variations we say that the length of path is a saddle point. If a great circle path terminates beyond the kinetic focus of its initial point, the length of the great circle path is a saddle point.

(a) On a sphere the great circle line starting from the north pole at is the shortest distance between two points as long as it does not reach the south pole at . On a slippery sphere a rubber band stretched between and will snap back if displaced either locally, as at , or by pulling the entire line aside, as along . The point is called the *antipode* of or in general the *kinetic focus* of . We say that if a great circle path terminates before the kinetic focus of its initial point, the length of the great circle path is a minimum. (b) If the great circle passes through antipode of the initial point , then the resulting line has a minimum length only when compared with some alternative lines. For example on a slippery sphere a rubber band stretched along this path will still snap back from local distortion, as at . However, if the entire rubber band is pulled to one side, as along , then it will not snap back, but rather slide over to the portion of a great circle down the backside of the sphere. With respect to paths like , the length of the great circle line is a maximum. With respect to all possible variations we say that the length of path is a saddle point. If a great circle path terminates beyond the kinetic focus of its initial point, the length of the great circle path is a saddle point.

From the common initial event we draw a true worldline 0 and a second true worldline 1 that terminates at some event on the original worldline 0. The event nearest to at which worldline 1 coalesces with worldline 0 is the kinetic focus .

From the common initial event we draw a true worldline 0 and a second true worldline 1 that terminates at some event on the original worldline 0. The event nearest to at which worldline 1 coalesces with worldline 0 is the kinetic focus .

Several true harmonic oscillator worldlines with initial event and initial velocity . Starting at the initial fixed event at the origin, all worldlines pass through the same event . That is, is the kinetic focus for all worldlines of the family starting at the initial event . Worldlines 1 and 0 differ infinitesimally; worldlines 2 and 0 differ by a finite amount. This oscillator is discussed in detail in Sec. VIII.

Several true harmonic oscillator worldlines with initial event and initial velocity . Starting at the initial fixed event at the origin, all worldlines pass through the same event . That is, is the kinetic focus for all worldlines of the family starting at the initial event . Worldlines 1 and 0 differ infinitesimally; worldlines 2 and 0 differ by a finite amount. This oscillator is discussed in detail in Sec. VIII.

Schematic space-time diagram of a family of true worldlines for the quartic oscillator starting at with . The kinetic focus occurs at a fraction 0.646 of the half-period , illustrated here for worldline 0. The kinetic foci of all worldlines of this family lie along the heavy gray line, the caustic. Squares indicate recrossing events of worldline 0 with the other two worldlines. This oscillator is discussed in detail in Sec. IX.

Schematic space-time diagram of a family of true worldlines for the quartic oscillator starting at with . The kinetic focus occurs at a fraction 0.646 of the half-period , illustrated here for worldline 0. The kinetic foci of all worldlines of this family lie along the heavy gray line, the caustic. Squares indicate recrossing events of worldline 0 with the other two worldlines. This oscillator is discussed in detail in Sec. IX.

Schematic space-time diagram of a family of true worldlines for a piecewise-linear oscillator , with initial event and initial velocity . The kinetic focus of worldline 0 occurs at of its half-period . Similarly, small circles and are the kinetic foci of worldlines 1 and 2, respectively. The heavy gray curve is the caustic, the locus of all kinetic foci of different worldlines of this family (originating at the origin with positive initial velocity). Squares indicate events at which the other worldlines recross worldline 0. This oscillator is discussed in detail in Sec. IX.

Schematic space-time diagram of a family of true worldlines for a piecewise-linear oscillator , with initial event and initial velocity . The kinetic focus of worldline 0 occurs at of its half-period . Similarly, small circles and are the kinetic foci of worldlines 1 and 2, respectively. The heavy gray curve is the caustic, the locus of all kinetic foci of different worldlines of this family (originating at the origin with positive initial velocity). Squares indicate events at which the other worldlines recross worldline 0. This oscillator is discussed in detail in Sec. IX.

Schematic space-time diagram for the repulsive inverse square potential , with a family of worldlines starting at with various initial velocities. Intersections are events where two worldlines cross. The heavy gray straight line , where , is the caustic, the locus of kinetic foci (open circles) and envelope of the indirect worldlines. Worldline 2, with zero initial velocity, is asymptotic to the caustic, with kinetic focus at infinite space and time coordinates. The caustic divides space-time: each final event above the caustic can be reached by two worldlines of this family of worldlines, each final event on the caustic by one worldline of the family, and each final event below the caustic by no worldline of the family. This system is discussed in detail in Sec. X.

Schematic space-time diagram for the repulsive inverse square potential , with a family of worldlines starting at with various initial velocities. Intersections are events where two worldlines cross. The heavy gray straight line , where , is the caustic, the locus of kinetic foci (open circles) and envelope of the indirect worldlines. Worldline 2, with zero initial velocity, is asymptotic to the caustic, with kinetic focus at infinite space and time coordinates. The caustic divides space-time: each final event above the caustic can be reached by two worldlines of this family of worldlines, each final event on the caustic by one worldline of the family, and each final event below the caustic by no worldline of the family. This system is discussed in detail in Sec. X.

For the Maupertuis action, the heavy line envelope (the “parabola of safety”) is the locus of spatial kinetic foci , or spatial caustic, of the family of parabolic orbits of energy originating from the origin with various directions of the initial velocity . The potential is . The horizontal and vertical axes are and , respectively, and the caustic/envelope equation is , found by Johann Bernoulli in 1692. The caustic divides space. Each final point inside the caustic can be reached from initial point by two orbits of the family, each final point on the caustic by one orbit of the family, and each point outside the caustic by no orbit of the family. is the vertex (highest reachable point ) of the caustic and , denote the maximum range points . This system is discussed in detail in Appendix B. (Figure adapted from Ref. 43.)

For the Maupertuis action, the heavy line envelope (the “parabola of safety”) is the locus of spatial kinetic foci , or spatial caustic, of the family of parabolic orbits of energy originating from the origin with various directions of the initial velocity . The potential is . The horizontal and vertical axes are and , respectively, and the caustic/envelope equation is , found by Johann Bernoulli in 1692. The caustic divides space. Each final point inside the caustic can be reached from initial point by two orbits of the family, each final point on the caustic by one orbit of the family, and each point outside the caustic by no orbit of the family. is the vertex (highest reachable point ) of the caustic and , denote the maximum range points . This system is discussed in detail in Appendix B. (Figure adapted from Ref. 43.)

The coffee-cup optical caustic. The caustic shape in panel (b) (a nephroid) was derived by Johann Bernoulli in 1692 (Ref. 44).

The coffee-cup optical caustic. The caustic shape in panel (b) (a nephroid) was derived by Johann Bernoulli in 1692 (Ref. 44).

An original true worldline, labeled 0, starts at initial event . We draw an arbitrary adjacent curve, labeled 1, anchored at two ends on and a later event on the original worldline. The variational function is chosen to vanish at the two ends and .

An original true worldline, labeled 0, starts at initial event . We draw an arbitrary adjacent curve, labeled 1, anchored at two ends on and a later event on the original worldline. The variational function is chosen to vanish at the two ends and .

Let be the earliest event along the true worldline , labeled 0, such that for worldline along . The unique variational function achieving for worldline 0 corresponds to a varied curve labeled 1. We show that for this location of , curve 1 is a true worldline and is the kinetic focus . The arbitrary curve 2 is used to verify that curve 1 is a true worldline.

Let be the earliest event along the true worldline , labeled 0, such that for worldline along . The unique variational function achieving for worldline 0 corresponds to a varied curve labeled 1. We show that for this location of , curve 1 is a true worldline and is the kinetic focus . The arbitrary curve 2 is used to verify that curve 1 is a true worldline.

Schematic illustration of the topological evolution of the minimum C of the action for two “directions” in function space as the final time increases from to to , respectively, where is the kinetic focus time. (Figure adapted from Ref. 59.)

Schematic illustration of the topological evolution of the minimum C of the action for two “directions” in function space as the final time increases from to to , respectively, where is the kinetic focus time. (Figure adapted from Ref. 59.)

By definition the kinetic focus of the initial event is the first event at which two adjacent true worldlines and coalesce. We show that at the kinetic focus for this variational function in the limit , and that the action is a saddle point when the terminal event lies anywhere on the worldline beyond the kinetic focus . (The lower has the opposite sign from the upper , and the upper and lower functions are slightly different due to the end-events and being slightly different.)

By definition the kinetic focus of the initial event is the first event at which two adjacent true worldlines and coalesce. We show that at the kinetic focus for this variational function in the limit , and that the action is a saddle point when the terminal event lies anywhere on the worldline beyond the kinetic focus . (The lower has the opposite sign from the upper , and the upper and lower functions are slightly different due to the end-events and being slightly different.)

A family of elliptical trajectories starting at with the same speed (the directions of differ), and hence the same energy , the same major axis , and the same period , in a gravitational potential. The value of exceeds that necessary to generate a circular orbit. The center of force is the Earth (heavy circle). The dashed circle gives the locus of the second focus of the ellipses (a circle centered at ). The outer ellipse, with foci at and the Earth, is the envelope of the family of ellipses and the locus of the spatial kinetic foci relevant for action . The space-time kinetic foci of the family, relevant for action , all occur at time . (Figure adapted from Butikov, Ref. 101.)

A family of elliptical trajectories starting at with the same speed (the directions of differ), and hence the same energy , the same major axis , and the same period , in a gravitational potential. The value of exceeds that necessary to generate a circular orbit. The center of force is the Earth (heavy circle). The dashed circle gives the locus of the second focus of the ellipses (a circle centered at ). The outer ellipse, with foci at and the Earth, is the envelope of the family of ellipses and the locus of the spatial kinetic foci relevant for action . The space-time kinetic foci of the family, relevant for action , all occur at time . (Figure adapted from Butikov, Ref. 101.)

Two different elliptical trajectories typically can connect to in the same time for the attractive potential. (Adapted from Bate *et al.*, Ref. 108.)

Two different elliptical trajectories typically can connect to in the same time for the attractive potential. (Adapted from Bate *et al.*, Ref. 108.)

An elliptical orbit (tilted ellipse) in a 2D isotropic harmonic oscillator potential with force center at . A family of trajectories is launched from with equal initial speeds and various directions . One member of the family is shown. The envelope of the family is the outer ellipse (heavy line), with foci at and (coordinates , ). Points on the envelope are the kinetic foci for the spatial orbits. Point , occurring at time , where is the period, is the kinetic focus for the space-time trajectories (worldlines). (Figure adapted from French, Ref. 101.)

An elliptical orbit (tilted ellipse) in a 2D isotropic harmonic oscillator potential with force center at . A family of trajectories is launched from with equal initial speeds and various directions . One member of the family is shown. The envelope of the family is the outer ellipse (heavy line), with foci at and (coordinates , ). Points on the envelope are the kinetic foci for the spatial orbits. Point , occurring at time , where is the period, is the kinetic focus for the space-time trajectories (worldlines). (Figure adapted from French, Ref. 101.)

A periodic orbit of a 2D anisotropic harmonic oscillator with commensurate frequencies (here ) (Ref. 113). The space-time kinetic focus occurs at time , where is the period for -motion, for any initial event (see text), and the spatial kinetic focus for initial position is located on a parabolic spatial caustic (see Ref. 105).

A periodic orbit of a 2D anisotropic harmonic oscillator with commensurate frequencies (here ) (Ref. 113). The space-time kinetic focus occurs at time , where is the period for -motion, for any initial event (see text), and the spatial kinetic focus for initial position is located on a parabolic spatial caustic (see Ref. 105).

A quasiperiodic orbit of a 2D anisotropic harmonic oscillator with incommensurate frequencies. The outer ellipse is the equipotential contour . The rectangle delimits the region of space actually reached by the particular orbit (Ref. 114). The space-time kinetic focus occurs at time , where is the period for -motion, for any initial event .

A quasiperiodic orbit of a 2D anisotropic harmonic oscillator with incommensurate frequencies. The outer ellipse is the equipotential contour . The rectangle delimits the region of space actually reached by the particular orbit (Ref. 114). The space-time kinetic focus occurs at time , where is the period for -motion, for any initial event .

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