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.

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,”
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E. F.
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,
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C. G.
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J. L.
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(
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Paris
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Kluwer
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Dordrecht
,
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183
and
219
. The same erroneous statement occurs in work published in 1760–61, ibid, p.
xxxiii
.
14.
We easily found about two dozen texts using the erroneous term “maximum.” See, for example,
E.
Mach
,
The Science of Mechanics
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La Salle, IL
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;
A.
Sommerfeld
,
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(
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New York
,
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P. M.
Morse
and
H.
Feshbach
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Methods of Theoretical Physics
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McGraw Hill
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New York
,
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D.
Park
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Springer
,
Berlin
,
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L. H.
Hand
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J. D.
Finch
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Analytical Mechanics
(
Cambridge U. P.
,
Cambridge
,
1998
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53
;
G. R.
Fowles
and
G. L.
Cassiday
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Analytical Mechanics
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Saunders
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R. K.
Cooper
and
C.
Pellegrini
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(
Kluwer
,
New York
,
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), p.
34
.
A similar error occurs in
A. J.
Hanson
,
Visualizing Quaternions
(
Elsevier
,
Amsterdam
,
2006
), p.
368
, which states that geodesics on a sphere can have maximum length.
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E. F.
Taylor
and
J. A.
Wheeler
,
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Addison-Wesley Longman
,
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,
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), pp.
1
7
and
3
4
.
16.
The number of authors of books and papers using “extremum” and “extremal” is endless. Some examples include
R.
Baierlein
,
Newtonian Dynamics
(
McGraw-Hill
,
New York
,
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), p.
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;
L. N.
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J. D.
Finch
, Ref. 14, p.
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;
J. D.
Logan
,
Invariant Variational Principles
(
Academic
,
New York
,
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) mentions both “extremal” and “stationary,” p.
8
;
L. D.
Landau
and
E. M.
Lifschitz
,
Mechanics
, 3rd ed. (
Butterworth Heineman
,
Oxford
,
2003
), pp.
2
and
3
;
see also their
Classical Theory of Fields
, 4th ed. (
Butterworth Heineman
,
Oxford
,
1999
), p.
25
;
S. T.
Thornton
and
J. B.
Marion
,
Classical Dynamics of Particles and Systems
, 5th ed. (
Thomson, Brooks/Cole
,
Belmont, CA
,
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), p.
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;
R. K.
Cooper
and
C.
Pellegrini
, Ref. 14.
17.
F. L.
Pedrotti
,
L. S.
Pedrotti
, and
L. M.
Pedrotti
,
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(
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,
Upper Saddle River, NJ
,
2007
), p.
22
;
J. R.
Taylor
,
Classical Mechanics
(
University Science Books
,
Sausalito
,
2005
), Problem 6.5;
P. J.
Nahin
,
When Least is Best
(
Princeton U. P.
,
Princeton, NJ
,
2004
), p.
133
;
D. S.
Lemons
,
Perfect Form
(
Princeton U. P.
,
Princeton, NJ
,
1997
), p.
8
;
D.
Park
, Ref. 14, p.
13
;
F. A.
Jenkins
and
H. E.
White
,
Fundamentals of Optics
, 4th ed. (
McGraw Hill
,
New York
,
1976
), p.
15
;
R.
Guenther
,
Modern Optics
(
Wiley
,
New York
,
1990
), p.
135
;
R. W.
Ditchburn
,
Light
, 3rd ed. (
Academic
,
London
,
1976
), p.
209
;
R. S.
Longhurst
,
Geometrical and Physical Optics
, 2nd ed. (
Longmans
,
London
,
1967
), p.
7
;
J. L.
Synge
,
Geometrical Optics
(
Cambridge U. P.
,
London
,
1937
), p.
3
;
G. P.
Sastry
, “
Problem on Fermat’s principle
,”
Am. J. Phys.
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,
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(
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M. V.
Berry
, review of
The Optics of Rays, Wavefronts and Caustics
by
O. N.
Stavroudis
(
Academic
,
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,
1972
),
Sci. Prog.
61
,
595
597
(
1974
);
V.
Lakshminarayanan
,
A. K.
Ghatak
, and
K.
Thyagarajan
,
Lagrangian Optics
(
Kluwer
,
Boston
,
2002
), p.
16
;
V.
Perlick
,
Ray Optics, Fermat’s Principle, and Applications to General Relativity
(
Springer
,
Berlin
,
2000
), pp.
149
and
152
.
18.
The seminal work on second variations of general functionals by Legendre, Jacobi, and Weierstrass and many others is described in the historical accounts of Refs. 19 and 20. Mayer’s work (Ref. 19) was devoted specifically to the second variation of the Hamilton action. In our paper, we adapt Culverwell’s work (Ref. 21) for the Maupertuis action W to the Hamilton action. Culverwell’s work was preceded by that of Jacobi (Ref. 22) and Kelvin and Tait (Ref. 23).
19.
H. H.
Goldstine
,
A History of the Calculus of Variations From the 17th Through the 19th Century
(
Springer
,
New York
,
1980
).
20.
I.
Todhunter
,
A History of the Progress of the Calculus of Variations During the Nineteenth Century
(
Cambridge U. P.
,
Cambridge
,
1861
) and (
Dover
,
New York
,
2005
).
21.
E. P.
Culverwell
, “
The discrimination of maxima and minima values of single integrals with any number of dependent variables and any continuous restrictions of the variables, the limiting values of the variables being supposed given
,”
Proc. London Math. Soc.
23
,
241
265
(
1892
).
22.
C. G. J.
Jacobi
, “
Zür Theorie der Variationensrechnung und der Differential Gleichungen
.”
J. f.Math.
XVII
,
68
82
(
1837
). An English translation is given in Ref. 20, p. 243, and a commentary is given in Ref. 19, p. 156.
23.
W.
Thomson (Lord Kelvin)
and
P. G.
Tait
,
Treatise on Natural Philosophy
(
Cambridge U. P.
,
Cambridge
,
1879
,
1912
), Part I;
reprinted as
Principles of Mechanics and Dynamics
(
Dover
,
New York
,
1962
), Part I, p.
422
.
24.
Gelfand and Fomin (Ref. 25) and other more recent books on calculus of variations are rigorous but rather sophisticated. A previous study (Ref. 26) of the nature of the stationarity of worldline action was based on the Jacobi-Morse eigenfunction method (Ref. 27), rather than on the more geometrical Jacobi-Culverwell-Whittaker approach.
25.
I. M.
Gelfand
and
S. V.
Fomin
,
Calculus of Variations
, translated by R. A. Silverman (
Prentice Hall
,
Englewood Cliffs, NJ
,
1963
), Russian edition 1961, reprinted (
Dover
,
New York
,
2000
).
26.
M. S.
Hussein
,
J. G.
Pereira
,
V.
Stojanoff
, and
H.
Takai
, “
The sufficient condition for an extremum in the classical action integral as an eigenvalue problem
,”
Am. J. Phys.
48
,
767
770
(
1980
).
Hussein
 et al. make the common error of assuming S can be a true maximum. See also
J. G.
Papastavridis
, “
An eigenvalue criterion for the study of the Hamiltonian action’s extremality
,”
Mech. Res. Commun.
10
,
171
179
(
1983
).
27.
M.
Morse
,
The Calculus of Variations in the Large
(
American Mathematical Society
,
Providence, RI
,
1934
).
28.
As we shall see, the nature of the stationary value of Hamilton’s action S (and also Maupertuis’ action W) depends on the sign of second variations δ2S and δ2W (defined formally in Sec. IV), which in turn depends on the existence or absence of kinetic foci (see Secs. II and VII). The same quantities (signs of the second variations and kinetic foci) are also important in classical mechanics for the question of dynamical stability of trajectories (Refs. 23 and 29–31), and in semiclassical mechanics where they determine the phase loss term in the total phase of the semiclassical propagator due to a particular classical path (Refs. 32 and 33). The phase loss depends on the Morse (or Morse-Maslov) index, which equals the number of kinetic foci between the end-points of the trajectory (see Ref. 34). Further, in devising computational algorithms to find the stationary points of the action (either S or W), it is useful to know whether we are seeking a minimum or a saddle point, because different algorithms (Ref. 35) are often used for the two cases. As we discuss in this paper, it is the sign of δ2S (or δ2W) that determines which case we are considering. Practical applications of the mechanical focal points are mentioned at the end of Ref. 37.
29.
E. J.
Routh
,
A Treatise on the Stability of a Given State of Motion
(
Macmillan
,
London
,
1877
), p.
103
;
reissued as
Stability of Motion
, edited by
A. T.
Fuller
(
Taylor and Francis
,
London
,
1975
).
30.
J. G.
Papastavridis
, “
Toward an extremum characterization of kinetic stability
,”
J. Sound Vib.
87
,
573
587
(
1983
).
31.
J. G.
Papastavridis
, “
The principle of least action as a Lagrange variational problem: Stationarity and extremality conditions
,”
Int. J. Eng. Sci.
24
,
1437
1443
(
1986
);
J. G.
Papastavridis
,“
On a Lagrangean action based kinetic instability theorem of Kelvin and Tait
,”
Int. J. Eng. Sci.
24
,
1
17
(
1986
).
32.
L. S.
Schulman
,
Techniques and Applications of Path Integration
(
Wiley
,
New York
,
1981
), p.
143
.
33.
M. C.
Gutzwiller
,
Chaos in Classical and Quantum Mechanics
(
Springer
,
New York
,
1990
), p.
184
.
34.
In general, saddle points can be classified (or given an index (Ref. 27)) according to the number of independent directions leading to maximum-type behavior. Thus the point of zero-gradient on an ordinary horse saddle has a Morse index of unity. The Morse index for an action saddle point is equal to the number of kinetic foci between the end-points of the trajectory (Ref. 32, p. 90). Readable introductions to Morse theory are given by
R.
Forman
, “
How many equilibria are there? An introduction to Morse theory
,” in
Six Themes on Variation
, edited by
R.
Hardt
(
American Mathematical Society
,
Providence, RI
,
2004
), pp.
13
36
,
and
B.
Van Brunt
,
The Calculus of Variations
(
Springer
,
New York
,
2004
), p.
254
.
35.
F.
Jensen
,
Introduction to Computational Chemistry
(
Wiley
,
Chichester
,
1999
), Chap. 14.
36.
E. T.
Whittaker
,
A Treatise on the Analytical Dynamics of Particles and Rigid Bodies
, 4th ed. (originally published in 1904) (
Cambridge U. P.
,
Cambridge
,
1999
), p.
253
.
The same example was treated earlier by
C. G. J.
Jacobi
,
Vorlesungen Über Dynamik
(
Braunschweig
,
Vieweg
,
1884
), reprinted (
Chelsea
,
New York
,
1969
), p.
46
.
37.
A closer mechanics-optics analogy is between a kinetic focus (mechanics) and a caustic point (optics) (Ref. 38). The locus of limiting intersection points of pairs of mechanical spatial orbits is termed an envelope or caustic (see Fig. 7 for an example), just as the locus of limiting intersection points of pairs of optical rays is termed a caustic. In optics, the intersection point of a bundle of many rays is termed a focal point; a mechanical analogue occurs naturally in a few systems, for example, the sphere geodesics of Fig. 1 and the harmonic oscillator trajectories of Fig. 3, where a bundle of trajectories recrosses at a mechanical focal point. In electron microscopes (Refs. 39 and 40), mass spectrometers (Ref. 41), and particle accelerators (Ref. 42), electric and magnetic field configurations are designed to create mechanical focal points.
38.
M.
Born
and
E.
Wolf
,
Principles of Optics
, 4th ed. (
Pergamon
,
Oxford
,
1970
), pp.
130
and
734
;
J. A.
Luck
and
J. H.
Andrews
, “
Optical caustics in natural phenomena
,”
Am. J. Phys.
60
,
397
407
(
1992
).
39.
M.
Born
and
E.
Wolf
, Ref. 38, p.
738
;
L. A.
Artsimovich
and
S.
Yu. Lukyanov
,
Motion of Charged Particles in Electric and Magnetic Fields
(
MIR
,
Moscow
,
1980
).
40.
P.
Grivet
,
Electron Optics
, 2nd ed. (
Pergamon
,
Oxford
,
1972
);
A. L.
Hughes
, “
The magnetic electron lens
,”
Am. J. Phys.
9
,
204
207
(
1941
);
J. H.
Moore
,
C. C.
Davis
, and
M. A.
Coplan
,
Building Scientific Apparatus
, 3rd ed. (
Perseus
,
Cambridge, MA
,
2003
), Chap. 5.
41.
P.
Grivet
, Ref. 40, p.
822
;
J. H.
Moore
 et al., Ref. 40.
42.
M. S.
Livingston
,
The Development of High-Energy Accelerators
(
Dover
,
New York
,
1966
);
M. L.
Bullock
, “
Electrostatic strong-focusing lens
,”
Am. J. Phys.
23
,
264
268
(
1955
);
L. W.
Alvarez
,
R.
Smits
, and
G.
Senecal
, “
Mechanical analogue of the synchrotron, illustrating phase stability and two-dimensional focusing
,”
Am. J. Phys.
43
,
292
296
(
1975
);
A.
Chao
 et al., “
Experimental investigation of nonlinear dynamics in the Fermilab Tevatron
,”
Phys. Rev. Lett.
61
,
2752
2755
(
1988
).
[PubMed]
For recent texts, see
A. W.
Chao
, “
Resource Letter PBA-1: Particle beams and accelerators
,”
Am. J. Phys.
74
,
855
862
(
2006
).
43.
V. G.
Boltyanskii
,
Envelopes
(
MacMillan
,
New York
,
1964
).
44.
P. T.
Saunders
,
An Introduction to Catastrophe Theory
(
Cambridge U. P.
,
Cambridge
,
1980
), p.
62
.
45.

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.

46.
M. C.
Gutzwiller
, “
The origins of the trace formula
,” in
Classical, Semiclassical and Quantum Dynamics in Atoms
, edited by
H.
Friedrich
and
B.
Eckhardt
(
Springer
,
New York
,
1997
), pp.
8
28
.
47.
This type of variation, δx=αϕ, δẋ=αϕ̇, where δx and δẋ vanish together for α0, is termed a weak variation. See, for example,
C.
Fox
,
An Introduction to the Calculus of Variations
(
Oxford U. P.
,
Oxford
,
1950
), reprinted (
Dover
,
New York
,
1987
), p.
3
.
48.
H.
Goldstein
,
C.
Poole
, and
J.
Safko
,
Classical Mechanics
, 3rd ed. (
Addison-Wesley
,
San Francisco
,
2002
), p.
44
.
49.
As discussed in Sec. VI, for a true worldline x0(t) or PQ, where Q is the (first) kinetic focus, we have δ2S=0 for one special variation (and δ2S>0 for all other variations) as well as δS=0 for all variations. Further, we show that δ3S, etc., all vanish for the special infinitesimal variation for which δ2S vanishes, and that SS0=0 to second-order for larger such variations. In the latter case, typically δ3S is nonvanishing due to a single coalescing alternative worldline. In atypical (for one dimension (Ref. 58)) cases, more than one coalescing alternative worldline occurs, and the first nonvanishing term is δ4S or higher order—see Ref. 32, pp. 122–127. The harmonic oscillator is a limiting case where δkS=0 for all k for the special variation around worldline PQ, which reflects the infinite number of true worldlines which connect P to Q, and which can all coalesce by varying the amplitude (see Fig. 3). In Morse theory (Refs. 27 and 34) the worldline PQ is referred to as a degenerate critical (stationary) point.
50.
D.
Morin
,
Introductory Classical Mechanics
, ⟨www.courses.fas.harvard.edu/~phys16/Textbook/⟩ Chap. 5, p.
V
8
.
51.
This statement and the corresponding one in Ref. 52 must be qualified. It is in general not simply a matter of the time interval (tRtP) being short. The spatial path of the worldline must be sufficiently short. When, as usually happens, more than one actual worldline can connect a given position xP to a given position xR in the given time interval (tRtP), for short time intervals only the spatially shortest worldline will have the minimum action. For example, the repulsive power-law potentials U(x)=Cxn (including the limiting case of a hard-wall potential at the origin for C0) and the repulsive exponential potential U(x)=U0exp(xa) have been studied (Ref. 53). No matter how short the time interval (tRtP), two different worldlines can connect given position xP to given position xR. The fact that two different true worldlines can connect the two points in the given time interval leads to a kinetic focus time tQ occurring later than tR for the shorter of the worldlines and a (different) kinetic focus time tQ occurring earlier than tR for the other worldline. For the first worldline S is a minimum and for the other worldline S is a saddle point. Another example is the quartic oscillator discussed in Sec. IX, where an infinite number of actual worldlines can connect given terminal events (xP,tP) and (xR,tR), no matter how short the time interval (tRtP). Only for the shortest of these worldlines is S a minimum. The situation is different in 2D (see Appendix  B).
52.
E. T.
Whittaker
, Ref. 36, pp.
250
253
. Whittaker deals with the Maupertuis action W discussed in Appendix  A, whereas we adapt his analysis to the Hamilton action S. In more detail, our Eq. (1) corresponds to the last equation on p. 251 of Ref. 36, with q1t, q2x and q2ẋ.
53.
L. I.
Lolle
,
C. G.
Gray
,
J. D.
Poll
, and
A. G.
Basile
, “
Improved short-time propagator for repulsive inverse power-law potentials
,”
Chem. Phys. Lett.
177
,
64
72
(
1991
). In Sec. X and Ref. 54 further analytical and numerical results are given for the inverse-square potential, U(x)=Cx2. For given end positions xP and xR, there are two actual worldlines (xP,tP)(xR,tR) for given short times (tRtP). There is one actual worldline for tR=tQ when the two worldlines have coalesced into one, and there is no actual worldline for longer times (remember xP and xR are fixed).
54.
A. G.
Basile
and
C. G.
Gray
, “
A relaxation algorithm for classical paths as a function of end points: Application to the semiclassical propagator for far-from-caustic and near-caustic conditions
,”
J. Comput. Phys.
101
,
80
93
(
1992
).
55.

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

56.

This argument can be refined. In Sec. 8 we show that δ2S becomes O(α3) for RQ but is still larger in magnitude than the δ3S term, which is also O(α3). See Ref. 60 for δ2S and Eq. (40a) for δ3S.

57.

There are other systems for which 3Ux3, etc., vanish, for example, U(x)=C, U(x)=Cx, U(x)=Cx2. For these systems the worldlines cannot have δ2S=0 [see Eq. (19)], so that kinetic foci do not exist.

58.
In higher dimensions more than one independent variation, occurring in different directions in function space, can occur due to symmetry. In Morse theory the number of these independent variations satisfying δ2S=0 is called the multiplicity of the kinetic focus (Van Brunt, Ref. 34, p.
254
; Ref. 32, p.
90
). If the mulitplicity is different from unity, Morse’s theorem is modified from the statement in Ref. 34 to read as follows: The Morse index of the saddle point in action of worldline PR is equal to the number of kinetic foci between P and R, with each kinetic focus counted with its multiplicity.
59.
J. M. T.
Thompson
and
G. W.
Hunt
,
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(
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,
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,
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), p.
20
.
60.

The fact that δ2S00 for α0 is not surprising because δ2S0 is proportional to α2. The surprising fact is that here δ2S0 vanishes as α3 as α0 because the integral involved in the definition, Eq. (36a), of δ2S0 is itself O(α). We can see directly that δ2S0 becomes O(α3) for R near Q for this special variation αϕ by integrating the ϕ̇ term by parts in Eq. (36a) and using ϕ=0 at the end-points. The result is δ2S0=(α22)PRϕ[mϕ̈+U(x0)ϕ]dt. Because x0 and x1=x0+αϕ are both true worldlines, we can apply the equation of motion mẍ+U(x)=0 to both. We then subtract these two equations of motion and expand U(x0+αϕ) as U(x0)+U(x0)αϕ+(12)U(x0)(αϕ)2+O(α3), giving mϕ̈+U(x0)ϕ=(12)U(x0)αϕ2+O(α2). (If the ϕ2, 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 δ2S0, we find to lowest nonvanishing order δ2S0=(α34)PRdtU(x0)ϕ3, which is O(α3). This result and Eq. (40a) for δ3S0 give the desired result (42) for S1S0.

61.

If we use arguments similar to those of this section and Sec. VI, we can show that δ2S vanishes again at the second kinetic focus Q2, and that for R beyond Q2 the wordline PR has a second, independent variation leading to δ2S<0, in agreement with Morse’s general theory (Ref. 34).

62.

If we use L(x,ẋ)=pẋH(x,p), we can rewrite the Hamilton action as a phase-space integral, that is, S=PR[pẋH(x,p)]dt. We set δS=0 and vary x(t) and p(t) independently and find (Ref. 63) the Hamilton equations of motion ẋ=Hp, ṗ=Hx. 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)=p22m+U(x).

63.
Reference 48, p.
353
.
64.
M. R.
Hestenes
, “
Elements of the calculus of variations
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(
McGraw Hill
,
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,
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), pp.
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91
.
65.
O.
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,
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104
(
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).
66.
J. G.
Papastavridis
, “
On the extremal properties of Hamilton’s action integral
,”
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956
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67.
C. W.
Misner
,
K. S.
Thorne
, and
J. A.
Wheeler
,
Gravitation
(
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,
San Francisco
,
1973
), p.
318
. These authors use the Rayleigh-Ritz direct variational method (see Ref. 12 for a detailed discussion of this method) with a two-term trial trajectory x(t)=a1sin(ωt2)+a2sin(ωt), where ω=2πtR and a1 and a2 are variational parameters, to study the half-cycle (tR=T02) and one-cycle (tR=T0) trajectories. Because the kinetic focus time tQ>T02 for this oscillator, they find, in agreement with our results, that S is a minimum for the half-cycle trajectory (with a10, a2=0) and a saddle point for the one-cycle trajectory (with a1=0, a20). However, in the figure accompanying their calculation, which shows the stationary points in (a1,a2) space, they label the origin (a1,a2)=(0,0) a maximum. The point (a1,a2)=(0,0) represents the equilibrium trajectory x(t)=0. As we have seen, a true maximum in S cannot occur, so that other “
directions
” in function space not considered by the authors must give minimum-type behavior of S, leading to an overall saddle point.
68.
The two-incline oscillator potential has the form U(x)=Cx, with C=mgsinαcosα and α the angle of inclination. Here x is a horizontal direction. A detailed discussion of this oscillator is given by
B. A.
Sherwood
,
Notes on Classical Mechanics
(
Stipes
,
Champaign, Il
,
1982
), p.
157
.
69.

Other constant force or linear potential systems include the 1D Coulomb model (Ref. 70) U(x)=q1q2x, the bouncing ball (Ref. 71), and for x0 the constant force spring (Ref. 72).

70.
I. R.
Lapidus
, “
One- and two-dimensional Hydrogen atoms
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Am. J. Phys.
49
,
807
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K.
Andrew
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J.
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Am. J. Phys.
58
,
1177
1183
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71.
I. R.
Gatland
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Theory of a nonharmonic oscillator
,”
Am. J. Phys.
59
,
155
158
(
1991
) and references therein;
W. M.
Hartmann
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The dynamically shifted oscillator
,”
Am. J. Phys.
54
,
28
32
(
1986
).
72.
A.
Capecelatro
and
L.
Salzarulo
,
Quantitative Physics for Scientists and Engineers: Mechanics
(
Aurie Associates
,
Newark, NJ
,
1977
), p.
162
;
C.-Y.
Wang
and
L. T.
Watson
, “
Theory of the constant force spring
,”
Trans. ASME, J. Appl. Mech.
47
,
956
958
(
1980
);
H.
Helm
, “
Comment on ‘A constant force generator for the demonstration of Newton’s second law’
,”
Am. J. Phys.
52
,
268
(
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73.
C. G.
Gray
,
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, and
V. A.
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,
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208
(
2004
).
74.

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).

75.
R. P.
Bell
, “
The occurrence and properties of molecular vibrations withV(x)=ax4
,”
Proc. R. Soc. London, Ser. A
183
,
328
337
(
1945
);
J.
Laane
, “
Origin of the ring-puckering potential energy function for four-membered rings and spiro compounds. A possibility of pseudorotation
,”
J. Phys. Chem.
95
,
9246
9249
(
1991
).
76.
C. G.
Joslin
,
J.
Yang
,
C. G.
Gray
,
S.
Goldman
, and
J. D.
Poll
, “
Infrared rotation and vibration-rotation bands of endohedral Fullerene complexes. K+@C60.
Chem. Phys. Lett.
211
,
587
594
(
1993
).
77.
A. S.
Barker
, “
Far infrared dispersion and the Raman spectra of ferroelectric crystals
,” in
Far-Infrared Properties of Solids
, edited by
S. S.
Mitra
and
S.
Nudelman
(
Plenum
,
New York
,
1970
), pp.
247
296
.
78.
J.
Thomchick
and
J. P.
McKelvey
, “
Anharmonic vibrations of an ‘ideal’ Hooke’s law oscillator
,”
Am. J. Phys.
46
,
40
45
(
1978
).
79.
R.
Baierlein
, Ref. 16, p.
73
.
80.
In two dimensions with x=(x1,x2) and v0=(v01,v02), our analytic condition (4) for the kinetic focus of worldline x(t,v0) becomes det(xiv0j)=0, where det(Aij) denotes the determinant of matrix Aij. The generalization to other dimensions is obvious. This condition (in slightly different form) is due to Mayer, Ref. 19, p.
269
. For a clear discussion, see
J. G.
Papastavridis
,
Analytical Mechanics
(
Oxford U. P.
,
Oxford
,
2002
), p.
1061
.
For multidimensions a caustic becomes in general a surface in space-time. The analogous theory for multidimensional spatial caustics, relevant for the action W, is discussed in Ref. 105. A simple example of a 2D surface spatial caustic is obtained by revolving the pattern of Fig. 7 about the vertical axis, thereby generating a paraboloid of revolution surface caustic/envelope. Due to axial symmetry, the caustic has a second (linear) branch, that is, the symmetry axis from y=0 to y=Y. An analogous optical example is discussed by
M. V.
Berry
, “
Singularities in waves and rays
,” in
Physics of Defects
,
Les Houches Lectures XXXIV
, edited by
R. D.
Balian
,
M.
Kleman
, and
J.-P.
Poirier
(
North Holland
,
Amsterdam
,
1981
), pp.
453
543
.
81.
A dynamics problem can be formulated as an initial value problem. For example, find x(t) from Newton’s equation of motion with initial conditions (xP,ẋP). It can also be formulated as a boundary value problem; for example, find x(t) from Hamilton’s principle with boundary conditions (xP,tP) and (xR,tR). Solving a boundary value problem with initial value problem methods (for example, the shooting method) is standard (Ref. 82). Solving an initial value problem with boundary value problem methods is much less common (Ref. 83). For an example of a boundary value problem with mixed conditions (prescribed initial velocities and final positions) for about 107 particles, see
A.
Nusser
and
E.
Branchini
, “
On the least action principle in cosmology
,”
Mon. Not. R. Astron. Soc.
313
,
587
595
(
2000
).
82.
See, for example,
W. H.
Press
,
S. A.
Teukolsky
,
W. T.
Vetterling
, and
B. P.
Flannery
,
Numerical Recipes in Fortran
, 2nd ed. (
Cambridge U. P.
,
Cambridge
,
1992
), p.
749
.
83.
H. R.
Lewis
and
P. J.
Kostelec
, “
The use of Hamilton’s principle to derive time-advance algorithms for ordinary differential equations
,”
Computer Phys. Commun.
96
,
129
151
(
1996
);
D.
Greenspan
, “
Approximate solution of initial value problems for ordinary differential equations by boundary value techniques
,”
J. Math. Phys. Sci.
15
,
261
274
(
1967
).
84.

The converse effect cannot occur: a time-dependent potential U(x,t) with U<0 at all times always has δ2S>0 as seen from Eq. (19). If U(x,t) is such that U 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).

85.
M. H.
Friedman
,
J. E.
Campana
,
L.
Kelner
,
E. H.
Seeliger
, and
A. L.
Yergey
, “
The inverted pendulum: A mechanical analog of the quadrupole mass filter
,”
Am. J. Phys.
50
,
924
931
(
1982
).
86.
P. K.
Gosh
,
Ion Traps
(
Oxford U. P.
,
Oxford
,
1995
), p.
7
.
87.
J. J.
Stoker
,
Nonlinear Vibrations in Mechanical and Electrical Systems
(
Wiley
,
New York
,
1950
), p.
112
.
Stoker’s statements on series convergence need amendment in light of the Kolmogorov-Arnold-Moser (KAM) theory (Ref. 89). See
J.
Moser
, “
Combination tones for Duffing’s equation
,”
Commun. Pure Appl. Math.
18
,
167
181
(
1965
);
T.
Kapitaniak
,
J.
Awrejcewicz
and
W.-H.
Steeb
, “
Chaotic behaviour in an anharmonic oscillator with almost periodic excitation
,”
J. Phys. A
20
,
L355
L358
(
1987
);
A. H.
Nayfeh
,
Introduction to Perturbation Techniques
(
Wiley
,
New York
,
1981
), p.
216
;
A. H.
Nayfeh
and
B.
Balachandran
,
Applied Nonlinear Dynamics
(
Wiley
,
New York
,
1995
), p.
234
;
S.
Wiggins
, “
Chaos in the quasiperiodically forced Duffing oscillator
,”
Phys. Lett. A
124
,
138
142
(
1987
).
88.
G.
Seifert
, “
On almost periodic solutions for undamped systems with almost periodic forcing
,”
Proc. Am. Math. Soc.
31
,
104
108
(
1972
);
J.
Moser
, “
Perturbation theory of quasiperiodic solutions and differential equations
,” in
Bifurcation Theory and Nonlinear Eigenvalue Problems
, edited by
J. B.
Keller
and
S.
Antman
(
Benjamin
,
New York
,
1969
), pp.
283
308
;
J.
Moser
, “
Perturbation theory for almost periodic solutions for undamped nonlinear differential equations
,” in
International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics
, edited by
J. P.
Lasalle
and
S.
Lefschetz
(
Academic
,
New York
,
1963
), pp.
71
79
;
M. S.
Berger
, “
Two new approaches to large amplitude quasi-periodic motions of certain nonlinear Hamiltonian systems
,”
Contemp. Math.
108
,
11
18
(
1990
).
89.
G. M.
Zaslavsky
,
R. Z.
Sagdeev
,
D. A.
Usikov
, and
A. A.
Chernikov
,
Weak Chaos and QuasiRegular Patterns
(
Cambridge U. P.
,
Cambridge
,
1991
), p.
30
.
90.
See, for example,
M.
Tabor
,
Chaos and Integrability in Nonlinear Dynamics
(
Wiley
,
New York
,
1989
), p.
35
;
J. M. T.
Thompson
and
H. B.
Stewart
,
Nonlinear Dynamics and Chaos
, 2nd ed. (
Wiley
,
Chichester
,
2002
), pp.
310
.
91.
For example, the equilibrium position can be modulated. A somewhat similar system is a ball bouncing on a vertically oscillating table. The motion can be chaotic. See, for example,
J.
Guckenheimer
and
P.
Holmes
,
Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields
(
Springer
,
New York
,
1983
), p.
102
;
N. B.
Tufillaro
,
T.
Abbott
, and
J.
Reilly
,
An Experimental Approach to Nonlinear Dynamics and Chaos
(
Addison-Wesley
,
Redwood City, CA
,
1992
), p.
23
;
A. B.
Pippard
,
The Physics of Vibration
(
Cambridge U. P.
,
Cambridge
,
1978
), Vol.
1
, pp.
253
271
.
92.
The forced Duffing oscillator with U(x,t)=(12)kx2+Cx4xF0cosωt is studied in Ref. 30. For k=0 the Duffing oscillator reduces to the quartic oscillator.
93.
R. H. G.
Helleman
, “
Variational solutions of non-integrable systems
,” in
Topics in Nonlinear Dynamics
, edited by
S.
Jorna
(
AIP
,
New York
,
1978
), pp.
264
285
. This author studies the forced Duffing oscillator with U(x,t)=(12)kx2Cx4xF0cosωt (note the sign change in C compared to Ref. 92), and the Henon-Heiles oscillator with the potential in Eq. (83).
94.
In Ref. 54 the harmonic potential U(x,t)=(12)k[xxc(t)]2 with an oscillating equilibrium position xc(t) is studied. The worldlines for this system are all nonchaotic.
95.
M.
Henon
and
C.
Heiles
, “
The applicability of the third integral of the motion: Some numerical experiments
,”
Astron. J.
69
,
73
79
(
1964
).
96.
There have been a few formal studies of action for chaotic systems, but few concrete examples seem to be available. See, for example,
S.
Bolotin
, “
Variational criteria for nonintegrability and chaos in Hamiltonian systems
,” in
Hamiltonian Mechanics
, edited by
J.
Seimenis
(
Plenum
,
New York
,
1994
), pp.
173
179
.
97.
Reference 48, p.
434
.
98.
H.
Poincaré
,
Les Méthodes Nouvelles de la Mécanique Céleste
(
Gauthier-Villars
,
Paris
,
1899
), Vol.
3
;
New Methods of Celestial Mechanics
(
AIP
,
New York
,
1993
), Part 3,
p
958
.
99.
The situation is complicated because, as Eq. (A1) shows, there are two forms for W, that is, the time-independent (first) form and the time-dependent (last) form. Spatial kinetic foci (discussed in Appendix  B) occur for the time-independent form of W. Space-time kinetic foci occur for the time-dependent form of W, as for S. Typically the kinetic foci for the two forms for W differ from each other (Refs. 30 and 98) and from those for S.
100.
E. J.
Routh
,
A Treatise on Dynamics of a Particle
(
Cambridge U. P.
,
Cambridge
,
1898
), reprinted (Dover, New York,
1960
), p.
400
.
101.
A. P.
French
, “
The envelopes of some families of fixed-energy trajectories
,”
Am. J. Phys.
61
,
805
811
(
1993
);
E. I.
Butikov
, “
Families of Keplerian orbits
,”
Eur. J. Phys.
24
,
175
183
(
2003
).
102.
We assume that we are dealing with bound orbits. Similar comments apply to scattering orbits (hyperbolas). Just as for the orbits in the linear gravitational potential discussed in the preceding paragraph, here too there are restrictions and special cases (Ref. 19, p.
164
; Ref. 103, p.
122
). If the second point (xR,yR) lies within the “ellipse of safety” (the envelope (French, Ref. 101)) of the elliptical trajectories of energy E originating at (xP,yP), then two ellipses with energy E can connect (xP,yP) to (xR,yR). If (xR,yR) lies on the ellipse of safety, then one ellipse of energy E can connect (xP,yP) to (xR,yR), and if (xR,yR) lies outside the ellipse of safety, then no ellipse of energy E can connect the two points. Usually the initial and final points (xP,yP) and (xR,yR) together with the center of force at (0,0) (one focus of the elliptical path) define the plane of the orbit. If (xP,yP), (xR,yR), and (0,0) lie on a straight line, the plane of the orbit is not uniquely defined, and there is almost always an infinite number of paths of energy E in three dimensions that can connect (xP,yP,zP=0) to (xR,yR,zR=0). A particular case of the latter is a periodic orbit where (xR,yR)=(xP,yP). Because the orbit can now be brought into coalescence with an alternative true orbit by a rotation around the line joining (xP,yP) to (0,0), a third kinetic focus arises for elliptical periodic orbits in three dimensions (see Ref. 33, p.
29
).
103.
N. G.
Chetaev
,
Theoretical Mechanics
(
Springer
,
Berlin
,
1989
).
104.
For the repulsive 1r potential, the hyperbolic spatial orbits have a (parabolic shaped) caustic/envelope (French, Ref. 101).
105.
If the orbit equation has the explicit form y=y(x,θ0), or the implicit form f(x,y,θ0)=0, the spatial kinetic focus is found from yθ0=0 or fθ0=0, respectively. Here θ0 is the launch angle at (xP,yP) (see Fig. 15 for an example). The derivation of these spatial kinetic focus conditions is similar to the derivation of the space-time kinetic focus condition of Eq. (4) (see Ref. 106, p. 59). In contrast, if the orbit equation is defined parametrically by the trajectory equations x=x(t,θ0) and y=y(t,θ0), the spatial kinetic focus condition is (x,y)(t,θ0)=0. This Jacobian determinant condition is similar to that of Ref. 80 for the space-time kinetic focus (see Ref. 106, p. 73 for a derivation). As an example, consider a family of figure-eight-like harmonic oscillator orbits of Fig. 16, launched from the origin (xP,yP)=(0,0) at time tP=0, all with speed v0 (and therefore the same energy E), at various angles θ0. The trajectory equations are x=(v0ω1)cosθ0sinω1t and y=(v0ω2)sinθ0sinω2t, where ω1=2ω2. The determinant condition for the spatial kinetic focus reduces to cosθ0tanω2t=1, which locates the kinetic focus (in time) for the orbit with launch angle θ0. Elimination of θ0 and t from these three equations leads to the locus of the (first) spatial kinetic foci, the spatial caustic/envelope equation y2=(v0ω2)22(v0ω2)x, which is a parabolic shaped curve with two cusps on the y-axis.
106.
R. H.
Fowler
,
The Elementary Differential Geometry of Plane Curves
(
Cambridge U. P.
,
Cambridge
,
1920
).
107.
The finding of the two elliptical (or hyperbolic or parabolic) shaped trajectories from observations giving the two end-positions and the time interval is a famous problem of astronomy and celestial mechanics, solved by Lambert (
1761
), Gauss (1801–1809), and others (Ref. 108).
108.
R. R.
Bate
,
D. D.
Mueller
, and
J. E.
White
,
Fundamentals of Astrodynamics
(
Dover
,
New York
,
1971
), p.
227
;
H.
Pollard
,
Celestial Mechanics
(
Mathematical Association of America
,
Washington
,
1976
), p.
28
;
P. R.
Escobal
,
Methods of Orbit Determination
(
Wiley
,
New York
,
1965
), p.
187
. For the elliptical orbits, more than two trajectories typically become possible at sufficiently large time intervals; these additional trajectories correspond to more than one complete revolution along the orbit (Ref. 109).
109.
R. H.
Gooding
, “
A procedure for the solution of Lambert’s orbital boundary-value problem
,”
Celest. Mech. Dyn. Astron.
48
,
145
165
(
1990
).
110.
It is clear from Fig. 13 that a kinetic focus occurs after time T0. To show rigorously that this focus is the first kinetic focus (unlike for W where it is the second), we can use a result of Gordon (Ref. 111) that the action S is a minimum for time t=T0. If one revolution corresponds to the second kinetic focus, the trajectory PP would correspond to a saddle point. The result tQ=T0 can also be obtained algebraically by applying the general relation (4) to the relation r=r(t,L) for the radial distance, where we use angular momentum L as the parameter labeling the various members of the family in Fig. 13. We obtain tQ from (rL)t=0. The latter equation implies that (tL)r=0, because (rL)t=(rt)L(tL)r. At fixed energy E (or fixed major axis 2a), the period T0 is independent of L for the attractive 1r potential, so that the solution of (tL)r=0 occurs for t=T0, which is therefore the kinetic focus time tQ.
111.
W. B.
Gordon
, “
A minimizing property of Keplerian orbits
,”
Am. J. Math.
99
,
961
971
(
1977
).
112.

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.

113.
A. P.
French
,
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(
Norton
,
New York
,
1966
), p.
36
.
114.
J. C.
Slater
and
N. H.
Frank
,
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(
McGraw Hill
,
New York
,
1933
), p.
85
.
115.
C. G.
Gray
,
G.
Karl
, and
V. A.
Novikov
, “
The four variational principles of mechanics
,”
Ann. Phys. (N.Y.)
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,
1
25
(
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).
116.
C. G.
Gray
,
G.
Karl
, and
V. A.
Novikov
, “
From Maupertuis to Schrödinger. Quantization of classical variational principles
,”
Am. J. Phys.
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,
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961
(
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).
117.
E.
Schrödinger
, “
Quantisierung als eigenwert problem I
,”
Ann. Phys.
79
,
361
376
(
1926
),
translated in
E.
Schrödinger
,
Collected Papers on Wave Mechanics
(
Blackie
,
London
,
1928
), Chelsea reprint 1982.
For modern discussions and applications, see, for example,
E.
Merzbacher
,
Quantum Mechanics
, 3rd ed. (
Wiley
,
New York
,
1998
), p.
135
;
S. T.
Epstein
,
The Variation Method in Quantum Chemistry
(
Academic
,
New York
,
1974
).
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