Poincaré's contribution to the development of the theory of
canonical transformation1
Michiyo Nakane2（Preparation）What is canonical transformation?
Before our historical examination, we will confirm the terminology
related to canonical transformation.
Let us consider the transformation of variables that transform
so-called canonical equations
dqi H dpi
H

,

, H  H (t , q1 , , qn , p1 , , pn ) (1)dt pidt
qi
to other canonical equations
dQi K dPi
K

,

, K  K (t , Q1 ,
dt
Pi
dt
Qi

, Qn , P1 ,, Pn ). (2)In addition to a notably symmetrical form, canonical equations haveimportant mathematical properties. It then is useful if one transforms
equations by maintaining their canonical form. In this report, the term
canonical transformation is used to describe the transformation of variables
by maintaining the canonical form of the equations involved.
In modern textbooks, old variables (q1 ,variables (Q1 ,, qn , p1 , , pn ) and new, Qn , P1, , Pn ) are shown as bound by equations with anappropriate function S :pi qi  H  PQi i K dS
dt

or dS  ( pi dqi  PdQii )  ( K  H )dt(3)

where ・ means a derivative with respect to t, and the transformation
become canonical. The function S is called a generating function. Since
canonical equations (1) are derived from the so-called Hamilton's principlet1

  L(t , q1 , , qn , q1 , , qn )dt  0 , (4)t0and equations (2) are drawn from principle (4), which is written as

1

2

This work was supported by JSPS KAKENHI Grant Number P17K01181.
Part time lecturer, Seijo University.



(Q1 ,

, Qn , Q1 ,

, Q n ) , relation (3) is obtained. We adopt a complete solution of

the Hamilton–Jacobi equation as the generating function:


S
S
S 
 H  q1 ,, qn ,,,, t   0.
t

q

q
1
n


In some cases, we call canonical equations Hamiltonian equations,and we call functions in the equations, such as H and K, Hamiltonian
functions. Today we often write equations of motion in canonical forms. In
dynamics, qi is generalized coordinates, pi is generalized momentum, andH is total energy, which does not explicitly depend on time. In this report,
we mainly discuss the case where H does not explicitly depend on time, andH is not changed after the transformation of variables; namely H = K. To
distinguish them from time dependent case, In this report, time
independent canonical equations corresponding to (1) and (2) are called (1’)
and (2’).
1. Introduction
At the end of the nineteenth century and the beginning of thetwentieth century, various influential textbooks on celestial mechanics were
published. Examples are the following:








E. W. Brown (1896) An Introductory Treatise on the Lunar Theory.
F. F. Tisserand (1889-1896) Traité de Mécanique Céleste, 4 vols.
H. Poincaré (1892-1899) Les Méthodes Nouvelles de la MécaniqueCéleste, 3 vols.
H. Poincaré (1905-1910) Leçons de Mécanique Céleste, 3 vols.
C. V. L. Charlier (1902, 1907) Die Mechanik des Himmels, 2 vols.
F. Moulton (1902), An Introduction to Celestial Mechanics.
E.T. Whittaker (1904) A Treatise on the Analytical Dynamics ofParticles and Rigid Bodies.
Some authors, such as Tisserrand, Poincaré, Charlier, and Whittaker,
used canonical equations rather than Newtonian equations of motion in
discussing the mathematical properties of canonical equations. Amongthese works, Poincaré’s two series of books are notable. The idea ofcanonical transformation was introduced at the beginning, whereas in otherbooks, it was introduced in the middle or latter parts. This fact suggeststhat Poincaré paid particular attention to canonical transformation.
This report examines Poincaré’s articles that were published around
1900, which include discussions related to canonical transformation and,
which are listed chronologically as follows:(1) “Sur la problème des trois corps et les équation de la dynamique,"Acta Mathematica, 13, 1890, pp.1-270.
(2) Les Méthodes Nouvelles, Vol.1, 1892.
(3) “Sur une forme nouvelle des équations du problème des trois
corps," Comptes rendus , 123, 1896, pp.1031-1035.
(4) “Sur une forme nouvelle des équations du problème des trois corps,"
Bulletin Astronomique, 14 ,1897, pp.53-67, Acta Mathematica, 21,
pp.83-97.
(5) Les Méthodes Nouvelles, Vol.3, 1899.
(6) Leçons de Mécanique Céleste, Vol.1, 1905.
In the following section, we will clarify Poincaré’s essential
contribution to constructing the modern theory of canonical transformation.
2. A Brief History of Canonical Equations
2.1. Jacobi's general theory of canonical transformation
First, we outline the history of canonical equations before 1890. The
canonical form of equations originated in Poisson’s 1809 paper,3 in which hewrote the equation of perturbation function as follows:d i 
d i


,

(i  1,, n) (2.1-1)
dt
 i
dt
 i
where   (t , 1 ,,  n , 1,,  n ) is the perturbation function, (1 ,,  n )are initial positions, and (1 ',,  n ') are initial velocities. Lagrange

S. D. Poisson, “ Mémoire sur la Variation des Constantes arbitraires dansles Questions de la Mécanique," Journal de l'École Polytechnique 8 (15cahier), 1809, pp.266-344.
3

introduced these equations in the second version of his famous Mécaniqueanalytique. 4 W. R. Hamilton noted Poisson and Lagrange’s forms andderived equations of motions in the form of (1’) in 1835.5 He applied his
original dynamical theory to what he called the “Problem of Perturbation.”
In 1837, Jacobi named equations that have a symmetrical form
“canonical”. 6 He seemed to have the idea that the transformation ofvariables in which the canonical form of equations is maintained. He
derived the following theorem, which was published in 1837. However, its
proof was published after his death.7

Jacobi's second theorem
The canonical form of equations is preserved if the old and newvariables are related as follows:


 Qi
 qi (i  1, , n) (2.1–2)
Pipi
where    ( P1 ,, Pn , p1 , , pn )In this case, the Hamiltonian function is also preserved, namely K  H inequations (2’). We call function  a generating function in accordance withmodern terminology. Except equation (2.1-2), Jacobi did not give any
information about  . However, the function  is tended to be confusedwith the complete solution of the Hamilton–Jacobi equation associated withthe original canonical equations. The other Jacobi’s theorem, which was
also derived around 1837, presumably caused such confusion.

Jacobi's first theorem8
J. L. Lagrange, Mécanique analytique (Second edition) Vol.1 (1811) ,Vol.2(1815). Reproduced in Œuvres Vol. 11, Vo.12.
5 W. R. Hamilton, “Second Essay on a General Method in Dynamics",
Philosophical Transaction of the Royal Society, Part 1, 1835, pp.95-144.
= in the Mathematical Papers of Sir William Rowan Hamilton, vol.2,
pp.212- 216.
6 C.G.J. Jacobi, “Note sur l'intégration des équations différentielles de la
dynamique", Comptes Rendus 5, 1837 pp.61-67.= in C.G.J. Jacobi'sGesammelte Werke, Vol. 4, pp.131-136.
7 C.G.J. Jacobi, “Über diejenigen Probleme der Mechanik in welchem eine
Kräftefunction existirt und über die Theorie der Störungen ," =in Werke Vol.
5, pp.219-395.
8 Jacobi showed his idea in the framework of Newtonian mechanics in“Über die Reduction der Integration der partiellen Differentialgleichungen4

If a complete solution

S  S (t , q1 , , qn , 1 , ,  n )(where 1 ,,  n are arbitrary constants) of the Hamilton–Jacobi equation
S H  t , q1 ,t

, qn ,

S
,
q1

,

S 
0
qn 

are obtained, solutions to canonical equations (1) are given bySS pi ,
 i
qi
 i

where 1 ,(i  1,, n)（2.1-3）, n are new arbitrary constants.

Jacobi’s first theorem states that solutions to canonical equations,which are a system of ordinary differential equations, are reduced to thoseof partial differential equations. Although the statements of Jacobi’s twotheorems are different, both involve the properties of canonical equations.
Furthermore, according to modern knowledge, a complete solution of the
Hamilton–Jacobi equation S  S (q1 ,qn , 1 , ,  n ) works as the generating

function that transforms old variables (q1 ,qn , p1 ,pn ) to new ones(1 , , n , 1, ,  n ) . Hence, it is not surprising that function  is confusedwith the complete solution. As we will see, Poincaré confused them in 1892.
In accordance with Poincaré’s numbering, in this report, we consider
Jacobi’s first theorem that in which the reduction of the solution to thecanonical equation is the partial differential equation. We consider Jacobi’s
second theorem that in which the condition of the canonical transformation
erster Ordnung zwischen irgend einer Zahl Variabeln auf die Integrationeines einzigen Systems gewohnlicher Differentialgleichungen", Journal für
die reine und angewandte Mathematik, 17, 1837, pp.97-162.=in Werke,
vol.4, pp.57-127. He developed this theorem in 1842 and 1843, which wasedited and published by Clebsch in Vorlesungen über Dynamik in
1866:Werke Supplementband.

is given.
2.2 Canonical equations in the middle of the nineteenth century
In the middle of the nineteenth century, mathematicians did not
develop Jacobi’s general theory of canonical transformation. However, wefind a kind of canonical transformation in the procedure they used to reducethe degree of freedom in the three-body problem. Whittaker’s report of 1899
includes examples by Bour (1856), Scheibner (1866, 1868), Mathieu (1874),
and so on. We now examine Whittaker’s description of Schreiber’s
reduction.
Scheibner set q1 , q2 , q3 to be the mutual distances of three bodies,
p1 

T
T
T
where T is kinetic energy, q4 is the angle that
, p2 
, p3 
q1
q2
q3

the node (of the plane of the bodies on the invariant plane) makes with oneof the principal axes of the bodies at their center of gravity, and set

p4  k cos i , where k is the constant of angular momentum on the invariant
plane, and i is the angle between the plane of the bodies and theinvariable plane. Through these transformations of variables, the canonicalequations in the three-body problem are reduced todqi H dpi
H

,

(i  1, 2,3, 4)
dt pidtqi
where H is a certain function of qi and pi , and H = const is an integralof the system. Whittaker’s main interest is Scheibner’s success in reducing
the number of degrees in the three-body problem from 18 to 8. However,from our perspective, Scheibner performed the canonical transformation ofvariables for particular problem.
The contemporary mathematicians tried to find a way to reduce
their problem independently of the general transformation theory. Moreover,their accumulated thought influenced Poincaré’s consideration of thisproblem.
2.3 Delaunay’s equation (1860)99C. E. Delaunay, “Théorie du mouvement de la lune I", Mémoire deIn 1860, Charles Delaunay derived his original canonical equationsof perturbation function F as follows:dL F dG F d  F

,

,

,
dt
l dt g dt 
dl
F dg
F d
F

,

,

dt
L dt
G dt


where a is the major axis, e is eccentricity, i is inclination, l is themean anomaly, g is the longitude of the ascending node; g   : mean
anomaly, L   a , G  L a(1  e2 ),   G cos i ,  : sum of three masses.
These canonical equations, which involve orbital elements, worked
effectively in his lunar theory. and they are included in modern textbooks ofcelestial mechanics. The readers of these textbooks may infer that Delaunaytransformed the canonical equations of motion and derived his equations.
However, Delaunay, while noting Poisson–Lagrange’s equations of theperturbation function, derived these equations from Newtonian equations of
motion, which are independent of the concept of canonical transformation.
In any case, the appearance of new canonical equations led to the general
theory of canonical transformation.
3. Poincaré's Consideration of the Canonical Transformation
3.1 Poincaré's description of the canonical transformation in 1890
To the best of my knowledge, Poincaré's first note on canonicaltransformation is included in his famous 1890 paper on the three-bodyproblem. Section 4 in the second part of this paper was devoted to studying
a case with only two degrees of freedom. Poincaré transformed Tisserand's
equations of the perturbation function, which are Delaunay’s equations in
the two-dimensional case, dL R
,
 
dt

l


 dG R, dt g

dl
R

dt
L
dg
R

dt
G

to those of the canonical equation of motion,l'Academie des Sciences, 28, 1860, pp.1-883.

dqi H dpi
H

,

,
dt pidtqi(i  1, 2) H  H (q1 , q2 , p1 , p2 )by setting (1890, pp.170–171)

q1  G, q2  L, p1  g  t , p2  l.
Poincaré also noted that if old variables (q1 , q2 , p1 , p2 ) and newvariables (Q1 , Q2 , P1 , P2 ) are related byQ1  q1  q2 , Q2  q1  q2 ,

P1 

1
1
( p1  p2 ), P2  ( p1  p2 )22Then the canonical form of equations is preserved (1890, p. 175).
Poincaré provided very simple examples of the two-dimensional case.
However, these two examples have different aspects: the first relates
Delaunay’s equations to equations of motion; the second is an example ofthe linear transformation of variables that retain the canonical form ofequations. Poincaré seemed to be seeking a general theory of canonicaltransformation. However, there is no evidence that indicates whether
Poincaré noted Jacobi’s second theorem or not at that time.
3.2 Poincaré's attitude in 1892
Poincaré's famous book, Les Méthodes Nouvelles (vol. 1, 1892)
begins by introducing canonical equations of motion that were derived from
Newtonian equations of motion. Next, Poincaré introduced Jacobi's first
theorem without proof. Subsequently, he mentioned what he called Jacobi’s
second theorem without proof. Poincaré’s version of Jacobi’s second theorem
is differs slightly from Jacobi’s original theorem. According to Poincaré, anarbitrary function S ( p1 ,, pn , Q1 ,, Qn ) leads the canonical transformation
if the following relations hold:
S
S
qi 
, Pi 
pi
Qi(i  1,..., n) (3.2-1)

In the modern view, minus signs should be put in front of these two
equations.
However, Poincaré’s developed his idea based on a serious
misunderstanding. Neither Jacobi nor Poincaré mentioned that function S

is a complete solution of the Hamilton–Jacobi equation. Instead, Poincaré
adopted the complete solution as a generating function and derived
Delaunay's equations of perturbation function from the canonical equationof motion. He believed that he succeeded in transforming the equation of
motion by deriving it from Delaunay’s equation. However, his procedure hassome gaps and cannot be followed. Subsequently, Poincaré found thisweakness in his argument, and his revision of Jacobi’s theorems was
published in 1905.
3.3 Poincaré's understanding of the canonical transformation in 1896–1897
In his two papers entitled “Sur une forme nouvelle des équations duproblème des trois corps," Poincaré described his new original properties ofcanonical transformation. Because the 1897 paper developed the idea
demonstrated in the 1896 paper, we focus on the former.
Similar to other mathematicians in the mid-nineteenth century,
Poincaré sought the transformation of variables that would reduce thedegrees of freedom in three-body problems. He tried to find transformations
that met the following conditions; linear transformations that retain the
canonical form of equations and equations of areas. He set the three-bodyproblem as follows.
Let A, B, and C be three masses ofm1 ( m2  m3) , m4 ( m5  m6) ,m7 ( m8  m9 ) whose generalized coordinates are A(q1 , q2 , q3 ), B(q4 , q5 , q6 ),C (q7 , q8 , q9 ) and pi  midxi
. Then the canonical equations of motion (1’),
dtwhich is composed of 18 equations, describes the three-body problem.
Poincaré noted that the canonical form is preserved if there are
relationships between the old and new variables hold without proof: Q dP   q dp  exact. (3.3-1)
i

i

i

i

After he mentioned the conditions that keep equations of areas
without proof,10 he provided two examples of the transformations.

Poincaré’s conditions of preserving the equation of area are(1) Q3k , Q3k 1 , Q3k 2 depend on only q3k , q3k 1 , q3k 2 , while P3k , P3k 1 , P3k 2 only10

p3k , p3k 1 , p3k 2 ,

Poincaré's transformation (  )p1  P1 , p4  P4 , q7  Q7 , q1  q7  Q1 , q4  q7  Q4 , P7  p1  p4  p7
(Similar relations hold for q2 , q3 , q5 , q6 , q8 , q9 . )Poincaré's transformation (  )Let G(Q7 , Q8 , Q9 ) be the center of gravity of the three bodies,D(1 , 2 , 3 ) be that of A and C ,Q1  q1  q7 , Q2  q2  q8 , Q3  q3  q9 , Q4  q4  1 , Q5  q5  2 , Q6  q6  3
m1  m2  m3 

and Pi  mi

m (m  m7 )m1m2
, m4  m5  m6  4 1, m7  m8  m9  m1  m4  m7
m1  m7
m1  m4  m7

dQi
where mi ' are masses after transformation.
dt

After he reduced the degrees of freedom from 18 to 12 while
preserving the canonical form of equations through these twotransformations, Poincaré performed a further reduction. He transformed(Qi , Pi ) (i  1,,6) to L, G, , l , g ,  and L, G, , l , g ,   , which are thevariables that appear in Delaunay’s equations. He defined transformation
as follows:Qk  k ( L, G, , l , g ,  ),
Qk  k ( L, G, , l , g ,  ),Pk  d kL3 dl(k  1, 2,3)
(3.3-2)Pk   d k
L3 dl (k  4,5, 6)

where  ,   are constants.
He then stated that the canonical form of equations is preserved if(2) The same relations between Q3k , Q3k 1 , Q3k 2 and q3k , q3k 1 , q3k 2 hold for
all k ,
(3) The same relations between P3k , P3k 1 , P3k 2 and p3k , p3k 1 , p3k 2 hold for
all k ,
where k  0,1, 2 .

PdQ
1
1  P2 dQ2  P3dQ3   ( Ldl  Gdg  d ) (3.3-3a)P4 dQ4  P5dQ5  P6dQ6   ( Ldl   Gdg   d ) (3.3-3b)
are exact without proof. He concluded the transformations defined by (3.3-2)
retain the canonical form of equations. Equations (1’) are finallytransformed todL
H dl H dt   dL
dt
 dl


 dl 
H
dL
H
 

dt
 dl  dt  ' dL
If we write old variables as (qi , pi ) and new variables as (Qi , Pi ) , and weneglect constants β and β’ , equations (3-2.3ab) are written as

 PdQ   p dq  exact. (3.3-4)
i

i

i

i

which are the modern relationships in canonical transformation.
Poincaré neither insisted that his original conditions of canonicaltransformation were given by 1-form nor proved them. However, historically,
it is remarkable that he provided the origin of the modern condition. He
gave proofs of relations (3.3-1) and (3.3-4), and he used them to develop his
idea of canonical transformation in subsequent publications.
3.4 Poincaré's attitude in 1899
In volume 3 of Les Méthodes Nouvelles (1899), chapter 29 is devoted
to a discussion of the principle of least action, which is the so-called
Hamilton’s principle of dynamics. Because Poincaré had written two papers
that were related to the principle of least action in 1896–189711, he devoted
one chapter to introducing related topics.
In his attention to the principle of least action, Poincaré provided
proof of his characterization of the canonical of transformation by 1-form
11“Sur les solutions périodiques et le principe de moindre action,” Comptes

rendus, CXXIII, 1896, pp.915-918, and “Les solutions périodiques et leprincipe de moindre action", Comptes rendus, CXXIV, 1897, pp.713-716.

(3.3.-4). His approach was different from the canonical transformation involume 1 of Les Méthodes Nouvelles.
Poincaré introduced the so-called Hamilton’s principle as follows
t1 
dq 
 J      H   pi i  dt  0. (3.4-1)
t0
dt 

He derived canonical equations of motion (1’) from this principle in a
way that differed from that shown in volume 1. In addition to the canonicalequations, in this principlet t1

 J   pi qi t t (3.4-2)
0should be 0 at the end points. We chose new variables (Q1 ,, Qn , P1, , Pn )such that PdQ   p dq  dS (3.4-3)
i

i

i

i

for some function S and set
t1 
dQi 
J      H   Pi
dt .
t0
dt 


Since
J J  dS
dt  St1  St0
dt

then

 J    J   S t t

t t1
0

or  J    Pi Qi 

t t1
t t0

（3.4-4）

Principle (3.4-1) holds in dynamical systems that are written using the newvariables. Poincaré then wrote that equation (3.4-4) is “equivalent” toequations (2’) for the same reason that (3.4-2) is “equivalent” to (1’) andequation (3.4-2) is “equivalent” to (3.4-4), and then equation (1’) is“equivalent” to (2’); that is, the canonical form of equations is preserved if(3.4-3) holds.
In these descriptions, we find that Poincaré attained almost thesame understanding that we have today. He gave the condition of thecanonical transformation in terms of exact differential 1-form (3.4-3) andproved it based on Hamilton's principle. However, it is not Poincaré’s final
attitude toward canonical transformation. We continue to follow hisarguments on canonical transformation.

In addition, in Les Méthodes Nouvelles, Poincaré described 1-form(3.3-1) in relation to invariant theory but not in relation to canonicaltransformation.
3.5 Poincaré’s final comprehension of the canonical transformationa) Poincaré’s final characterization of canonical transformation
Different from his work in Les Méthodes Nouvelles, Poincaré
discussed the practical usage of the theory of celestial mechanics in his
three-volume Leçons de Mécanique Céleste (1905–1910). However, as in Les
Méthodes Nouvelles, he discussed canonical transformation in the
introductory part of Leçons. He wrote that the understanding of certain
analytical transformations was indispensable in studying dynamics. Indeed,
he devoted an entire chapter to explaining the theory of canonicaltransformation.
Poincaré first considered not equations of motion but any equation
written in the canonical form:dxi H dyi
H

,

, H  H ( x1 , , xn , y1 , , yn ). (3.5-1)
dt yidt
xi
He set solutions to equations (3.5-1) as follows:xi  xi (t , 1 ,,  n )yi  yi (t, 1,,  n ) ,where 1 ,,  n are integral constants. By differentiating equations (3.5-1)
Poincaré showed that they are equivalent to
y
d
y
d
F
xi

xi i 
. (3.5-2)dt k d kt  kHe introduced new variables

X i  X i ( x1 ,, xn , y1 , yn ), Yi  Yi ( x1 ,, xn , y1 , yn ) (i  1,, n)and sought the transformation of variables ( x1 ,, xn , y1 , yn ) to newvariables. Then he demonstrated that if the relationship X dY   x dq  exact., (3-5.3)
i

i

i

i

holds, the new variables hold the relationship that corresponds to (3.5-2) bydifferentiating (3-5.3) with respect to  k (k  1,, n) and t . Poincaré thenconcluded that canonical transformation occurs if the old and new variables
are bounded by relationships (3-5.3); that is, the relationship he describedin 1897 paper without demonstrating it.
Next, Poincaré introduced the canonical equations of motion as
follows. Let ( X1 , X 2 , X 3 ),,( X n2 , X n1 , X n ) be positions of (n / 3) mass points
in orthogonal coordinates. Then kinematic energy is given as
2

1
 dX 
T   mi  i  ,2
 dt 

and total energy is H  T  U , where U is potential energy, and equationsof motions are given as
dX i2
U
.
mi 2  dt
X i
Poincaré set Yi  midX i
, noting that U depends only on X i and T
dt

only on X i . He indicated that X i , Yi and H construct canonical equationsof motion.
Poincaré defined curvilinear coordinates q1 ,, qn , which are
generalized coordinates in modern terminology, as follows:

X i   i (q1 ,, qn ) (3.5-4)
He differentiated (3.5-4), noting relationships among X i , X i , qi , qi
and setting
T pi .
qi

Since T is the second degree of the homogenous function of X i or pi ,Poincaré derived the following: p dq  Y dX .
i

Therefore,

i

i

i

 q dp   X dY  d  p q   X Y 
i

i

i

i

i i

i i

or  qi dpi   X i dYi  exact.

He then concluded that the transformation from ( X i , Yi ) to ( qi , pi ) was
canonical.
b) Poincaré’s version of Jacobi’s method
Poincaré entitled section Jacobi’s method. Let S be an unknown
Sfunction and set pi . Since total energy H was constant, he derived aqi
non-linear first order differential equation, which is the so-called Hamilton–
Jacobi equation:


S
S 
H  q1 ,, qn ,,,  const. (3.5-5)

q

q
1
n 

He adopted a complete solution to equation (3.5-5) that involved
arbitrary constants 1 ,, n as the “unknown function.” The total energy
H depends on these arbitrary constants, and (3.5-5) becomesH  q1 ,, qn ,

S
,
q1

,

S 
   ( 1 ,
qn 

, n ) .

In contrast, one obtains
SSdS   dqi  
d i
qi
i
and
S
 pi ,
qi

where  1 ,

S
 i
i

,  n are new arbitrary constants. Poincaré noted that there are

2n relationships among 4n variables and decided that 2n variables could be
regarded as functions of another 2n variable. For example,(1 , , n ,  1 , ,  n ) could be regarded as functions of (q1 , , qn , p1 , , pn ) . If
one set the former as the new variable and the latter as the old variable, thefollowing relationship holds: d    q dp  d (S   q p )
i

i

i

i

i

i

Then the transformation is canonical. The canonical equations forqi , pi are transformed tod  i H  d i
H


,

 0. (i  1,
dt i idt i, n) (3.5-6)

Poincaré easily obtained the integral of equations (3.5-6) as follows:i  const.  i t  iiwhere  1 ,, n are integral constants. He noted that the solutions to theoriginal equations written as (qi , pi ) are obtained through (3.5-6). That is,
he transformed the equations to facilitate their solutions. In other words,
Poincaré solved canonical equations using the complete solution to the
Hamilton–Jacobi equation.
Poincaré termed this procedure Jacobi’s method. It is true that
Jacobi had solved the canonical equations through a complete solution to
the Hamilton–Jacobi equation. Consequently, mathematicians may find it
difficult to call Poincaré’ procedure Jacobi’s method. However, Poincaré’s
procedure has a different aspect: one at first solves the Hamilton–Jacobi
equation to transform the canonical equation to a simple one using acomplete solution of the Hamilton–Jacobi equation. One then obtains thesolution to original canonical equations through the solutions to simpler
equations. Different from Jacobi’s method, Poincaré’s procedure involves theidea of canonical transformation.
It was Jacobi’s idea to generate a function of the canonicaltransformation. However, he did not develop a complete solution to thegenerating function of the canonical transformation. However, Poincaré
noted this relationship. Based on this understanding, Poincaré
demonstrated his original method, calling it Jacobi’s method.
Subsequently, Poincaré discussed the case where H explicitly
depends on time, noting the 1-form condition and showing that function H
is not retained in the transformed canonical equation.
In demonstrating Poincaré’s version of Jacobi’s method, Poincarésucceeded in demonstrating a complete solution, which became a generating
function, which was the property he had suspected in 1892. However, basedon our modern knowledge, Poincaré’s demonstration was theoretically weak.
He vaguely grasped that the generating function gives 2n relations among
4n old and new variables, and he did not examine it further. However, in
1907, Chariler distinguished four types of generating function:
S (q, Q), S (q, P), S (p, Q), S (p, P)

where (q,p) are old variables and (Q, P) are new variables. FromCharlier’s framework, Poincaré had chosen the case of S (q, P) and naturally
had considered it the complete solution. Charlier’s approach, although it
was based on Jacobi’s second theorem, was quite different from Poincaré’s
approach. Charlier’s books contain ideas that were important in thedevelopment of the theory of canonical transformation. For example, hederived the origin of the action-angle variables, which Schwarzschildeffectively used to explain the Stark effect in 1916. However, here we only
indicate that his approach differed from Poincaré’s approach, and we note
his four types of the generating function.
4. Concluding Remarks: How Did Poincaré's Idea in 1899 Circulate Among
Physicists?
Today we accept the results that Poincaré achieved in 1899 but not
his final results for the following reason. Max Born‘s, influential textbook onmechanics, 12 at first adopted the same approach. Born derived the 1-form
condition of (3.4-3) from Hamilton’s principle. In fact, Born referred toPoincaré's Les Méthodes Nouvelles in his Atommechanik. Around 1916,
physicists realized the Hamilton–Jacobi theory offered effective
mathematical tools to the old quantum theory. Since the Hamilton–Jacobi
theory had been developed in celestial mechanics, mainly German-speaking
physicists had read and referred to Charlier's Die Mechanik des Himmels instudying this theory.
Compared with Charlier's books in German, Poincaré's books in the
French language were not as popular. However, Born needed themathematical method to analyze a degeneration system in discussing the12

Vorlesungen über Atommechanik , 1925, Julius Springer.

structure of the helium atom. In Göttingen, he read Poincaré's Les
Méthodes Nouvelles with Heisenberg around 1922–1923.13 However, Born’scolleague David Hilbert, in his lectures entitled “Mathematical Foundationof Quantum Theory” (1922–1923), constructed a theory of geometrical optics
and dynamics based on the variational principle.14 In his lectures, Hilbert’smechanics is based on Hamilton’s principle. Born then decided to construct
his theory based on Hamilton’s principle, and he chose Poincaré's 1899
approach. In due course, Born inserted Charlier’s results, including the fourtypes of generating functions. His description resembles that provided inmodern textbooks.
Lothar Nordheim, Born’s assistant, with Fues published an article15
that introduced the basis of the Hamilton–Jacobi theory. Their introduction
canonical transformation was almost the same as in Born’s book.
The influential modern English textbook on mechanics, HerbertGoldstein’s Classical Mechanics, refers to Nordheim and Fues’s descriptionof canonical transformations. Because of this process, Poincaré's 1899
approach should become the standard today.

D. C. Cassidly, Uncertainty: The life and Science of Werner Heisenberg,
1992, W.H. Freewan & Company, Chapter 8.
14 D. Hilbert, Mathematische Grundlagen der Quantentheorie (T. Sauer and
U. Majer eds.) David Hilbert's Lectures on the Foundations of Physics
1915-1927. 2009, Springer, pp. 507-601.
15 N. Nordheim and E.Fues, Die Hamilton-Jacobische Theorie derDynamik“ in Handbuch der Physik (1927), pp. 91-130.
13