- Email: [email protected]
Physics and pseudogroups
Vol. 13 (1978)
REPORTS
PHYSICS
ON MATHEMATICAL
PHYSICS
No. 3
AND PSEUDOGROUPS* J. F. POMMARET
Coll&ge de France, Paris, France (Received
September
20, 1976)
The purpose of this paper is to give a new interpretation of the famous “Equivalence Problem” of E. Cartan, within the framework of differential invariants and bundles of geometric objects. The first example is the well-known theorem of G. Darboux for closed 2-forms. A second striking example is related to the so-called “Helmholtz postulate” in classical thermodynamics. Finally, we work out a counter-example of V. Guillemin and S. Sternberg, exhibiting geometric object involved and giving a new insight into the deformation theory of such structures.
The history of the theory of differentiable groups of transformations, solutions of systems of partial differential equations, formerly called in$nite groups, today Lie pseudogroups (the Lie groups were known as finite groups), began in 1897 with the Leipziger Berichte of S. Lie. Our purpose has been to use the work of a student of Lie, E. Vessiot, dating back to 1903 and forgotten from this time on. For a detailed study, including many examples, we refer the reader to [2]. For a brief but complete account of the theory and its relation to the formal theory of linear systems of p.d.e., we refer him to [4], [6]. We shall first recall some basic definitions. Then we shall focus on a problem of local solvability of some non-linear sequence, the so-called equivalence problem of E. Cartan. Finally we shall give examples; one of them is a key to a better understanding of axiomatic thermodynamics. We also present a counterexample adapted from [5]. Let X be a differentiable connected paracompact manifold with dimX = n and let Y be a copy of X. All the manifolds and maps that we shall consider in the sequel will be differentiable. DEFINITION. By a finite transformation f:
U c X --+ V c Y with source x E U and target y E V, we mean a local automorphism of X, denoted by yk = fk(xi), with convenient local coordinates. By an inJinitesimaI transformation we mean the map exp(tQ where t is a vector field, that is to say a section of the tangent bundle T = T(x), and t is a small parameter. We have yk = x’+~[~(x)+ ... * Presented
at the Symposium
on Methods
of Differential
June 1976, Warsaw. [3451
Geometry
in Physics and Mechanics,
346
J. F. POMMARET
Remark. We shall identify f with its graph which is also a local section of the trivial bundle Xx Y over U c X. Let 8 be a fibered manifold or a bundle over X with local coordinates (xi, vk), we shall denote by J,(8) the bundle of q-jets of d using local coordinates (xi, $‘, pi) with EC=@ i, . . . . P,) and 1 < IpI = pi + . . . +pu, < q.
DEFINITION. By j,: 8 -+ J,(8) we mean a map that transforms any section f of &’over an open set U c X into the section j,(,‘J): (x) + (.xJk(x), 13,J”(x)) of J,(8) over U, with 1 < 1~’ < q. DEFINITION. If @: J,(8) --+9 is a morphism of fibered manifolds over X, we call the composition 29 = @ o j4: & + 9 a non-linear operator of order q. By the kernel of 9 with respect to a section co of 9, we mean the set of all sections f of B over U c X such that 9 -f= coon U. Using local coordinates, we have @‘(x, fk(x), dpfk(x)) = d(x)
Vx E II.
DEFINITION. By a non-linear system of partial diferential equations we understand a fibered submanifold 92!eof J,(8), that is to say a submanifold with surjective induced projection: W, + X. From now on we shall be only concerned with involutive systems ([l], [6]). DEFINITION. By a Lie pseudogroup of transformations of X, we understand a group of transformations, solutions of a non-linear system $A?4 c J,(Xx Y). In the sequel we shall deal only with transitive Lie pseudogroups, that is to say those defined by systems W, with surjective induced projection 9, + Xx Y. We shall now introduce the Lie group GL,(n, R) with the coordinates p: or simply p or even 2i.
It can be considered as an open subset of the fiber of J,(Xx
det[p:] # 0 with a composition in the first order jet coordinates
law identical to that of differentiation,
ay az = _az ._. _
ax
Y) defined by
that is to say,
ay ax
DEFINITION. By a bundle 9 of geometric objects, of order q, we mean a bundle 9 over X with local coordinates (x’, u”), provided with an action of GL,(n, R) on its fiber, giving rise to the transition laws:
x = y(x),
ii =
q&g).
EXAMPLE. Transformation laws of Christoffel’s symbols in the Riemannian geometry. The key trick is to notice that any section w of 9 gives rise to a non-linear system of p.d.e. defining the finite transformations of a Lie pseudogroup I’.
PHYSICS AND PSEUDOGROUPS
347
then
Conversely, it can be shown ([21, [4], [6]) that we can relate a bundle F of geometric objects and a section o of it to any transitive Lie pseudogroup F. Writing G-‘(~(JJ),~) = m(x) as @(~(~),p) = (0“(x) we are able to get out the source, to find the differential invariants @‘(y,p) of .F and to obtain a non-linear operator 9 with kernel F with respect to LC). We come now to the main point of this paper. Let v: X -+ X be any invertible map. DEFINITION. The Lie pseudogroup 7 = v 0 ro q-l is said to be similar to I’. Moreover we define the section 0 = v(m) to be the section of 9 such that G@(x)) .
PROPOSITION. The Jinite equations of T are @(of, Proof:
p) = O(x).
Using the evident, symbolic form, we have: &%i) = (fJ90 I’0 @-l)(Z) = (470 r)(c0) = &u) = 0
for any finite transformation of ry because, by definition, for any finite transformation of .F we have F(o) = or). Conversely, if we are given two sections u) and W of 9, an important problem will be to know if there exists q such that 0 = Q)(O).In fact, we have to solve, at least locally, the non-linear p.d. equations :
9.f=G
with
f=qr’.
Of course, W must satisfy, in general, some integrability conditions. However, it should be too long to speak about them and we refer the reader to [2], [4], [6]. We shall now give two examples and a counterexample in order to study the possibility of solving this problem, even locally. EXAMPLEI.
dimX = n = 2m, 9
= A2T*.
A section o of A2F can be viewed as a 2-form o = orj(x)dxi ~dxj. It is easy to check that the non-linear system W1 f(w> = w is involutive when o has a maximal rank, that is to say det [mij(x)] # 0, and when it is closed, that is to say do = 0 or a;$) +aaki(x)
: am&y)
I
o
-XT=.
But we know that these conditions are satisfied by the 2-form W = dxl A dx2 + . . . + +dP1 A dx” and our problem will be to look for a nice transformation of coordinates 9
J. F. POMMARET
348 such that pi(m) = W. A positive,
local answer
to thjfi question
was given by G. Darboux
in 1876 and is currently used in theoretical mechanics. The pseudogroup to U, describes the change of states of a given dynamical system. EXAMPLE II. Let
X = R3.
us consider
we introduce
corresponding
r:
R3 -+ R3 : y’ = x1 +a,
the set o of exterior SI = dx’,
forms
y2 = x2 +f(x’),
c:f(x’) y3 = x3 + L-. ax1
If
a, /?, y:
,8 = dx*-x3dx1,
y = dx1Adx3
then it is easy to check that r is defined by the non-linear system W1 f(m) = u). It is then possible to show that the bundle F is such that a section m of it can be viewed as a set of exterior forms CC,j3, y such that : a A y = 0, a A /3 # 0, /I A y # 0 and that the correspondif and only if there exist constants cl, cl, ing non-linear system f(w) = COis involutive c2, cl and c3 such that: da = c,uA/~+c;~, d/l = c,ar\/?+cS.y, dy = c~#?AY. At this time, taking the ordinary exterior derivative, we must have: 4(c,--3)
In particular However,
=
0
and
c;c*-c;c3
= 0
the above example gives: da = 0, dp = y, dy = 0. we can choose the constants c such that: da = y,
d@ = -Y,
dy = 0
and it is easy to find a set of forms, solutions of this analytic system of p.d.e. In fact, as we must have a~ y = 0, it is just necessary to solve the equations CZA da = 0, d(a+/l) = 0 together with the inequality u A p # 0. The reader will discover that it is the kind of situation one finds in thermodynamics, where a is the l-form of heat exchange and /I the l-form of work exchange of a thermodynamical system with the internal state variables xi (n = 3 in the example above). Of course it is known that, when aA da = 0, then locally a = f(x)dg(x) and it is possible to make a transformation of coordinates such that, in the new coordinates, a = T(x)dxl where T is the absolute temperature of the system and x1 the entropy variable. It is also well known that, if d(a+/l) = 0, then locally we can write cl+B = du where U(X) is the internal energy of the system. Finally there is another property of the system, which is known as Helmholtz postulate, saying that it is always possible to find new coordinates, called normal coordinates, such that
one can write
c1 = T(x)dxl
and B = i$2 pr( x )d xi or equivalently
such
that
T(x)
&4(x) =T’ We shall prove that it is no longer a postulate, responding problem of local solvability.
giving
a positive
answer
to the cor-
PHYSICS
Proof
AND
PSEUDOGROUPS
349
When n = 2, as
a/\/j =
= T(X)dXlAdU
ar\(a+@)
=
T(x)-gg$jdxl A dx2 >
we can take the new variables y’ = x1, y2 = U(X) and get ,8 = dy2 - T(y)dyl. v(y) be a non-constant
Let now
dy2 = T(y). Then integral of the differential equation __dy’
am F+T(Y)T =o. It follows that ,!I =
-av’o,dv
with
av(Y) # 0
ayz.
w
Using the new variables z1 = yl, z2 = v(y), we finally obtain a = T(z)dz’, /? = b(z)dz2. Similarly, when it > 2, we can choose the new variables: y’ = x1, y2 = u, y3 = x3, . . . . . . . y” = X” to obtain a = T(y)dy’, /I = dy2- Tb)dy’. Let us now consider y3, . . . , y” to be parameters and let us use the same trick as above. We get dy3 +
Finally,
taking
= /92(z)dz2+
z1 = yl,
.z2 = v(y),
. + ___ av(y)dy” ..
ay"
1) .
z3 = y3, . . . , z” = y” we have
tl = T(z)dz’,
/3
. . . +/!?,(z)dz”.
In our proof we have only been using the condition
CIA/I # 0.
COUNTEREXAMPLE. The example exhibited by V. Guillemin and S. Stemberg [5] is that of a pseudogroup .P defined by eliminating the arbitrary parameters a, b, c, d from the equations (X = R~):
dy’
_
dy2 y2dy’-y’dy2+dy3
=
h4 dy’
_
dxl
- 1 0 0 0 o- -
_-b
01000
dx2
00100
x2dx1 - x’dx2 + dx3
abcl0
dx4
a d 0
1
dx5
. _
We have to use the methods of [3] in order to construct the bundle F. A very (! !) tedious computation shows that r is the Lie pseudogroup that leaves the following exterior forms unchanged : 1’ = dx’,
12 = dx2,
A3 = x2dx1 - x’dx2 + dx3,
v1 = - x2dx’ A dx2 A dx4 - x’dx’ A dx2 A dx5 + d.x’ A dx3 A dx5 + dx2 A dx3 A dx4, v2 = -x’dx’r\dx2~dx4+x2dx1~dx2~dx5+dx1~dx3~dx4-dx2~dx3/\dx5.
J. F. POMMARET
350
Moreover, in the general situation, we have to know five exterior forms ill, A2, 13, vl, v2 such that: ;11Av’+?.2Av2 .
with the integrability d?.’ = 0,
= 03
A2AV1-A1AV2
= 0,
A3AV’ = 0,
A3AV2 = 0
conditions : dA2 = 0,
dA3+2A1/,A2
= 0,
dv’ = 0,
dv2 = 0.
Another set of solutions may be obtained from the one already given, by keeping Iz’, L2, A3 unchanged and adding A(x’, x2, x3)dx1 A dx2 A dx3 to vl, B(x’, x2, x3)dx1 A A dx2 A dx3 to v2. It can be shown that, for some A and B, there is no local change of coordinates that would allow one to pass from the first set of forms to the second one. REFERENCES
H.: J. Differential Geometry 1 (1967). 269. Pommaret, J. F.: Ann. Inst. K Poimare’ 18 (1973), 285. -: C. R. Acad. Sci. Paris 280 (1975), 1963. -: ibid. 282 (1976), 587, 635. Guillemin, V. and Stemberg, S.: J. Differential Geometry 1 (1967), 127. Pommaret, J. F: Systems of partial difirential equations and Lie pseudogroups, Breach, London, New-York 1978.
[I] Goldschmidt,
[2] [3] [4] [S] [6]
Gordon
and
REPORTS
PHYSICS
ON MATHEMATICAL
PHYSICS
No. 3
AND PSEUDOGROUPS* J. F. POMMARET
Coll&ge de France, Paris, France (Received
September
20, 1976)
The purpose of this paper is to give a new interpretation of the famous “Equivalence Problem” of E. Cartan, within the framework of differential invariants and bundles of geometric objects. The first example is the well-known theorem of G. Darboux for closed 2-forms. A second striking example is related to the so-called “Helmholtz postulate” in classical thermodynamics. Finally, we work out a counter-example of V. Guillemin and S. Sternberg, exhibiting geometric object involved and giving a new insight into the deformation theory of such structures.
The history of the theory of differentiable groups of transformations, solutions of systems of partial differential equations, formerly called in$nite groups, today Lie pseudogroups (the Lie groups were known as finite groups), began in 1897 with the Leipziger Berichte of S. Lie. Our purpose has been to use the work of a student of Lie, E. Vessiot, dating back to 1903 and forgotten from this time on. For a detailed study, including many examples, we refer the reader to [2]. For a brief but complete account of the theory and its relation to the formal theory of linear systems of p.d.e., we refer him to [4], [6]. We shall first recall some basic definitions. Then we shall focus on a problem of local solvability of some non-linear sequence, the so-called equivalence problem of E. Cartan. Finally we shall give examples; one of them is a key to a better understanding of axiomatic thermodynamics. We also present a counterexample adapted from [5]. Let X be a differentiable connected paracompact manifold with dimX = n and let Y be a copy of X. All the manifolds and maps that we shall consider in the sequel will be differentiable. DEFINITION. By a finite transformation f:
U c X --+ V c Y with source x E U and target y E V, we mean a local automorphism of X, denoted by yk = fk(xi), with convenient local coordinates. By an inJinitesimaI transformation we mean the map exp(tQ where t is a vector field, that is to say a section of the tangent bundle T = T(x), and t is a small parameter. We have yk = x’+~[~(x)+ ... * Presented
at the Symposium
on Methods
of Differential
June 1976, Warsaw. [3451
Geometry
in Physics and Mechanics,
346
J. F. POMMARET
Remark. We shall identify f with its graph which is also a local section of the trivial bundle Xx Y over U c X. Let 8 be a fibered manifold or a bundle over X with local coordinates (xi, vk), we shall denote by J,(8) the bundle of q-jets of d using local coordinates (xi, $‘, pi) with EC=@ i, . . . . P,) and 1 < IpI = pi + . . . +pu, < q.
DEFINITION. By j,: 8 -+ J,(8) we mean a map that transforms any section f of &’over an open set U c X into the section j,(,‘J): (x) + (.xJk(x), 13,J”(x)) of J,(8) over U, with 1 < 1~’ < q. DEFINITION. If @: J,(8) --+9 is a morphism of fibered manifolds over X, we call the composition 29 = @ o j4: & + 9 a non-linear operator of order q. By the kernel of 9 with respect to a section co of 9, we mean the set of all sections f of B over U c X such that 9 -f= coon U. Using local coordinates, we have @‘(x, fk(x), dpfk(x)) = d(x)
Vx E II.
DEFINITION. By a non-linear system of partial diferential equations we understand a fibered submanifold 92!eof J,(8), that is to say a submanifold with surjective induced projection: W, + X. From now on we shall be only concerned with involutive systems ([l], [6]). DEFINITION. By a Lie pseudogroup of transformations of X, we understand a group of transformations, solutions of a non-linear system $A?4 c J,(Xx Y). In the sequel we shall deal only with transitive Lie pseudogroups, that is to say those defined by systems W, with surjective induced projection 9, + Xx Y. We shall now introduce the Lie group GL,(n, R) with the coordinates p: or simply p or even 2i.
It can be considered as an open subset of the fiber of J,(Xx
det[p:] # 0 with a composition in the first order jet coordinates
law identical to that of differentiation,
ay az = _az ._. _
ax
Y) defined by
that is to say,
ay ax
DEFINITION. By a bundle 9 of geometric objects, of order q, we mean a bundle 9 over X with local coordinates (x’, u”), provided with an action of GL,(n, R) on its fiber, giving rise to the transition laws:
x = y(x),
ii =
q&g).
EXAMPLE. Transformation laws of Christoffel’s symbols in the Riemannian geometry. The key trick is to notice that any section w of 9 gives rise to a non-linear system of p.d.e. defining the finite transformations of a Lie pseudogroup I’.
PHYSICS AND PSEUDOGROUPS
347
then
Conversely, it can be shown ([21, [4], [6]) that we can relate a bundle F of geometric objects and a section o of it to any transitive Lie pseudogroup F. Writing G-‘(~(JJ),~) = m(x) as @(~(~),p) = (0“(x) we are able to get out the source, to find the differential invariants @‘(y,p) of .F and to obtain a non-linear operator 9 with kernel F with respect to LC). We come now to the main point of this paper. Let v: X -+ X be any invertible map. DEFINITION. The Lie pseudogroup 7 = v 0 ro q-l is said to be similar to I’. Moreover we define the section 0 = v(m) to be the section of 9 such that G@(x)) .
PROPOSITION. The Jinite equations of T are @(of, Proof:
p) = O(x).
Using the evident, symbolic form, we have: &%i) = (fJ90 I’0 @-l)(Z) = (470 r)(c0) = &u) = 0
for any finite transformation of ry because, by definition, for any finite transformation of .F we have F(o) = or). Conversely, if we are given two sections u) and W of 9, an important problem will be to know if there exists q such that 0 = Q)(O).In fact, we have to solve, at least locally, the non-linear p.d. equations :
9.f=G
with
f=qr’.
Of course, W must satisfy, in general, some integrability conditions. However, it should be too long to speak about them and we refer the reader to [2], [4], [6]. We shall now give two examples and a counterexample in order to study the possibility of solving this problem, even locally. EXAMPLEI.
dimX = n = 2m, 9
= A2T*.
A section o of A2F can be viewed as a 2-form o = orj(x)dxi ~dxj. It is easy to check that the non-linear system W1 f(w> = w is involutive when o has a maximal rank, that is to say det [mij(x)] # 0, and when it is closed, that is to say do = 0 or a;$) +aaki(x)
: am&y)
I
o
-XT=.
But we know that these conditions are satisfied by the 2-form W = dxl A dx2 + . . . + +dP1 A dx” and our problem will be to look for a nice transformation of coordinates 9
J. F. POMMARET
348 such that pi(m) = W. A positive,
local answer
to thjfi question
was given by G. Darboux
in 1876 and is currently used in theoretical mechanics. The pseudogroup to U, describes the change of states of a given dynamical system. EXAMPLE II. Let
X = R3.
us consider
we introduce
corresponding
r:
R3 -+ R3 : y’ = x1 +a,
the set o of exterior SI = dx’,
forms
y2 = x2 +f(x’),
c:f(x’) y3 = x3 + L-. ax1
If
a, /?, y:
,8 = dx*-x3dx1,
y = dx1Adx3
then it is easy to check that r is defined by the non-linear system W1 f(m) = u). It is then possible to show that the bundle F is such that a section m of it can be viewed as a set of exterior forms CC,j3, y such that : a A y = 0, a A /3 # 0, /I A y # 0 and that the correspondif and only if there exist constants cl, cl, ing non-linear system f(w) = COis involutive c2, cl and c3 such that: da = c,uA/~+c;~, d/l = c,ar\/?+cS.y, dy = c~#?AY. At this time, taking the ordinary exterior derivative, we must have: 4(c,--3)
In particular However,
=
0
and
c;c*-c;c3
= 0
the above example gives: da = 0, dp = y, dy = 0. we can choose the constants c such that: da = y,
d@ = -Y,
dy = 0
and it is easy to find a set of forms, solutions of this analytic system of p.d.e. In fact, as we must have a~ y = 0, it is just necessary to solve the equations CZA da = 0, d(a+/l) = 0 together with the inequality u A p # 0. The reader will discover that it is the kind of situation one finds in thermodynamics, where a is the l-form of heat exchange and /I the l-form of work exchange of a thermodynamical system with the internal state variables xi (n = 3 in the example above). Of course it is known that, when aA da = 0, then locally a = f(x)dg(x) and it is possible to make a transformation of coordinates such that, in the new coordinates, a = T(x)dxl where T is the absolute temperature of the system and x1 the entropy variable. It is also well known that, if d(a+/l) = 0, then locally we can write cl+B = du where U(X) is the internal energy of the system. Finally there is another property of the system, which is known as Helmholtz postulate, saying that it is always possible to find new coordinates, called normal coordinates, such that
one can write
c1 = T(x)dxl
and B = i$2 pr( x )d xi or equivalently
such
that
T(x)
&4(x) =T’ We shall prove that it is no longer a postulate, responding problem of local solvability.
giving
a positive
answer
to the cor-
PHYSICS
Proof
AND
PSEUDOGROUPS
349
When n = 2, as
a/\/j =
= T(X)dXlAdU
ar\(a+@)
=
T(x)-gg$jdxl A dx2 >
we can take the new variables y’ = x1, y2 = U(X) and get ,8 = dy2 - T(y)dyl. v(y) be a non-constant
Let now
dy2 = T(y). Then integral of the differential equation __dy’
am F+T(Y)T =o. It follows that ,!I =
-av’o,dv
with
av(Y) # 0
ayz.
w
Using the new variables z1 = yl, z2 = v(y), we finally obtain a = T(z)dz’, /? = b(z)dz2. Similarly, when it > 2, we can choose the new variables: y’ = x1, y2 = u, y3 = x3, . . . . . . . y” = X” to obtain a = T(y)dy’, /I = dy2- Tb)dy’. Let us now consider y3, . . . , y” to be parameters and let us use the same trick as above. We get dy3 +
Finally,
taking
= /92(z)dz2+
z1 = yl,
.z2 = v(y),
. + ___ av(y)dy” ..
ay"
1) .
z3 = y3, . . . , z” = y” we have
tl = T(z)dz’,
/3
. . . +/!?,(z)dz”.
In our proof we have only been using the condition
CIA/I # 0.
COUNTEREXAMPLE. The example exhibited by V. Guillemin and S. Stemberg [5] is that of a pseudogroup .P defined by eliminating the arbitrary parameters a, b, c, d from the equations (X = R~):
dy’
_
dy2 y2dy’-y’dy2+dy3
=
h4 dy’
_
dxl
- 1 0 0 0 o- -
_-b
01000
dx2
00100
x2dx1 - x’dx2 + dx3
abcl0
dx4
a d 0
1
dx5
. _
We have to use the methods of [3] in order to construct the bundle F. A very (! !) tedious computation shows that r is the Lie pseudogroup that leaves the following exterior forms unchanged : 1’ = dx’,
12 = dx2,
A3 = x2dx1 - x’dx2 + dx3,
v1 = - x2dx’ A dx2 A dx4 - x’dx’ A dx2 A dx5 + d.x’ A dx3 A dx5 + dx2 A dx3 A dx4, v2 = -x’dx’r\dx2~dx4+x2dx1~dx2~dx5+dx1~dx3~dx4-dx2~dx3/\dx5.
J. F. POMMARET
350
Moreover, in the general situation, we have to know five exterior forms ill, A2, 13, vl, v2 such that: ;11Av’+?.2Av2 .
with the integrability d?.’ = 0,
= 03
A2AV1-A1AV2
= 0,
A3AV’ = 0,
A3AV2 = 0
conditions : dA2 = 0,
dA3+2A1/,A2
= 0,
dv’ = 0,
dv2 = 0.
Another set of solutions may be obtained from the one already given, by keeping Iz’, L2, A3 unchanged and adding A(x’, x2, x3)dx1 A dx2 A dx3 to vl, B(x’, x2, x3)dx1 A A dx2 A dx3 to v2. It can be shown that, for some A and B, there is no local change of coordinates that would allow one to pass from the first set of forms to the second one. REFERENCES
H.: J. Differential Geometry 1 (1967). 269. Pommaret, J. F.: Ann. Inst. K Poimare’ 18 (1973), 285. -: C. R. Acad. Sci. Paris 280 (1975), 1963. -: ibid. 282 (1976), 587, 635. Guillemin, V. and Stemberg, S.: J. Differential Geometry 1 (1967), 127. Pommaret, J. F: Systems of partial difirential equations and Lie pseudogroups, Breach, London, New-York 1978.
[I] Goldschmidt,
[2] [3] [4] [S] [6]
Gordon
and