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On the multimomentum bundles and the Legendre maps in field theories
Vol. 45 (2000)
REPORTS ON MATHEMATICAL
No. I
PHYSICS
ON THE MULTIMOMENTUM BUNDLES AND THE LEGENDRE MAPS IN FIELD THEORIES A. ECHEVERR~A-ENR~QUEZ, M. C. Mu~~oz-LECANDAand N. ROM,&ROY Departamento de Matematica Aplicada y Telematica Edificio C-3, Campus Norte UPC C/ Jordi Girona I, E-08034 Barcelona, Spain (e-mails: [email protected], [email protected]) (Received
March
18.
1999)
We study the geometrical background of the Hamiltonian formalism of first-order classical field theories. In particular, different proposals of multimomentum bundles existing in the usual literature (including their canonical structures) are analyzed and compared. The corresponding Legendre maps are introduced. As a consequence, the definition of regular and almost-regular Lagrangian systems is reviewed and extended in different but equivalent ways. Key words: AMS
1.
jet bundles,
s. c. (1991):
53C80,
classical
field theories,
Legendre
55R10,
58A20,70G50,
70H99.
map,
Hamiltonian
formalism
Introduction The standard
geometric
structures
underlying
the covariant
Lagrangian
descrip-
tion of first-order field theories are first-order jet bundles J’E 5 E 1 A4 and their canonical structures [5, 7, 9, 18, 19, 23, 271. Nevertheless, for the covariant Hamiltonian formalism of these theories there are several choices for the phase space where this formalism takes place. Among all of them, only the multisymp&tic models will deserve our attention in this work. So, in [ 11, 12, 17, 241 (see also [15-171) the multimomentum phase space is taken to be Mn I A’;T* E, the bundle of m-forms on E (m being the dimension of M) vanishing by the action of two n-vertical vector fields. In [3, 4, 20, 211 J’n* = AyT*E/ArT”E is used as the multimomentum phase space (where ArT*E is the bundle of .nsemibasic m-forms in E). Finally, in [4, 8, 9, 25, 261 the basic choice is the bundle l-l = n*TM @ V*(n) 8 rr*A”T*M (here V*(x) denotes the dual bundle of V(n): the n-vertical subbundle of TE) which, in turn, is related to J ’ E* s rr*TM 8 T*E @In*AmT*M. The origin of all these multimomentum bundles is related to the different Legendre maps which arise essentially from the fiber derivative of the Lagrangian density (see Section 4). All these choices have special features. So, Mn and JIE* are endowed with natural multisymplectic forms. In papers such as [ 121, the former is used for ob-
A. ECHEVERRiA-ENRfQUEZ, M. C. MUNOZ-LECANDA and
86
N.
ROMAN-ROY
taining the Poincark-Curtan form in J’E, which is needed for the Lagrangian formalism. This is done by defining the suitable Legendre map connecting J’ E and M7t. On the other hand, the dimensions of J’ E, Mrr and 1’ E” are not equal: in fact, dim Mn = dim J’E + 1 and dim J’E* = dim J’E + (dim M)*; but dim Il = dim J’n* = dim J1 E and the choice of both J ‘rr* and Il as multimomenturn phase spaces allows us to formulate coherent covariant Hamiltonian formalisms for field theories. Finally, the construction of J1rr* is closely related to Mrc, but their relation to fl and J’ E* is not evident at all. Hence, the aim of this work is to carry out a comparative study of these multimomentum bundles, and introduce the canonical geometrical structures of Mn and J ’ E*. In every case, the corresponding Legendre map is also defined. An interesting conclusion of this study is that the multimomentum bundles J’n* and Il are canonically diffeomorphic. Finally, using the different Legendre maps, we can classify Lagrangian systems in field theory into (hyper)regular and almost-regular, in different but equivalent ways, attending to the characteristics of these maps. All manifolds are real, paracompact, connected and C”. All maps are Co3 and summation over crossed repeated indices is understood.
2.
Multimomentum
bundles
Let n : E -+ A4 be a fiber bundle (dimE = N+m, dim44 = m), x1 : J’E + E the first-order jet bundle of local sections of n, and 17’ = r o n ‘. .I’ E is an affine bundle modelled on n*T*M @ V(n). A local chart of natural coordinates in E adapted to the bundle E += M will be denoted by (x@, yA). The induced local chart in J’E is denoted by (xc”, yA, $). DEFINITION 1. The bundle (over E) J’E*
:= n*TM @‘ET*E 8’~ n*A”T*M
is called the generalized multimomentum M. We denote the natural projections J’E* + M. The
local
(xw, yA, p;, pz) that
system
(x@, yA)
in J’E*
bundle associated with the bundle n : E + by 5’ : J’E” -+ E and 3 := Tro;i :
induces
as follows:
a local
if y E J’E*,
yj& y@(f:dx” + g;dyA),
J’ =
system
of
with y 5
63 d”xly,
where dmx G dx’ A _. . A dx”‘, and therefore X’“(Y)
=
x’“((n
0
?>(y)),
VA(Y)
= YA(;l(Y)>l
natural
coordinates
y 5
x, we have
(1)
ON THE MULTIMOMENTUM BUNDLES AND THE LEGENDRE MAPS IN FIELD THEORIES
p:(y)
87
= y
a a A...A axm ax1
where i3”‘x = -
DEFINITION 2. The bundle
l-l := n*TM @‘EV*(r)
* (over
E)
@3En*A”T*M
= u
T,(,y,M @ V;(z)
@ LI”‘T;~,,,M
YCE
is called the reduced multimomentum bundle associated with the bundle n : E + We denote the natural projections by p’ : II + E and 3’ := n o p’ : I7 -+ M.
hf.
From the local system (xl*, yA) we can construct a natural system of coordinates (xp, yA, pz) in l7 as follows: considering y E l7, we write
x’“(Y) = p;(y)
= 9
dxCL,-!-
A...
-!-
ayA’ ad
the dual basis of in V*(n) aYA
as
The relation between these multimomentum 6:
which
= YA(P'(rN;
YAW
a
and denoting expressed
x%‘(Y));
is induced
DEFINITION
A-
a axrn >
by {CA), an element
j E l-l is
bundles is given by the (onto) map
J’E*
---+
n,
(xIL, YA, P$ P;)
*
(xP”, YA*PZ),
by the natural restriction
’
T* E -+ V* (R.).
3. Consider the multicotangent
bundle A”T*E.
Then, for every YE:E
we define A;ITcE := {v E A”T;E
: i(ul)i(uz)y
= 0 , ~1,
~2
E
V,(X)}.
The bundle (over E) Mrr s AyT*E
:= u Y~E
AyT;E
= U{(Y+)
: a E AyT;E)
YEE
will be called the extended multimomentum bundle associated with the bundle :ir : E --+ M. We denote the natural projections by r3’ : Mrr --, E and I?’ :Mrr-+,M.
88
A. ECHEVERRfA-ENRiQUEZ,M. C. MUfiOZ-LECANDA andN. ROMAN-ROY
The
local
chart
(x@, yA)
in
E
induces
(xF, yA, p, pi) in Mn. Hence, if j E Mn whose expression in a natural chart is
a natural
(with j s
system
of
coordinates
y -% x), it is an m-covector
j = h drnx + hZdyA A dm--lxP, and we have that x”(9) = xp”(y>,
YAG> = YA(Y>,
p,“(Y) = j
(
-t
A
a9
where am-lx~ z i(dx”)Px. Another usual characterization
P(i)
am-lxp
= A.,
= A;,
>
of the bundle Mrt
PROPOSITION 1. Mn=A;“T*E
= W”x>
is the following:
is canonically isomorphic to Aff(JIE,rr*AmT*M).
Proof: It is a consequence of Lemma 3 of the appendix, taking G = T,E, H = T,M, F = V,(n), and C = JJ E, and then observing that the sequence (6) is 0 --+ V,(n)
3
T,E
TY”, T,M +
0 0
(see also [3]).
Given a section f : E + J’ E of r~I, in the same way as it is in Lemma 2 of Appendix, we have a splitting
COMMENT.
commented
Aff (J’E, n*AmT*M) and dim(Mn),
” n*AmT*M ~3 (n*AmT*M @ V*(n)),
= dim Il,, + 1 for every y E E.
We can introduce the canonical contraction in J’E*, which is defined map 1: J’E* = n*TM 8 T+E 18 n*A”‘T*M ---+ A”‘T*E, y=ul&Wk@x
w
as the
ak A r* i(Uk)x.
In a natural chart (x@, yA, pi, p”,) in J ’ E *, bearing in mind (l), we obtain
l(Y) =
where dm-lxP = i
gzdyA),A i d”x
(&)dmXiY
. Remember
Therefore, the relation between given by the proposition below.
= (f,fdmx+@yA
Adm-‘x,),,
that f,” denotes
the multimomentum
2 f;L”, /.ki bundles Mrr and J’E*
is
ON THE MULTIMOMENTUM BUNDLESAND THE LEGENDREMAPSIN FIELDTHEORIES 89 FQOPOSITION2. MIT = l(J’E*). 1 onto its image Mnr). Proof:
(We will denote 10 : J’E* + Mlr
For every y E E we must prove that
l(J’E*)?, = {y E A”T(GE : i(ut)i(uz)y In fact, if y E J’E”, i(ul) i(uz)(l(y))
= 0 , ~11,~2 E V,,(n)) = AYTTE.
y = l(y) and ul, u2 E V,n,
in a local chart we have
+ P~(YVYA A d’“-‘.rlLILm = 0.
= i(ul) i(u2)(pi(y)dmx
and conversely, if y E AmT;E satisfies the above condition, h;dyA A d”-‘x,)Y, therefore y = l(y), with h(dx’ m (Observe
the restriction
that AyT*E
then y = (hd”x
+
+. . . + dx”) + A;dyA )lY@d”‘-+
is canonically
Note that, in natural coordinates,
isomorphic
n
to T*E A n*A’“-‘T*M).
we have
l:y~(“~,yA,p~,p~)H~E(x~,VA,pi;,p=P~). The sections of the bundle rr*A”‘T*M -+ E are the rr-semibasic E. Therefore we introduce the notation AtT*E = n*A”‘T*M.
m-forms
on
DEFINITION 4. The bundle (over E) J1x* := AyT*E/A;T*E
= Mrc/A;T*E
will be called the restricted multimomentum E -+ M. We denote the natural projections J’n* + 44.
bundle associated with the bundle :r : by K ’ : J ’ x* + E and K’ := it CIK ’ :
The natural coordinates in J1xTT*will be denoted as (x~, y”, pi). The relation between the bundles Mn and J’n* is given by the natural projection p: Mlr = A\‘;T*E + (xl*, YA, p;,
p> *
The relation between the multimomentum following theorem.
AyT*EfAyT”E w,
bundles
= J’x*,
.VA, p;>.
l7 and J’n*
is described
THEOREM 1. The multimomentum bundles J’n* and II are canonically morphic. We will denote this diffeomorphism by * : J ’ n * -+ ll.
by the difeo-
90
A. ECHEVERRfA-ENRfQUEZ, M. C. MUROZ-LECANDA and N. ROMAN-ROY
Proof: By Proposition 1, we have that hyT*E = Mn is canonically isomorphic to Aff(J’E, hmT*M). On the other hand, taking G = T,E, H = T,M, F = V,(n), and C = Jj E in Lemma 4 of Appendix, we obtain Aff(J; E, AmT;M)/AmT]ZIM Therefore, J’n*
extending
N Aff(J’E,
2: T,M @ V;(rr) @ AmT;M.
these constructions
to the bundles we have
x*A~T*M)/~*A”T*M
21 rr*TM @ V*(r) 8 rc*A”T*M
:= II.
REMARK. As is known, a connection in the bundle n : E + M, that is, a section f : E + J’ E of TT‘, induces a linear map r : V*(n) + T* E and, as a consequence, another one
? : rr*TM @IV*(n) @ AmT*M +
AyT*E,
H F(a)
U@,cr@~
A i(u)6.
In this way we can get the inverse map !&I-’ by means of a connection: it is the composition of p with J_L: AyT*E -+ AyT*E/ArT*E. Nevertheless, 9-l is connection independent because, given two connections fi, f,, the image of J’i -J” is in AtT*E c AyT*E. Next, we give the coordinate expression of this diffeomorphism. First, let us recall that the natural coordinates in E produce an affine reference frame in J’E as follows: if n(y) = x, take j&, j$ E JiE as
jo=
4:M+
E:@(x)=y
,
c$:M+
E:@(x)=y
,
i
Observe that ~IJ is the l-jet of critical sections with target y; then we obtain V - jo = v,“(j>(j$ To), for every j E JJE. An affine map ~0: J’E + ArT*E is given by (p = (I, $), where @ denotes the linear part of 9. Then p(Y) = 900) + $(V,A(F)(UA”- Yo)) = (P(Y0) + v,“(Y)$(Y,” - Yo). Denoting
by 4 the fiber coordinate 4(4p(Y)) = (~(Yo))(a”X)
in n*AmT*M
= AzT*E,
we obtain
+ u,A(Y)W$ - yO)(amX) = A + v,A(r)$
ON THE MULTIMOMENTUM BUNDLESAND
THE LEGENDRE MAPS IN FLELD THEORIES
‘31
Then we have an affine coordinate system in Aff(J’E, AtT*E), denoted by (q, q::), with q:(V) = 4(@@ - Jo)), 4(V) = 4(P((Yo)), and hence cp = (Z,f$, (p(u,A) = (I + I$$PX.
with 1 = q(q),
12 = q:(q)
NOW, let j E J ’ E, with 9 < y 5 x, Consider Y : AyT*E + rl
Aff(J’E,
the same,
the map
A;;T*E), -r(q)
*
or, what means
: Y ++ (+*rl),,
(introduced in Lemma 3 of Appendix), where $ : M + E is a representative with 4(x) = y. If /I = (hPx + hzdyA A dm-lx,), E AyT*E, then
Therefore,
of j,
we obtain
that is. Y (/I) = (h, hI;LA) or equivalently 4(Y(B))
qz(y(B))
= P(B) = A,
= p:(B)
or
= Ax
Thus Y is a diffeomorphism in the fiber, and therefore is the identity on the base. Finally, we have the (commutative) diagram AyT*E
Aff(J’E,
+
-r
= p,
r*q;
a diffeomorphism,
= p;;. since it
A;;T*E) 1
1 AyT*E/A;T*E
r*q
5
Aff(.J’E,
A;T*E)/AzT*E
and, for proving that r goes to the quotient, it suffices to see that Y(ArT*E) c AgT*E. To show this we must identify AtT*E as a subset of both AYT’E and Aff(J’E, ArT*E). In AyT*E we have that the elements of ArT*E are characterized as follows: & E ArT* E iff, for a natural coordinate system (x@“,yA, p, pz), we have p:(p) = 0. On the other hand, as a subset of Aff(J’E, ArT*E), ArT*E
= {cp E Aff(J’E,
ArT*E)
: cp= constant}
= (4p E Aff(J’E,
A,“T*E) : I$ = 0)
92
A. ECHEVERRIA-ENRiQUEZ, M. C. MUROZ-LECANDA and N. ROMAN-ROY
($ denotes the linear part of q), or equivalently, system (4, qi), A;;T*E = {p E Aff(J’E,
for an affine natural coordinate
A,mT*E) : q;(p)
= 0).
From the local expression of T, we obtain that T(ArT*E) = AtT*E. As a consequence of this, if pi are the fiber coordinates in J17r* = AyT*E/AcT*E, and pz are those in l-l = Aff(J’E, AtT*E)/ArT*E, we have that
and, consequently, identity. 3.
Canonical
in these natural coordinate
systems, the diffeomorphism
forms in the multimomentum
The multimomentum ential forms:
bundles J’E*
bundles
and MIT are endowed with canonical
DEFINITION 5. The canonical m-form of J’E* is the form 6 defined as follows: if y E J’E* and X1, . . . , X, E 3C(J’E*), then
The canonical
6(y;X1,...,
Xm) :=
(m + l)-firm
of JIE*
!# is the
differ-
E B”(J’E*)
W(Ty~‘(Xd,. . . , Ty$(X,)). is b := -dG
E Dnm+l(JIE*).
In a natural chart in J1 E* we have 0 = p;dmx + pZdyA A dm-lxP 6 = -dp;
A d”x - dp’;l A dyA A d”‘-!xW.
F&MARK. As is known [2], the multicotangent bundle A”T*E is endowed with canonical forms 0 E JP(A”T*E) and the multisymplectic form fi := -dO E Pfl(AnT*E). Then, it can be easily proved that $ = L*@. Observe that Mrr 3 A;lT* E is a subbundle of the multicotangent bundle AmT*E. Let c : A;“T*E c, A”T*E be the natural imbedding (hence < o 10 = 1). DEFINITION 6. The canonical m-form of MT is 0 := <*O E V(Mrr). It is also called the multimomentum Liouville m-form. The cunonicaE (m + I)-form of Mn is 52 = -dO E J~“+‘(MJT), and it is called the multimomentum Liouville (m + 1)-form. The expressions
of these forms in a natural chart in MIT are 0 = pd”x + p$dyA A dm-‘xW,, 52 = -dp
A
dmx - dp;
A
dyA A dm-‘xW.
ON THE MULTIMOMENTUM BUNDLES AND THE LEGENDRE MAPS IN FIELD THEORIES
Then, a simple calculation allows us to prove that S2 is 1-nondegenerate, (MIT, !22) is a multisymplectic manifold. REMARK. Considering @((Y,
a);
Xl,.
the natural projection . . , L>
:=
4~;
t1 :
AyT*E
T~Y.~)~l(X,)r
+
and hence
E, we have
. . . , T~,,,&W
for every (y, a) E A;“T*E
(where y E E and (Yf AyTtE),
and Xi E x(A;“T*E).
The canonical lowing way.
6 and 0 can be characterized
alternatively
m-forms
‘33
in the fol-
PROPOSITION 3. 1. 6 is the unique j3’-semibasic m-form in J’ E* such that, jtir every section $ : E + J’E* of jj, the relation $*6 = I o $ holds. 2. 0 is the only i?’-semibasic m-form in Mn such that, for every section $I : E + ATT*E of k’, the relation @*C3= Cp holds.
Proof: 1. From the definition of 6 it is obvious that it is 6’-semibasic. for the second relation, let y E E and ~1, . . _, u, E T, E; therefore (Ij*&y;
Ul,
Conversely,
.
. . 9wn) = &j(y):
T&u,),
. . . >T&u,))
= &Y))(Y:
(T&’
= &(y))(y;
Ul, . ., ld,) = (I 0 $)(y;
suppose that 6 E D”(J’E*)
Then,
0 T&W,
. . ., (T$&+
0 T&b,,))
111, . . . , u,,).
verifies both conditions
in the statement.
We will prove that 6 is uniquely determined. Let y E J ’ E*, with y < ): -% x, and ul,..., u, E Ty J’E”. We can suppose that ~1. . . , u, are linearly independent and that (~1,. . . , II,) n V,(n o $‘) = (0) (where (~1, . . . , u,,) denotes the subspace generated by these vectors), for in other case 6(y; ul, _ . . , u,,) = 0. Then, by Lemma 1 (see Appendix), there exist ul, . . . , u,?, E T,M and a local section 4 : A4 -+ J’ E* of rr o 5’, such that 4(x) = y and T,$(u,) = ulr (h = 1,. . . , m), and we have that &y;q,...
1
urn> =&ii(y);
.
= ($*o)(y;
T,$h>, . . >T,hn)) ,.. LA{, . . . ,
u,,) = (10 @)(y;
Hence, 6(y; ~1, . . . , II,,) is uniquely determined. 2. The proof follows the same pattern as the one above. 4.
Legendre
Ul,
.
. .
u,,). cl
maps
From the Lagrangian point of view, a classical jield theory is described by its conjiguration bundle n : E -+ M (where M is an oriented manifold with the volume
A. ECHEVERRk-ENRiQUEZ, M. C. ML6JOZ-LECANDA and N. ROMAN-ROY
94
form w E am(M)), and a Lugrangian density which is a it-‘-semibasic m-form on J ’ B. A Lagrangian density is usually written as C = l(Tt I*&), where l c Coo{J1 E) is the hgrangian function associated with C and w. Then a Lugrangian system is a couple ((E, M; n), C). The PoincartGCartan m- and (m + I)-forms associated with the Lagrangian density _C are defined using the vertica2 endotnorphism V of the bundle J’E
@L:=
i(V)L
In a natural chart in J’E, 0,: = $dyA CL S&C--
in which it-‘*& = d”x,
A dm-‘xF
av;2,“,,A
d$ /L
V
SAL := -dOL
+ 1: = BL: + L E Qm(J’E);
we have
-
A dyA A d”-lx,,
a2L apav;
- pdyB
a2z
A
FijE$~ For
a more detailed description 281. To construct a Hamiltonian field theory, the Legendre maps the multimomentum bundle, we DEFINITION 7.
E LY+‘(J’E).
r\dyA A dm-‘xfi
-- ax •t ayB
a2E awav,B
dyB ~d~x.
of all these concepts see, for instance, [l
,
5, 7, 10,
formalism associated with a Lagrangian system in are introduced. Then, depending on the choice of can define different types of these maps as follows.
Let ((E, M; rr), C) be a Lagrangian
system, and v E J’E,
with
j&$X.
1. Let 2) c TJ 1E be the subbundle of natural coordinates x’*TE
= n’*V(n)
there is a natural diagram
in J’E
of total derivatives
& + vi & ). We have that 1 I @ D with T,E IJ?= V,rt ly $ Dj (see [28] for details). Hence projection
T,J;E
Then, the generalized
is generated
in J ’ E (which in a system
o : n’*TE
= V,n ’ Z(T;M
Legendre
by
-+ ~r’*V(n),
@ Vyn),
D,‘y
map is the Cm-map
6% J’E
+
JIE* 9
y
H
DyCy o (Id @ @ly.
and we can draw the c (A”T;M),
ON THE MULTIMOMENTUM
BUNDLES AND THE LEGENDRE MAPS IN FIELD THEORIES
!kf?
2. The reduced Legendre map is the C”-map FL:J’E-+
l-I,
.v k+ Tvc,. where the map T,;C, is defined from the following diagram (the vertical arrows are canonical isomorphisms given by the directional derivatives) T,J;E
---+ TLc,(+ImT;M
2: I qc, T.,L,
--I
T;M @ V,.n --+
A”‘T;M.
(FL is just the vertical derivative of L [ 131). 3. The @-st, extended Legendre map is the P-map
The (second) bY
.?? : J’E --+ .Mrr given by
extended Legendre map is the P-map
.?k : J’E
+ MX
given
3-L = F-L + n*c. 4. The restricted Legendre map is the Coo-map 3L : J’E + I7 given by 3L := /lo FL = /A 0 3-c. If j E J1 E, with x’(j) = y, using multimomentum bundles, we have 3L(j)
= s(y)dyA
A dm-lxg
CL E(j)
a,>
E(j)
coordinates
A dm-‘xM 1 + CL - ~:)o$&4drn~~
= $(j)dyA lr
A d”-‘x,l
Y
= $+)
P
s’
P
- u;($$(‘)d”ll I’
= s(y)
in the different
, Y
= $(j)dyA cc
w FL(y)
the natural
P
-&
@ (dyA - uf(y)dx”)
&
@ CA @ dmx
( .\-’ ’
@ d”x
, I
y’
A. ECHEVERRiA-ENRiQUEZ, M. C. MUfiOZ-LECANDA and
96
that is, the local expressions 3c*xp
= xWL,
3-c*xw = xp,
of the Legendre
N.
ROMAN-ROY
maps are the following:
ax
3C*yA = yA,
3c*p;
3-L*yA = y A,
3-L*p; = -$,
= $---$ P 3_c*p = L - l&$, W
CL 3-z*xp = xp,
3-z*+
= y A,
3-rp;
3-L”p = -?I;-$,
= $,
P
!J
FC*xw = x/1,
FC*yA = yA,
FC*p; = g. Y
REMARKS 1,Taking into account all the above results, it is immediate that FC = 6 o FL: (see diagrams (2) and (4)), and FL: = 9 o 3C.
to proye
2. It is interesting to point out that, as 0 L: and 8~ can be thought as m-forms on E along the projection x1 : J’ E -+ E, the extended Legendre maps can be defined as
where Zl,. . . , Z, E Tn~(y)E, and 21, . . . , .i?, E T,J’E are such that Tjn’2, = Z,. In addition, the (second) extended Legendre map can also be defined as the “firstorder vertical Taylor approximation to 2” [3, 121. Finally, we have the following relations between the Legendre maps and the PoincartGCartan (m + I)-form in J’E (which can be easily proved using natural systems of coordinates and the expressions of the Legendre maps). PROPOSITION4. Let ((E, M; n), C) be a Lagrangian Fk*o Kc*0
= OL,
= OL - c = 8L,
Gz*Gl= OL -L
= OL,
system. Then
3-c*s-J = s2& 3-C% = !& - dL = -dec, FT*ti = S& - dC = -dOr.
REMARK. Observe that the Hamiltonian formalism is essentially the dual formalism of the Lagrangian model, by means of the Lagrangian density. Then, as J’ E is an affine bundle, its affine dual can be identified with Aff (J ’ E, n*A”T*M) 2: Mn, whose dimension is greater than dim J’E, and hence Aff (J’E, n*AmT*M)/AtT*E 2 J’n* is more suitable as a dual bundle (from the dimensional point of view). Then, the canonical forms 0 and fi in Mrr, and 6 and 6 in J’E*, can be
ON THE MULTIMOMENTUM BUNDLES AND THE LEGENDRE MAPS IN FIELD THEORIES
97
pulled-back to the restricted and reduced multimomentum bundles J Ix* and II, using sections of the projections p : Mn --+ J lx * and 6 : J ’ E* + l7. respectively [3, 61 In this way, the reduced and restricted multimomentum bundles are endowed with (noncanonical) geometrical structures (HumiEton-Cartan forms) needed for stating the Hamiltonian formalism. In addition, connections in the bundle n : E + A4 induce linear sections of p (and S) [3, 6, 9, 251 and it can be proved that there is a bijective correspondence between the set of connections in the bundle JT : E + M, and the set of linear sections of the projection p. Hence, all these results, together with Theorem 1, allows us to relate two Iof the most usual Hamiltonian formalisms of field theories [6].
5.
Regular and singular
Following concepts. DEFINITION
I.
systems
the well-known
8.
terminology
of mechanics,
Let ((E, M; n), L) be a Lagrangian
we define
the following
system.
((E, M; n), C) is said to be a regular
or non-degenerate Lagrangian system if FL, and hence, FL are local diffeomorphisms. As a particular case, ((E, M; n), L) is said to be a hyper-regular Lagrangian system if F.C, and hence FL, are global diffeomorphisms.
2. In other cases ((E, M; n), L) is said to be a singular or degenerate Lagrangian system. PROPOSITION 5. 1.
Let ((E, M; n), C) be a hyper-regular
FC(J’E) is a m2-codimensional verse to the projection 6.
imbedded
submanifold
2. 3-L
J1x*,
3_L(J’E),
3-L(J’E),
Lugrangiun system. Then of J’E*
imbedded
which is tran.s-
submanifolds
of Mrr
FAL(J’ E) and II are difleomorphic.
Hence, F-Lk3-L and 3-L are_diffeomorphisms on their images; and_the maps ,.L, restricted to 3L(J’E) or to 3L(J’E), and 10 and 8, restricted to FC( J' E), are ulso diffeomorphisms. Prod: 1. If C is hyper-regular then FL is_a diffeomorphism and hence, as FL = FL o 8, we obtain that FL is injective and FC( J’E) is transverse to the fibers of 8. 2. The proof of this statement 3. It is a direct consequence
is like for the one above (see also [21]). of the above items.
0
98
A. ECHEVERRiA-ENRiQUEZ, M. C. M@JOZ-LECANDA andN. ROMAN-ROY In this way we have the following
(commutative)
diagram
(2)
Observe that there exists a map CL’: e(J’E)s Mn +s(J’E) c Mn, which iLa diffeomorphism defined by the relation 3,C = p’ o 3.C, and p o CL’= p on 3C(J’E). For dealing with singular Lagrangians we must assume minimal “regularity” conditions. Hence we introduce the following terminology. DEFINITION9. A singular Lagrangian system ((E, M; n), 13) is said to be almostregular if 1. P := FTC(J’E) and P := FC(J’E) are closed submanifolds of J’n* and l7, respectively. (We will denote by 10 : P c-) J’n* and Jo : P L) l7 the corresponding imbeddings). 2. 3L, and hence FL, are submersions onto their images. 3. For every j E J’E, the fibers 3’c-‘(3C(y)) and hence FC’(FC(y)) are connected submanifolds of J1 E. (This definition
is equivalent
to that in [21], but slightly different
[9, 251). Let ((E, M; r), L) be an almost-regular + := =(J’E), Let jo : F c, Mn,
50 : p c,
+ := E(J’
Lagrangian E),
Mrr, j o : k L, J’E*
from that in
system. Denote
i, := E(J’E). be the canonical
inclusions,
and
ON THE MULTIMOMENTUM BUNDLESAND THE LEGENDREMAPSIN FIELD THEORIES 99 be the restrictions of the maps p, ~0, 6 and the diffeomorphism Finally, define the restriction mappings 3,Co : J’E
+
P,
E.
FL0 : J’E
+
: J’E ?‘,
+
+,
?ko
FCo : J’E
= 3’c,‘(3Lo(~))
and $’ are submanifolds A P L) Mn, JO : 9 C, Mn,&
3. 9
4. The restriction
mappings
20
-+ $,.
Lagrangian
system. Then
= F,CC,‘(FCo(j>).
of Mrr, b is a submanifold ,. : P C, J’E* are imbeddings.
3L0,
respectively.
-+ P.
PROPOSITION6. Let ((E, M; 7t), C) be an almost-regular 1. The maps *O and /ll are difleomorphisms. 2. For every j E J’E, ?&-‘(Eo(9,
: J’E
9,
and f&
(3)
of J’E*,
are submersions
and
Jo :
with connected
$b ers. Proof:
*O is a diffeomorphism as is q. The second equality of (3) is a consequence of the relation F& = *O o 3Tco. For the proof of the first equality of (,3), and of the assertions concerning F, ? and 50, see [21] and [22]. Then, the proofs of the other assertions are simi1ar.U
Thus we have the (commutative)
diagram
J’E
(4)
. I P
where jIi’ : $ -+ ? is defined by the relation l
Jo
c
l-I
b’ := bILL-’ o b.
REMARKS. The fact that ii is a diffeomorphism is particularly relevant, since it allows us to construct a Hamiltonian formalism for an almost-regular Lagrangian system
VW.
A. ECHEVERRfA-ENRiQUEZ,
100
M. C. MUfiOZ-LECANDA
and N. ROM_&-ROY
It is insesting to point out that the map /2 (which is relatedxith the Legendre map 3Ca) is not a diffeomorphism in general, since rank 3Cc 2 rank3Ca = rank3C0, as is evident from the analysis of the corresponding Jacobian matrices.
l
The matrix of the tangent maps 3C, Id 0
a2_c axvav;
and FC, in a natural coordinate 0
Id
a2E apav;
where the sub-matrix
is the partial Hessian
the regularity
to demanding
of C is equivalent is regular
\
everywhere
system is
in J’E.
matrix of C. Obviously,
that the partial
This fact establishes
Hessian
matrix
the relation
to
I
the concept of regularity given in an equivalent way by saying that a Lagrangian (otherwise it is said to system ((E, 44; n), L) is regular if fi L is I-nondegenerate be singular or nonregular).
Conclusions We have reviewed the definitions of four different multimomentum bundles for the Hamiltonian formalism of first-order classical field theories (multisymplectic models). The so-called generalized and reduced multimomentum bundles are related straightforwardly from their definition, and the same happens with the generalized and restricted multimomentum bundles. The first goal of this work was to relate both couples, proving that the reduced and restricted multimomenturn bundles are, in fact, canonically diffeomorphic. In natural local coordinates, this diffeomorphism is just the identity. The canonical forms, which the generalized and the extended multimomentum bundles are endowed with, have been defined and characterized in several equivalent ways. Given a Lagrangian system in field theory, we have introduced the corresponding Legendre maps relating these multimomentum bundles to the first-order jet bundle associated with this system. Some of them, the generalized and reduced Legendre maps, are defined in a natural way as fiber derivatives of the Lagrangian density, being the other ones obtained from those. The relation among all these maps has been clarified. Regular and almost-regular Lagrangian systems are defined and studied, attending to the geometric features of the Legendre maps. In this way, the standard definitions existing in the standard literature are extended and completed.
ON THE MULTIMOMENTUM BUNDLESANDTHE LEGENDREMAPSIN FIELD THEORIES10 1
A.
Appendix
LEMMA 1. Let n : F + N be a d!ferentiable bundle, with dim N = n and dim F = n + r, and p E F with q = r(p). Let ~11,. . . , uh E T,F (h 5 n) be such that VI,..., vh are linearly independent, and (VI, . . . , uh) n V,(n) = {O}. Then: 1. There exist X1, . . . , xh E X(W), for a neighbourhood W C F of p, such that Xh are linearly independent at every point of W. (4 Xl,..., Xh generate an involutive distribution in W. (b) Xl,..., (c) Xi(p)= Ui,i = 1, . . . . h. = (o}, firevery x E w. (4 (X,(x), ...,xh(x))nvv,(n) 2. There exists a local section y of n, defined in a neighbourhood of q E N, and UI,.... uh E T,N such that y(q) = p, and T,,y(u;) = zli, i = 1,. . . . h. Proof: Let ?_&+I,. . . , v, E T,F be such that T, = V,(n) 61?(vi. . . . . v,). Let (W, 9) be a local chart of F at p, adapted to n. We have q : W + UI x U2 c BY x JR”. Let ei, . . , e, and e,+i, . . , en+r be local basis of Iw” and Iw’, respectively. Then
(TpvO(vd>. . .t T,cp(v,)) = h, . . . . en). T,dV,(n))
= (e,+l, . . . , en+r) ,
Vx
E
(5)
W.
Let Zi,..., Z, E X(Ui x cT2) be the constant extensions of T&vi), . . , T&v,) to Ui x U2. We have that [Zi, Zj] = 0, Vi, j. Finally, let Xi, . . . , X, E X(W). with Xi = (P*Zi. Therefore: 1. Taking Xl,..., Xh (h 5 n), conditions l(a), l(b) and l(c) hold trivially (by construction), and condition l(d) holds as a consequence of (5). 2. Observe that Xi, . . . , X, E X(W) generate the horizontal subspace of a connection defined in the bundle n : W --+ n(W), which is integrable because the distribution is involutive. Then, let y be the integral section of this connection at p. Therefore, y(q) = p, and the subspace tangent to the image of y at p is generated by ui, . . . , v,. Hence, there exist ui, . . . , u, such that T, y (ui) = ~1,. 0 Now, let A be an affine space modelled on a vector space S, and T another vector space, both over the same field K. Let Aff(A, T) be the set of affine maps from A to T; that is, maps q : A + T be such that there exists a linear map $J : S -+ T verifying that q(a) - q(b) = $(a - b); for a, b E A. LEMMA 2. 1. There is a natural isomorphism between Aff (A, T)/ T and S* C3T (and then dimAff(A, T) = dim(S* 63 T) +dimT = dimT(dimS+ 1)). 2. There is a canonical isomorphism between Aff (A, T) and Aff (A, K) ~?3T. Proof: Aff(A, T) is a vector space over K with the natural operations. The map A : Aff(A,T) --+ S* 8 T, which assigns $I to every cp, is linear and we have the exact sequence 0 ---+ T &
Aff(A, T) &
S* @ T ----+ 0,
102
A. ECHEVERRfA-ENRfQUEZ,
M. C. MUhOZ-LECANDA
and N. ROMAN-ROY
where, if t E T, then j(t) : A -+ T is the constant map (j(t))(a) = t, for every a E A. Therefore, we have a natural isomorphism Aff(A, T)/T 21 S* 8 T and then dimAff(A,
T) = dim(S* @ T) + dim T = dim T(dim S + 1).
Moreover, for ya E A, there exists an splitting Aff (A, T) z T CB(S* 81 T) given by the following retract of the above exact sequence j,,:S*@T+Aff(A,T), (P
-
Y b-3.(P(Y - Yo).
(PO:
On the other hand, we have the bilinear map Aff(A, K) x T + (a, t)
Aff(A, T), tct
H
and hence we can define the following
:
a I--+ a(a)t,
morphism
Aff(A, K) 63 T +
Aff(A, T),
OJz@ Ui I+ UjOJi, which is injective because we can assume that the vectors ui are linearly independent, and both spaces have the same dimension. Therefore, Aff (A, T) and 0 Aff (A, K) @ T are canonically isomorphic. If (St,... , s,) is a basis of S, (a’, . . . , am) is its dual basis and (tl, . . . , t,,J is a basis of T, then taking an affine reference in A, (ti, ti @ a’) is a basis of Aff (A, T), as a vector space. Now, let G, H be finite-dimensional vector spaces over K, and F a subspace of G. Consider the exact sequence O-+FFG?+HHO.
(6)
The set C-{a:H+G;
cr linear,rtoa=IdH}
is an affine space modelled on L(H, F) = H*@F. In fact, if 0 E C and ;I E H*@F, then a+h~E, since noh=O. Furthermore, if a,puEC, then rto(a-p)=O, and hence u - p E H* @ F. If dim H = m, taking the space Aff (C, A* H*), and according to the second item of the above Lemma, we have that Aff (C, Am H*) 2 Aff (C, K) ~3 Am H*, and then dim Aff (C, Am H*) = dim Aff (C, K) = dim(H* @ F) + 1. Now, consider the subspace A;lG* = [a E AmG*
: i(u) i(
= 0, U, 2, E F} c A”‘G*.
ON THE MULTIMOMENTUM
BUNDLES AND THE LEGENDRE MAPS IN FIELD THEORIES
LEMMA
3. The spaces Aff (C, A”‘H*)
Proof:
If (f’,
and Ay G* are canonically
103
isomorphic.
is a basis of F and g’ , . . . , gm E G are such that . . . , f’) g”) is a basis of G, then A;lG* is generated by (g’ A . . . A
(f’, . .,” j-;,s’,..., gm, fJ A g’l A . . . A gim-l) (with 1 5 il < . . . -c i,_l < m), therefore dimAyG* = 1 + dim F dim H; that is, dim ATG* = dim Aff (C, A” H*). Next, define the linear map Y: A;lG* ----+ Aff(C, A”H*), n
T(n)
*
: cr H o*q.
We want to prove that it is an isomorphism, for which it suffices to prove that it is injective. Thus, suppose that Y(n) = 0 (that is, a*~ = 0 for every ~7 E C), then we must prove that n = 0. Let g’ , . . . , g” E G be linearly independent (so dimL(g’, . . . , g”) = m); we are going to calculate q(g’, . . . , g”). 1. L(g’,... , g”) n ~(0 = KV. g’“) @ t(F) and therefore n : L(g’, . . , g”) -+ H is an Then G = L(g’,..., anda:H+G, withIsoa=IdH. isomorphism. Hence, there exist h’, . . . , hmEH such that a(h’) = g’; therefore q(g’, 2. L(g’,
. , g”) = q(a(h’),
. . . 2s”> n t(F)
. . ,a(h”))
= (a*q)(h’,
. . . , h”) = 0.
# WI.
(a) dim(L(g’, ...,grn) n t(F)) = 1. In this case there is u E L(g’ , . . . , g”) nt(F), with u # 0, such that L(g’, . . , 8’“) g”) (up to an ordering change), and then there is k E K such that = L(u, g*, . . . , q(g’,
. . . , g”)
= krl@, g*, . . . , g”),
therefore L(g*, . . . , g”)nt(F) = (0). Let (u, u’, . . . , u’) be a basis of F, and g’ be such that (u,ul ,..., u’,gl,g* ,..., g’“) is a basis of G. Then L(g’, g*, . . . , g’“) f? t(F) = {0), and hence, as a consequence of the above item, we conclude that , g”) = 0. Furthermore, L(g’ + u, g*, . . , g*) fl t(F) = {0} because, if V(s’,g*,... this is not true, then L(g’,g*, . . . ,g”)flt(F) # {0}, and hence r](g’+u, g*, , g”) = 0, by the above item. Thus, rl(g’, . . ., g”) = kq(u, g*, . . . , g”) = k(q@+u,
g*, . . . , gm)-_rl(g’,
g*, . . . , g”‘)) = 0.
(b) dim(L(g’, . . . , g”) f’ t(F)) = s > 1, with s 5 r = dim F. Let (f’ , . . . , f”, g’+l, . . . , g”) be a basis of L(g’, . . . , g”), with f’, . . . , f” Then, as rl E AmG*, rl(g’, . . . , g”) = krj(f’,
. . . , f”, gsfl,.
E F.
. , g”) = 0.
Then q = 0, so r is injective and is a canonical isomorphism and A;lG*.
between Aff (C , Am H *) 0
104
A. ECHEVERRfA-ENRfQUEZ,
M. C. MUNOZ-LECANDA and N. ROMAN-ROY
Using the first item of Lemma 2 and identifying sion. LEMMA
4. Aff(x,
AmH*)/AmH*
2 (H*
@ F)*
@
A = C, we make the concluAmH* E H 63 F* @ AmH*.
Acknowledgments We are grateful for the financial support of the CICYT TAP97-0969-CO3-01. We wish to thank Mr. Jeff Palmer for his assistance in preparing the English version of the manuscript. REFERENCES [ll E. Binz, J. Sniatycki, H. Fisher: The Geometry of Classical Fields, North Holland, Amsterdam 1988. 121 F. Cantrijn, L. A. Ibort and M. de Lebn: On the Geometry of Multisymplectic Manifolds, J. Austral. Math. Sot. Ser. 66 (1999) 303-330. I31 J. F. Carifiena, M. Crampin and L. A. Ibort: On the multisymplectic formalism for first order field theories, Diff Geon. Appl. 1 (1991), 345-374. [41 A. Echeverria-Enrfquez and M. C. Mufloz-Lecanda: Variational calculus in several variables: a Hamiltonian approach, Ann. Inst. Henri Poincare’ 56(l) (1992), 27-47. [51 A. Echeverria-Enrfquez, M. C. Mufioz-Lecanda and N. Roman-Roy: Forts. Phys. 44 (1996) 235-280. E61 A. Echevern’a-Enriquez, M. C. Mmioz-Lecanda and N. Roman-Roy: “Geometry of Hamiltonian First-Order Classical Field Theories”. DMAT-UPC (2000) (in preparation). [71 P. L. Garcia: “The Poincar&Cartan invariant in the calculus of variations”, Symp. Math. 14, (Convegno di Geometria Simplettica e Fisica Matematica, INDAM, Rome, 1973) Acad. Press, London 1974, p. 219-246. [81 G. Giachetta, L. Mangiarotti and G. Sardanashvily: Constraint Hamiltonian Systems and Gauge Theories, Int. J. Theor. Phys. 34(12) (1995), 2353-2371. [91 G. Giachetta, L. Mangiarotti and G. Sardanashvily: New Lagrangian and Hamiltonian Methods in Field Theory, World Scientific, Singapore (1997). [lOI H. Goldschmidt and S. Stemberg: The Hamilton-Cartan formalism in the calculus of variations, Ann. Inst. Fourier Grenoble 23(l) (1973), 203-267. [ill M. .I. Gotay: “A Multisymplectic Framework for Classical Field Theory and the Calculus of Variations I: Covariant Hamiltonian formalism”, Mechanics, Analysis and Geometry: 200 Years after Lagrange, M. Francaviglia Ed., Elsevier, Amsterdam 1991, 203-235. [121 M. J. Gotay, J. Isenberg and J. E. Marsden: Momentum maps and classical relativistic fields, Part I: Covariant Field Theory, Preprint November 18, 1997 (66 pages). II31 X. Gracia: “Fibre derivatives: some applications to singular lagrangians”, Proc. V Fall Workshop: Geometry and Physics, 1996, J. F. CariIiena, E. Martinez, M. F. Raiiada, (Eds.) Anales de Fisica Monografias RSEF 4 Madrid (1997). 43-58. 1141 I. Kanatchikov: “Novel algebraic structures from the polysymplectic form in field theory”, GROUP21, Physical Applications and Mathematical Aspects of Geometry. Groups and Algebras, Vol. 2, H. D. Doebner, W Scherer, C. Schulte (Eds.), World Scientific, Singapore 1997, p. 894. [151 J. Kijowski: A Finite-dimensional Canonical Formalism in the Classical Field Theory, Common. Math. Phys. 30 (1973), 99-128. [161 J. Kijowski and W. Szczyrba, “Multisymplectic manifolds and the geometrical construction of the Poisson Brackets in the classical field theory”, Geometric Symplectique et Physique Mathektatique, Coll. Int. C.N.R.S. 237 (1975), 347-378. I171 J. Kijowski and W. M. Tulczyjew: A Symplectic Framework for Field Theories, Lect. Notes Phys. 170, Springer, Berlin 1979. [181 D. Krupka: Geometry of Lagrangean Structures, Supp. Rend. Circ. Mat. Palermo, ser. II, no. 14 (1987), 187-224.
ON THE MULTIMOMENTUM BUNDLES AND THE LEGENDRE MAPS IN FIELD THEORIES ](I5 [ 191 B. A. Kupershmidt: Geometry of Jet Bundles and the Structure of Lagrangian and Hamiltonian Formalisms, Lect. Notes in Math. 775, Springer, New York 1980, pp. 162-218. [20] M. de I.&n, J. Marfn-Solano and J. C. Marrero: “Ehresmann connections in classical field theories”. Proc. III Fall Workshop: Differential Geometry and its Applicafions, Anales de Fisica, Monografias 2 (1995), 73-89. [21] M. de Leon, J. Marfn-Solano and J. C. Marrero: “A geometrical approach to classical field theories: in Differential Geometry. A constraint algorithm for singular theories”, Proc. on New Developments L. Tamassi, J. Szenthe (Eds.), Kluwer, Dordrecht 1996, pp. 291-312. [22] M. de Leon, J. Marfn and J. Marrero: The constraint algorithm in the jet formalism, Difl Geom. Appl. 6 (1996), 275-300. [23] M. de Le6n and P. R. Rodrigues: Hamiltonian structures and Lagrangian field theories, Bol. Acad. Gabga de Ciencias VII (1988), 69-81. 1241 J. E. Marsden and S. Shkoller: “Multisymplectic geometry, covariant Hamiltonians and water waves”, Math. Proc. Camb. Phil. Sot. 1251 G. Sardanashvily: Generalized
125 (1999), 553-575. Hamiltonian Formalism
for Field Theory. Constraint Systems. World Scientific, Singapore 1995. 1261 G. Sardanashvily and 0. Zakharov: On application of the Hamilton formalism in fibred manifolds to field theory, Difl Geom. Appl. 3 (1993), 245-263. [27] D. J. Saunders: The Cartan form in Lagrangian field theories, J. Phys. A: Math. Gen. 20 (1987). 333-319. [28] D. J. Saunders, The Geometry of Jet Bundles, London Math. Sot. Lect. Notes Ser. 142, Cambridge University Press, Cambridge 1989.
REPORTS ON MATHEMATICAL
No. I
PHYSICS
ON THE MULTIMOMENTUM BUNDLES AND THE LEGENDRE MAPS IN FIELD THEORIES A. ECHEVERR~A-ENR~QUEZ, M. C. Mu~~oz-LECANDAand N. ROM,&ROY Departamento de Matematica Aplicada y Telematica Edificio C-3, Campus Norte UPC C/ Jordi Girona I, E-08034 Barcelona, Spain (e-mails: [email protected], [email protected]) (Received
March
18.
1999)
We study the geometrical background of the Hamiltonian formalism of first-order classical field theories. In particular, different proposals of multimomentum bundles existing in the usual literature (including their canonical structures) are analyzed and compared. The corresponding Legendre maps are introduced. As a consequence, the definition of regular and almost-regular Lagrangian systems is reviewed and extended in different but equivalent ways. Key words: AMS
1.
jet bundles,
s. c. (1991):
53C80,
classical
field theories,
Legendre
55R10,
58A20,70G50,
70H99.
map,
Hamiltonian
formalism
Introduction The standard
geometric
structures
underlying
the covariant
Lagrangian
descrip-
tion of first-order field theories are first-order jet bundles J’E 5 E 1 A4 and their canonical structures [5, 7, 9, 18, 19, 23, 271. Nevertheless, for the covariant Hamiltonian formalism of these theories there are several choices for the phase space where this formalism takes place. Among all of them, only the multisymp&tic models will deserve our attention in this work. So, in [ 11, 12, 17, 241 (see also [15-171) the multimomentum phase space is taken to be Mn I A’;T* E, the bundle of m-forms on E (m being the dimension of M) vanishing by the action of two n-vertical vector fields. In [3, 4, 20, 211 J’n* = AyT*E/ArT”E is used as the multimomentum phase space (where ArT*E is the bundle of .nsemibasic m-forms in E). Finally, in [4, 8, 9, 25, 261 the basic choice is the bundle l-l = n*TM @ V*(n) 8 rr*A”T*M (here V*(x) denotes the dual bundle of V(n): the n-vertical subbundle of TE) which, in turn, is related to J ’ E* s rr*TM 8 T*E @In*AmT*M. The origin of all these multimomentum bundles is related to the different Legendre maps which arise essentially from the fiber derivative of the Lagrangian density (see Section 4). All these choices have special features. So, Mn and JIE* are endowed with natural multisymplectic forms. In papers such as [ 121, the former is used for ob-
A. ECHEVERRiA-ENRfQUEZ, M. C. MUNOZ-LECANDA and
86
N.
ROMAN-ROY
taining the Poincark-Curtan form in J’E, which is needed for the Lagrangian formalism. This is done by defining the suitable Legendre map connecting J’ E and M7t. On the other hand, the dimensions of J’ E, Mrr and 1’ E” are not equal: in fact, dim Mn = dim J’E + 1 and dim J’E* = dim J’E + (dim M)*; but dim Il = dim J’n* = dim J1 E and the choice of both J ‘rr* and Il as multimomenturn phase spaces allows us to formulate coherent covariant Hamiltonian formalisms for field theories. Finally, the construction of J1rr* is closely related to Mrc, but their relation to fl and J’ E* is not evident at all. Hence, the aim of this work is to carry out a comparative study of these multimomentum bundles, and introduce the canonical geometrical structures of Mn and J ’ E*. In every case, the corresponding Legendre map is also defined. An interesting conclusion of this study is that the multimomentum bundles J’n* and Il are canonically diffeomorphic. Finally, using the different Legendre maps, we can classify Lagrangian systems in field theory into (hyper)regular and almost-regular, in different but equivalent ways, attending to the characteristics of these maps. All manifolds are real, paracompact, connected and C”. All maps are Co3 and summation over crossed repeated indices is understood.
2.
Multimomentum
bundles
Let n : E -+ A4 be a fiber bundle (dimE = N+m, dim44 = m), x1 : J’E + E the first-order jet bundle of local sections of n, and 17’ = r o n ‘. .I’ E is an affine bundle modelled on n*T*M @ V(n). A local chart of natural coordinates in E adapted to the bundle E += M will be denoted by (x@, yA). The induced local chart in J’E is denoted by (xc”, yA, $). DEFINITION 1. The bundle (over E) J’E*
:= n*TM @‘ET*E 8’~ n*A”T*M
is called the generalized multimomentum M. We denote the natural projections J’E* + M. The
local
(xw, yA, p;, pz) that
system
(x@, yA)
in J’E*
bundle associated with the bundle n : E + by 5’ : J’E” -+ E and 3 := Tro;i :
induces
as follows:
a local
if y E J’E*,
yj& y@(f:dx” + g;dyA),
J’ =
system
of
with y 5
63 d”xly,
where dmx G dx’ A _. . A dx”‘, and therefore X’“(Y)
=
x’“((n
0
?>(y)),
VA(Y)
= YA(;l(Y)>l
natural
coordinates
y 5
x, we have
(1)
ON THE MULTIMOMENTUM BUNDLES AND THE LEGENDRE MAPS IN FIELD THEORIES
p:(y)
87
= y
a a A...A axm ax1
where i3”‘x = -
DEFINITION 2. The bundle
l-l := n*TM @‘EV*(r)
* (over
E)
@3En*A”T*M
= u
T,(,y,M @ V;(z)
@ LI”‘T;~,,,M
YCE
is called the reduced multimomentum bundle associated with the bundle n : E + We denote the natural projections by p’ : II + E and 3’ := n o p’ : I7 -+ M.
hf.
From the local system (xl*, yA) we can construct a natural system of coordinates (xp, yA, pz) in l7 as follows: considering y E l7, we write
x’“(Y) = p;(y)
= 9
dxCL,-!-
A...
-!-
ayA’ ad
the dual basis of in V*(n) aYA
as
The relation between these multimomentum 6:
which
= YA(P'(rN;
YAW
a
and denoting expressed
x%‘(Y));
is induced
DEFINITION
A-
a axrn >
by {CA), an element
j E l-l is
bundles is given by the (onto) map
J’E*
---+
n,
(xIL, YA, P$ P;)
*
(xP”, YA*PZ),
by the natural restriction
’
T* E -+ V* (R.).
3. Consider the multicotangent
bundle A”T*E.
Then, for every YE:E
we define A;ITcE := {v E A”T;E
: i(ul)i(uz)y
= 0 , ~1,
~2
E
V,(X)}.
The bundle (over E) Mrr s AyT*E
:= u Y~E
AyT;E
= U{(Y+)
: a E AyT;E)
YEE
will be called the extended multimomentum bundle associated with the bundle :ir : E --+ M. We denote the natural projections by r3’ : Mrr --, E and I?’ :Mrr-+,M.
88
A. ECHEVERRfA-ENRiQUEZ,M. C. MUfiOZ-LECANDA andN. ROMAN-ROY
The
local
chart
(x@, yA)
in
E
induces
(xF, yA, p, pi) in Mn. Hence, if j E Mn whose expression in a natural chart is
a natural
(with j s
system
of
coordinates
y -% x), it is an m-covector
j = h drnx + hZdyA A dm--lxP, and we have that x”(9) = xp”(y>,
YAG> = YA(Y>,
p,“(Y) = j
(
-t
A
a9
where am-lx~ z i(dx”)Px. Another usual characterization
P(i)
am-lxp
= A.,
= A;,
>
of the bundle Mrt
PROPOSITION 1. Mn=A;“T*E
= W”x>
is the following:
is canonically isomorphic to Aff(JIE,rr*AmT*M).
Proof: It is a consequence of Lemma 3 of the appendix, taking G = T,E, H = T,M, F = V,(n), and C = JJ E, and then observing that the sequence (6) is 0 --+ V,(n)
3
T,E
TY”, T,M +
0 0
(see also [3]).
Given a section f : E + J’ E of r~I, in the same way as it is in Lemma 2 of Appendix, we have a splitting
COMMENT.
commented
Aff (J’E, n*AmT*M) and dim(Mn),
” n*AmT*M ~3 (n*AmT*M @ V*(n)),
= dim Il,, + 1 for every y E E.
We can introduce the canonical contraction in J’E*, which is defined map 1: J’E* = n*TM 8 T+E 18 n*A”‘T*M ---+ A”‘T*E, y=ul&Wk@x
w
as the
ak A r* i(Uk)x.
In a natural chart (x@, yA, pi, p”,) in J ’ E *, bearing in mind (l), we obtain
l(Y) =
where dm-lxP = i
gzdyA),A i d”x
(&)dmXiY
. Remember
Therefore, the relation between given by the proposition below.
= (f,fdmx+@yA
Adm-‘x,),,
that f,” denotes
the multimomentum
2 f;L”, /.ki bundles Mrr and J’E*
is
ON THE MULTIMOMENTUM BUNDLESAND THE LEGENDREMAPSIN FIELDTHEORIES 89 FQOPOSITION2. MIT = l(J’E*). 1 onto its image Mnr). Proof:
(We will denote 10 : J’E* + Mlr
For every y E E we must prove that
l(J’E*)?, = {y E A”T(GE : i(ut)i(uz)y In fact, if y E J’E”, i(ul) i(uz)(l(y))
= 0 , ~11,~2 E V,,(n)) = AYTTE.
y = l(y) and ul, u2 E V,n,
in a local chart we have
+ P~(YVYA A d’“-‘.rlLILm = 0.
= i(ul) i(u2)(pi(y)dmx
and conversely, if y E AmT;E satisfies the above condition, h;dyA A d”-‘x,)Y, therefore y = l(y), with h(dx’ m (Observe
the restriction
that AyT*E
then y = (hd”x
+
+. . . + dx”) + A;dyA )lY@d”‘-+
is canonically
Note that, in natural coordinates,
isomorphic
n
to T*E A n*A’“-‘T*M).
we have
l:y~(“~,yA,p~,p~)H~E(x~,VA,pi;,p=P~). The sections of the bundle rr*A”‘T*M -+ E are the rr-semibasic E. Therefore we introduce the notation AtT*E = n*A”‘T*M.
m-forms
on
DEFINITION 4. The bundle (over E) J1x* := AyT*E/A;T*E
= Mrc/A;T*E
will be called the restricted multimomentum E -+ M. We denote the natural projections J’n* + 44.
bundle associated with the bundle :r : by K ’ : J ’ x* + E and K’ := it CIK ’ :
The natural coordinates in J1xTT*will be denoted as (x~, y”, pi). The relation between the bundles Mn and J’n* is given by the natural projection p: Mlr = A\‘;T*E + (xl*, YA, p;,
p> *
The relation between the multimomentum following theorem.
AyT*EfAyT”E w,
bundles
= J’x*,
.VA, p;>.
l7 and J’n*
is described
THEOREM 1. The multimomentum bundles J’n* and II are canonically morphic. We will denote this diffeomorphism by * : J ’ n * -+ ll.
by the difeo-
90
A. ECHEVERRfA-ENRfQUEZ, M. C. MUROZ-LECANDA and N. ROMAN-ROY
Proof: By Proposition 1, we have that hyT*E = Mn is canonically isomorphic to Aff(J’E, hmT*M). On the other hand, taking G = T,E, H = T,M, F = V,(n), and C = Jj E in Lemma 4 of Appendix, we obtain Aff(J; E, AmT;M)/AmT]ZIM Therefore, J’n*
extending
N Aff(J’E,
2: T,M @ V;(rr) @ AmT;M.
these constructions
to the bundles we have
x*A~T*M)/~*A”T*M
21 rr*TM @ V*(r) 8 rc*A”T*M
:= II.
REMARK. As is known, a connection in the bundle n : E + M, that is, a section f : E + J’ E of TT‘, induces a linear map r : V*(n) + T* E and, as a consequence, another one
? : rr*TM @IV*(n) @ AmT*M +
AyT*E,
H F(a)
U@,cr@~
A i(u)6.
In this way we can get the inverse map !&I-’ by means of a connection: it is the composition of p with J_L: AyT*E -+ AyT*E/ArT*E. Nevertheless, 9-l is connection independent because, given two connections fi, f,, the image of J’i -J” is in AtT*E c AyT*E. Next, we give the coordinate expression of this diffeomorphism. First, let us recall that the natural coordinates in E produce an affine reference frame in J’E as follows: if n(y) = x, take j&, j$ E JiE as
jo=
4:M+
E:@(x)=y
,
c$:M+
E:@(x)=y
,
i
Observe that ~IJ is the l-jet of critical sections with target y; then we obtain V - jo = v,“(j>(j$ To), for every j E JJE. An affine map ~0: J’E + ArT*E is given by (p = (I, $), where @ denotes the linear part of 9. Then p(Y) = 900) + $(V,A(F)(UA”- Yo)) = (P(Y0) + v,“(Y)$(Y,” - Yo). Denoting
by 4 the fiber coordinate 4(4p(Y)) = (~(Yo))(a”X)
in n*AmT*M
= AzT*E,
we obtain
+ u,A(Y)W$ - yO)(amX) = A + v,A(r)$
ON THE MULTIMOMENTUM BUNDLESAND
THE LEGENDRE MAPS IN FLELD THEORIES
‘31
Then we have an affine coordinate system in Aff(J’E, AtT*E), denoted by (q, q::), with q:(V) = 4(@@ - Jo)), 4(V) = 4(P((Yo)), and hence cp = (Z,f$, (p(u,A) = (I + I$$PX.
with 1 = q(q),
12 = q:(q)
NOW, let j E J ’ E, with 9 < y 5 x, Consider Y : AyT*E + rl
Aff(J’E,
the same,
the map
A;;T*E), -r(q)
*
or, what means
: Y ++ (+*rl),,
(introduced in Lemma 3 of Appendix), where $ : M + E is a representative with 4(x) = y. If /I = (hPx + hzdyA A dm-lx,), E AyT*E, then
Therefore,
of j,
we obtain
that is. Y (/I) = (h, hI;LA) or equivalently 4(Y(B))
qz(y(B))
= P(B) = A,
= p:(B)
or
= Ax
Thus Y is a diffeomorphism in the fiber, and therefore is the identity on the base. Finally, we have the (commutative) diagram AyT*E
Aff(J’E,
+
-r
= p,
r*q;
a diffeomorphism,
= p;;. since it
A;;T*E) 1
1 AyT*E/A;T*E
r*q
5
Aff(.J’E,
A;T*E)/AzT*E
and, for proving that r goes to the quotient, it suffices to see that Y(ArT*E) c AgT*E. To show this we must identify AtT*E as a subset of both AYT’E and Aff(J’E, ArT*E). In AyT*E we have that the elements of ArT*E are characterized as follows: & E ArT* E iff, for a natural coordinate system (x@“,yA, p, pz), we have p:(p) = 0. On the other hand, as a subset of Aff(J’E, ArT*E), ArT*E
= {cp E Aff(J’E,
ArT*E)
: cp= constant}
= (4p E Aff(J’E,
A,“T*E) : I$ = 0)
92
A. ECHEVERRIA-ENRiQUEZ, M. C. MUROZ-LECANDA and N. ROMAN-ROY
($ denotes the linear part of q), or equivalently, system (4, qi), A;;T*E = {p E Aff(J’E,
for an affine natural coordinate
A,mT*E) : q;(p)
= 0).
From the local expression of T, we obtain that T(ArT*E) = AtT*E. As a consequence of this, if pi are the fiber coordinates in J17r* = AyT*E/AcT*E, and pz are those in l-l = Aff(J’E, AtT*E)/ArT*E, we have that
and, consequently, identity. 3.
Canonical
in these natural coordinate
systems, the diffeomorphism
forms in the multimomentum
The multimomentum ential forms:
bundles J’E*
bundles
and MIT are endowed with canonical
DEFINITION 5. The canonical m-form of J’E* is the form 6 defined as follows: if y E J’E* and X1, . . . , X, E 3C(J’E*), then
The canonical
6(y;X1,...,
Xm) :=
(m + l)-firm
of JIE*
!# is the
differ-
E B”(J’E*)
W(Ty~‘(Xd,. . . , Ty$(X,)). is b := -dG
E Dnm+l(JIE*).
In a natural chart in J1 E* we have 0 = p;dmx + pZdyA A dm-lxP 6 = -dp;
A d”x - dp’;l A dyA A d”‘-!xW.
F&MARK. As is known [2], the multicotangent bundle A”T*E is endowed with canonical forms 0 E JP(A”T*E) and the multisymplectic form fi := -dO E Pfl(AnT*E). Then, it can be easily proved that $ = L*@. Observe that Mrr 3 A;lT* E is a subbundle of the multicotangent bundle AmT*E. Let c : A;“T*E c, A”T*E be the natural imbedding (hence < o 10 = 1). DEFINITION 6. The canonical m-form of MT is 0 := <*O E V(Mrr). It is also called the multimomentum Liouville m-form. The cunonicaE (m + I)-form of Mn is 52 = -dO E J~“+‘(MJT), and it is called the multimomentum Liouville (m + 1)-form. The expressions
of these forms in a natural chart in MIT are 0 = pd”x + p$dyA A dm-‘xW,, 52 = -dp
A
dmx - dp;
A
dyA A dm-‘xW.
ON THE MULTIMOMENTUM BUNDLES AND THE LEGENDRE MAPS IN FIELD THEORIES
Then, a simple calculation allows us to prove that S2 is 1-nondegenerate, (MIT, !22) is a multisymplectic manifold. REMARK. Considering @((Y,
a);
Xl,.
the natural projection . . , L>
:=
4~;
t1 :
AyT*E
T~Y.~)~l(X,)r
+
and hence
E, we have
. . . , T~,,,&W
for every (y, a) E A;“T*E
(where y E E and (Yf AyTtE),
and Xi E x(A;“T*E).
The canonical lowing way.
6 and 0 can be characterized
alternatively
m-forms
‘33
in the fol-
PROPOSITION 3. 1. 6 is the unique j3’-semibasic m-form in J’ E* such that, jtir every section $ : E + J’E* of jj, the relation $*6 = I o $ holds. 2. 0 is the only i?’-semibasic m-form in Mn such that, for every section $I : E + ATT*E of k’, the relation @*C3= Cp holds.
Proof: 1. From the definition of 6 it is obvious that it is 6’-semibasic. for the second relation, let y E E and ~1, . . _, u, E T, E; therefore (Ij*&y;
Ul,
Conversely,
.
. . 9wn) = &j(y):
T&u,),
. . . >T&u,))
= &Y))(Y:
(T&’
= &(y))(y;
Ul, . ., ld,) = (I 0 $)(y;
suppose that 6 E D”(J’E*)
Then,
0 T&W,
. . ., (T$&+
0 T&b,,))
111, . . . , u,,).
verifies both conditions
in the statement.
We will prove that 6 is uniquely determined. Let y E J ’ E*, with y < ): -% x, and ul,..., u, E Ty J’E”. We can suppose that ~1. . . , u, are linearly independent and that (~1,. . . , II,) n V,(n o $‘) = (0) (where (~1, . . . , u,,) denotes the subspace generated by these vectors), for in other case 6(y; ul, _ . . , u,,) = 0. Then, by Lemma 1 (see Appendix), there exist ul, . . . , u,?, E T,M and a local section 4 : A4 -+ J’ E* of rr o 5’, such that 4(x) = y and T,$(u,) = ulr (h = 1,. . . , m), and we have that &y;q,...
1
urn> =&ii(y);
.
= ($*o)(y;
T,$h>, . . >T,hn)) ,.. LA{, . . . ,
u,,) = (10 @)(y;
Hence, 6(y; ~1, . . . , II,,) is uniquely determined. 2. The proof follows the same pattern as the one above. 4.
Legendre
Ul,
.
. .
u,,). cl
maps
From the Lagrangian point of view, a classical jield theory is described by its conjiguration bundle n : E -+ M (where M is an oriented manifold with the volume
A. ECHEVERRk-ENRiQUEZ, M. C. ML6JOZ-LECANDA and N. ROMAN-ROY
94
form w E am(M)), and a Lugrangian density which is a it-‘-semibasic m-form on J ’ B. A Lagrangian density is usually written as C = l(Tt I*&), where l c Coo{J1 E) is the hgrangian function associated with C and w. Then a Lugrangian system is a couple ((E, M; n), C). The PoincartGCartan m- and (m + I)-forms associated with the Lagrangian density _C are defined using the vertica2 endotnorphism V of the bundle J’E
@L:=
i(V)L
In a natural chart in J’E, 0,: = $dyA CL S&C--
in which it-‘*& = d”x,
A dm-‘xF
av;2,“,,A
d$ /L
V
SAL := -dOL
+ 1: = BL: + L E Qm(J’E);
we have
-
A dyA A d”-lx,,
a2L apav;
- pdyB
a2z
A
FijE$~ For
a more detailed description 281. To construct a Hamiltonian field theory, the Legendre maps the multimomentum bundle, we DEFINITION 7.
E LY+‘(J’E).
r\dyA A dm-‘xfi
-- ax •t ayB
a2E awav,B
dyB ~d~x.
of all these concepts see, for instance, [l
,
5, 7, 10,
formalism associated with a Lagrangian system in are introduced. Then, depending on the choice of can define different types of these maps as follows.
Let ((E, M; rr), C) be a Lagrangian
system, and v E J’E,
with
j&$X.
1. Let 2) c TJ 1E be the subbundle of natural coordinates x’*TE
= n’*V(n)
there is a natural diagram
in J’E
of total derivatives
& + vi & ). We have that 1 I @ D with T,E IJ?= V,rt ly $ Dj (see [28] for details). Hence projection
T,J;E
Then, the generalized
is generated
in J ’ E (which in a system
o : n’*TE
= V,n ’ Z(T;M
Legendre
by
-+ ~r’*V(n),
@ Vyn),
D,‘y
map is the Cm-map
6% J’E
+
JIE* 9
y
H
DyCy o (Id @ @ly.
and we can draw the c (A”T;M),
ON THE MULTIMOMENTUM
BUNDLES AND THE LEGENDRE MAPS IN FIELD THEORIES
!kf?
2. The reduced Legendre map is the C”-map FL:J’E-+
l-I,
.v k+ Tvc,. where the map T,;C, is defined from the following diagram (the vertical arrows are canonical isomorphisms given by the directional derivatives) T,J;E
---+ TLc,(+ImT;M
2: I qc, T.,L,
--I
T;M @ V,.n --+
A”‘T;M.
(FL is just the vertical derivative of L [ 131). 3. The @-st, extended Legendre map is the P-map
The (second) bY
.?? : J’E --+ .Mrr given by
extended Legendre map is the P-map
.?k : J’E
+ MX
given
3-L = F-L + n*c. 4. The restricted Legendre map is the Coo-map 3L : J’E + I7 given by 3L := /lo FL = /A 0 3-c. If j E J1 E, with x’(j) = y, using multimomentum bundles, we have 3L(j)
= s(y)dyA
A dm-lxg
CL E(j)
a,>
E(j)
coordinates
A dm-‘xM 1 + CL - ~:)o$&4drn~~
= $(j)dyA lr
A d”-‘x,l
Y
= $+)
P
s’
P
- u;($$(‘)d”ll I’
= s(y)
in the different
, Y
= $(j)dyA cc
w FL(y)
the natural
P
-&
@ (dyA - uf(y)dx”)
&
@ CA @ dmx
( .\-’ ’
@ d”x
, I
y’
A. ECHEVERRiA-ENRiQUEZ, M. C. MUfiOZ-LECANDA and
96
that is, the local expressions 3c*xp
= xWL,
3-c*xw = xp,
of the Legendre
N.
ROMAN-ROY
maps are the following:
ax
3C*yA = yA,
3c*p;
3-L*yA = y A,
3-L*p; = -$,
= $---$ P 3_c*p = L - l&$, W
CL 3-z*xp = xp,
3-z*+
= y A,
3-rp;
3-L”p = -?I;-$,
= $,
P
!J
FC*xw = x/1,
FC*yA = yA,
FC*p; = g. Y
REMARKS 1,Taking into account all the above results, it is immediate that FC = 6 o FL: (see diagrams (2) and (4)), and FL: = 9 o 3C.
to proye
2. It is interesting to point out that, as 0 L: and 8~ can be thought as m-forms on E along the projection x1 : J’ E -+ E, the extended Legendre maps can be defined as
where Zl,. . . , Z, E Tn~(y)E, and 21, . . . , .i?, E T,J’E are such that Tjn’2, = Z,. In addition, the (second) extended Legendre map can also be defined as the “firstorder vertical Taylor approximation to 2” [3, 121. Finally, we have the following relations between the Legendre maps and the PoincartGCartan (m + I)-form in J’E (which can be easily proved using natural systems of coordinates and the expressions of the Legendre maps). PROPOSITION4. Let ((E, M; n), C) be a Lagrangian Fk*o Kc*0
= OL,
= OL - c = 8L,
Gz*Gl= OL -L
= OL,
system. Then
3-c*s-J = s2& 3-C% = !& - dL = -dec, FT*ti = S& - dC = -dOr.
REMARK. Observe that the Hamiltonian formalism is essentially the dual formalism of the Lagrangian model, by means of the Lagrangian density. Then, as J’ E is an affine bundle, its affine dual can be identified with Aff (J ’ E, n*A”T*M) 2: Mn, whose dimension is greater than dim J’E, and hence Aff (J’E, n*AmT*M)/AtT*E 2 J’n* is more suitable as a dual bundle (from the dimensional point of view). Then, the canonical forms 0 and fi in Mrr, and 6 and 6 in J’E*, can be
ON THE MULTIMOMENTUM BUNDLES AND THE LEGENDRE MAPS IN FIELD THEORIES
97
pulled-back to the restricted and reduced multimomentum bundles J Ix* and II, using sections of the projections p : Mn --+ J lx * and 6 : J ’ E* + l7. respectively [3, 61 In this way, the reduced and restricted multimomentum bundles are endowed with (noncanonical) geometrical structures (HumiEton-Cartan forms) needed for stating the Hamiltonian formalism. In addition, connections in the bundle n : E + A4 induce linear sections of p (and S) [3, 6, 9, 251 and it can be proved that there is a bijective correspondence between the set of connections in the bundle JT : E + M, and the set of linear sections of the projection p. Hence, all these results, together with Theorem 1, allows us to relate two Iof the most usual Hamiltonian formalisms of field theories [6].
5.
Regular and singular
Following concepts. DEFINITION
I.
systems
the well-known
8.
terminology
of mechanics,
Let ((E, M; n), L) be a Lagrangian
we define
the following
system.
((E, M; n), C) is said to be a regular
or non-degenerate Lagrangian system if FL, and hence, FL are local diffeomorphisms. As a particular case, ((E, M; n), L) is said to be a hyper-regular Lagrangian system if F.C, and hence FL, are global diffeomorphisms.
2. In other cases ((E, M; n), L) is said to be a singular or degenerate Lagrangian system. PROPOSITION 5. 1.
Let ((E, M; n), C) be a hyper-regular
FC(J’E) is a m2-codimensional verse to the projection 6.
imbedded
submanifold
2. 3-L
J1x*,
3_L(J’E),
3-L(J’E),
Lugrangiun system. Then of J’E*
imbedded
which is tran.s-
submanifolds
of Mrr
FAL(J’ E) and II are difleomorphic.
Hence, F-Lk3-L and 3-L are_diffeomorphisms on their images; and_the maps ,.L, restricted to 3L(J’E) or to 3L(J’E), and 10 and 8, restricted to FC( J' E), are ulso diffeomorphisms. Prod: 1. If C is hyper-regular then FL is_a diffeomorphism and hence, as FL = FL o 8, we obtain that FL is injective and FC( J’E) is transverse to the fibers of 8. 2. The proof of this statement 3. It is a direct consequence
is like for the one above (see also [21]). of the above items.
0
98
A. ECHEVERRiA-ENRiQUEZ, M. C. M@JOZ-LECANDA andN. ROMAN-ROY In this way we have the following
(commutative)
diagram
(2)
Observe that there exists a map CL’: e(J’E)s Mn +s(J’E) c Mn, which iLa diffeomorphism defined by the relation 3,C = p’ o 3.C, and p o CL’= p on 3C(J’E). For dealing with singular Lagrangians we must assume minimal “regularity” conditions. Hence we introduce the following terminology. DEFINITION9. A singular Lagrangian system ((E, M; n), 13) is said to be almostregular if 1. P := FTC(J’E) and P := FC(J’E) are closed submanifolds of J’n* and l7, respectively. (We will denote by 10 : P c-) J’n* and Jo : P L) l7 the corresponding imbeddings). 2. 3L, and hence FL, are submersions onto their images. 3. For every j E J’E, the fibers 3’c-‘(3C(y)) and hence FC’(FC(y)) are connected submanifolds of J1 E. (This definition
is equivalent
to that in [21], but slightly different
[9, 251). Let ((E, M; r), L) be an almost-regular + := =(J’E), Let jo : F c, Mn,
50 : p c,
+ := E(J’
Lagrangian E),
Mrr, j o : k L, J’E*
from that in
system. Denote
i, := E(J’E). be the canonical
inclusions,
and
ON THE MULTIMOMENTUM BUNDLESAND THE LEGENDREMAPSIN FIELD THEORIES 99 be the restrictions of the maps p, ~0, 6 and the diffeomorphism Finally, define the restriction mappings 3,Co : J’E
+
P,
E.
FL0 : J’E
+
: J’E ?‘,
+
+,
?ko
FCo : J’E
= 3’c,‘(3Lo(~))
and $’ are submanifolds A P L) Mn, JO : 9 C, Mn,&
3. 9
4. The restriction
mappings
20
-+ $,.
Lagrangian
system. Then
= F,CC,‘(FCo(j>).
of Mrr, b is a submanifold ,. : P C, J’E* are imbeddings.
3L0,
respectively.
-+ P.
PROPOSITION6. Let ((E, M; 7t), C) be an almost-regular 1. The maps *O and /ll are difleomorphisms. 2. For every j E J’E, ?&-‘(Eo(9,
: J’E
9,
and f&
(3)
of J’E*,
are submersions
and
Jo :
with connected
$b ers. Proof:
*O is a diffeomorphism as is q. The second equality of (3) is a consequence of the relation F& = *O o 3Tco. For the proof of the first equality of (,3), and of the assertions concerning F, ? and 50, see [21] and [22]. Then, the proofs of the other assertions are simi1ar.U
Thus we have the (commutative)
diagram
J’E
(4)
. I P
where jIi’ : $ -+ ? is defined by the relation l
Jo
c
l-I
b’ := bILL-’ o b.
REMARKS. The fact that ii is a diffeomorphism is particularly relevant, since it allows us to construct a Hamiltonian formalism for an almost-regular Lagrangian system
VW.
A. ECHEVERRfA-ENRiQUEZ,
100
M. C. MUfiOZ-LECANDA
and N. ROM_&-ROY
It is insesting to point out that the map /2 (which is relatedxith the Legendre map 3Ca) is not a diffeomorphism in general, since rank 3Cc 2 rank3Ca = rank3C0, as is evident from the analysis of the corresponding Jacobian matrices.
l
The matrix of the tangent maps 3C, Id 0
a2_c axvav;
and FC, in a natural coordinate 0
Id
a2E apav;
where the sub-matrix
is the partial Hessian
the regularity
to demanding
of C is equivalent is regular
\
everywhere
system is
in J’E.
matrix of C. Obviously,
that the partial
This fact establishes
Hessian
matrix
the relation
to
I
the concept of regularity given in an equivalent way by saying that a Lagrangian (otherwise it is said to system ((E, 44; n), L) is regular if fi L is I-nondegenerate be singular or nonregular).
Conclusions We have reviewed the definitions of four different multimomentum bundles for the Hamiltonian formalism of first-order classical field theories (multisymplectic models). The so-called generalized and reduced multimomentum bundles are related straightforwardly from their definition, and the same happens with the generalized and restricted multimomentum bundles. The first goal of this work was to relate both couples, proving that the reduced and restricted multimomenturn bundles are, in fact, canonically diffeomorphic. In natural local coordinates, this diffeomorphism is just the identity. The canonical forms, which the generalized and the extended multimomentum bundles are endowed with, have been defined and characterized in several equivalent ways. Given a Lagrangian system in field theory, we have introduced the corresponding Legendre maps relating these multimomentum bundles to the first-order jet bundle associated with this system. Some of them, the generalized and reduced Legendre maps, are defined in a natural way as fiber derivatives of the Lagrangian density, being the other ones obtained from those. The relation among all these maps has been clarified. Regular and almost-regular Lagrangian systems are defined and studied, attending to the geometric features of the Legendre maps. In this way, the standard definitions existing in the standard literature are extended and completed.
ON THE MULTIMOMENTUM BUNDLESANDTHE LEGENDREMAPSIN FIELD THEORIES10 1
A.
Appendix
LEMMA 1. Let n : F + N be a d!ferentiable bundle, with dim N = n and dim F = n + r, and p E F with q = r(p). Let ~11,. . . , uh E T,F (h 5 n) be such that VI,..., vh are linearly independent, and (VI, . . . , uh) n V,(n) = {O}. Then: 1. There exist X1, . . . , xh E X(W), for a neighbourhood W C F of p, such that Xh are linearly independent at every point of W. (4 Xl,..., Xh generate an involutive distribution in W. (b) Xl,..., (c) Xi(p)= Ui,i = 1, . . . . h. = (o}, firevery x E w. (4 (X,(x), ...,xh(x))nvv,(n) 2. There exists a local section y of n, defined in a neighbourhood of q E N, and UI,.... uh E T,N such that y(q) = p, and T,,y(u;) = zli, i = 1,. . . . h. Proof: Let ?_&+I,. . . , v, E T,F be such that T, = V,(n) 61?(vi. . . . . v,). Let (W, 9) be a local chart of F at p, adapted to n. We have q : W + UI x U2 c BY x JR”. Let ei, . . , e, and e,+i, . . , en+r be local basis of Iw” and Iw’, respectively. Then
(TpvO(vd>. . .t T,cp(v,)) = h, . . . . en). T,dV,(n))
= (e,+l, . . . , en+r) ,
Vx
E
(5)
W.
Let Zi,..., Z, E X(Ui x cT2) be the constant extensions of T&vi), . . , T&v,) to Ui x U2. We have that [Zi, Zj] = 0, Vi, j. Finally, let Xi, . . . , X, E X(W). with Xi = (P*Zi. Therefore: 1. Taking Xl,..., Xh (h 5 n), conditions l(a), l(b) and l(c) hold trivially (by construction), and condition l(d) holds as a consequence of (5). 2. Observe that Xi, . . . , X, E X(W) generate the horizontal subspace of a connection defined in the bundle n : W --+ n(W), which is integrable because the distribution is involutive. Then, let y be the integral section of this connection at p. Therefore, y(q) = p, and the subspace tangent to the image of y at p is generated by ui, . . . , v,. Hence, there exist ui, . . . , u, such that T, y (ui) = ~1,. 0 Now, let A be an affine space modelled on a vector space S, and T another vector space, both over the same field K. Let Aff(A, T) be the set of affine maps from A to T; that is, maps q : A + T be such that there exists a linear map $J : S -+ T verifying that q(a) - q(b) = $(a - b); for a, b E A. LEMMA 2. 1. There is a natural isomorphism between Aff (A, T)/ T and S* C3T (and then dimAff(A, T) = dim(S* 63 T) +dimT = dimT(dimS+ 1)). 2. There is a canonical isomorphism between Aff (A, T) and Aff (A, K) ~?3T. Proof: Aff(A, T) is a vector space over K with the natural operations. The map A : Aff(A,T) --+ S* 8 T, which assigns $I to every cp, is linear and we have the exact sequence 0 ---+ T &
Aff(A, T) &
S* @ T ----+ 0,
102
A. ECHEVERRfA-ENRfQUEZ,
M. C. MUhOZ-LECANDA
and N. ROMAN-ROY
where, if t E T, then j(t) : A -+ T is the constant map (j(t))(a) = t, for every a E A. Therefore, we have a natural isomorphism Aff(A, T)/T 21 S* 8 T and then dimAff(A,
T) = dim(S* @ T) + dim T = dim T(dim S + 1).
Moreover, for ya E A, there exists an splitting Aff (A, T) z T CB(S* 81 T) given by the following retract of the above exact sequence j,,:S*@T+Aff(A,T), (P
-
Y b-3.(P(Y - Yo).
(PO:
On the other hand, we have the bilinear map Aff(A, K) x T + (a, t)
Aff(A, T), tct
H
and hence we can define the following
:
a I--+ a(a)t,
morphism
Aff(A, K) 63 T +
Aff(A, T),
OJz@ Ui I+ UjOJi, which is injective because we can assume that the vectors ui are linearly independent, and both spaces have the same dimension. Therefore, Aff (A, T) and 0 Aff (A, K) @ T are canonically isomorphic. If (St,... , s,) is a basis of S, (a’, . . . , am) is its dual basis and (tl, . . . , t,,J is a basis of T, then taking an affine reference in A, (ti, ti @ a’) is a basis of Aff (A, T), as a vector space. Now, let G, H be finite-dimensional vector spaces over K, and F a subspace of G. Consider the exact sequence O-+FFG?+HHO.
(6)
The set C-{a:H+G;
cr linear,rtoa=IdH}
is an affine space modelled on L(H, F) = H*@F. In fact, if 0 E C and ;I E H*@F, then a+h~E, since noh=O. Furthermore, if a,puEC, then rto(a-p)=O, and hence u - p E H* @ F. If dim H = m, taking the space Aff (C, A* H*), and according to the second item of the above Lemma, we have that Aff (C, Am H*) 2 Aff (C, K) ~3 Am H*, and then dim Aff (C, Am H*) = dim Aff (C, K) = dim(H* @ F) + 1. Now, consider the subspace A;lG* = [a E AmG*
: i(u) i(
= 0, U, 2, E F} c A”‘G*.
ON THE MULTIMOMENTUM
BUNDLES AND THE LEGENDRE MAPS IN FIELD THEORIES
LEMMA
3. The spaces Aff (C, A”‘H*)
Proof:
If (f’,
and Ay G* are canonically
103
isomorphic.
is a basis of F and g’ , . . . , gm E G are such that . . . , f’) g”) is a basis of G, then A;lG* is generated by (g’ A . . . A
(f’, . .,” j-;,s’,..., gm, fJ A g’l A . . . A gim-l) (with 1 5 il < . . . -c i,_l < m), therefore dimAyG* = 1 + dim F dim H; that is, dim ATG* = dim Aff (C, A” H*). Next, define the linear map Y: A;lG* ----+ Aff(C, A”H*), n
T(n)
*
: cr H o*q.
We want to prove that it is an isomorphism, for which it suffices to prove that it is injective. Thus, suppose that Y(n) = 0 (that is, a*~ = 0 for every ~7 E C), then we must prove that n = 0. Let g’ , . . . , g” E G be linearly independent (so dimL(g’, . . . , g”) = m); we are going to calculate q(g’, . . . , g”). 1. L(g’,... , g”) n ~(0 = KV. g’“) @ t(F) and therefore n : L(g’, . . , g”) -+ H is an Then G = L(g’,..., anda:H+G, withIsoa=IdH. isomorphism. Hence, there exist h’, . . . , hmEH such that a(h’) = g’; therefore q(g’, 2. L(g’,
. , g”) = q(a(h’),
. . . 2s”> n t(F)
. . ,a(h”))
= (a*q)(h’,
. . . , h”) = 0.
# WI.
(a) dim(L(g’, ...,grn) n t(F)) = 1. In this case there is u E L(g’ , . . . , g”) nt(F), with u # 0, such that L(g’, . . , 8’“) g”) (up to an ordering change), and then there is k E K such that = L(u, g*, . . . , q(g’,
. . . , g”)
= krl@, g*, . . . , g”),
therefore L(g*, . . . , g”)nt(F) = (0). Let (u, u’, . . . , u’) be a basis of F, and g’ be such that (u,ul ,..., u’,gl,g* ,..., g’“) is a basis of G. Then L(g’, g*, . . . , g’“) f? t(F) = {0), and hence, as a consequence of the above item, we conclude that , g”) = 0. Furthermore, L(g’ + u, g*, . . , g*) fl t(F) = {0} because, if V(s’,g*,... this is not true, then L(g’,g*, . . . ,g”)flt(F) # {0}, and hence r](g’+u, g*, , g”) = 0, by the above item. Thus, rl(g’, . . ., g”) = kq(u, g*, . . . , g”) = k(q@+u,
g*, . . . , gm)-_rl(g’,
g*, . . . , g”‘)) = 0.
(b) dim(L(g’, . . . , g”) f’ t(F)) = s > 1, with s 5 r = dim F. Let (f’ , . . . , f”, g’+l, . . . , g”) be a basis of L(g’, . . . , g”), with f’, . . . , f” Then, as rl E AmG*, rl(g’, . . . , g”) = krj(f’,
. . . , f”, gsfl,.
E F.
. , g”) = 0.
Then q = 0, so r is injective and is a canonical isomorphism and A;lG*.
between Aff (C , Am H *) 0
104
A. ECHEVERRfA-ENRfQUEZ,
M. C. MUNOZ-LECANDA and N. ROMAN-ROY
Using the first item of Lemma 2 and identifying sion. LEMMA
4. Aff(x,
AmH*)/AmH*
2 (H*
@ F)*
@
A = C, we make the concluAmH* E H 63 F* @ AmH*.
Acknowledgments We are grateful for the financial support of the CICYT TAP97-0969-CO3-01. We wish to thank Mr. Jeff Palmer for his assistance in preparing the English version of the manuscript. REFERENCES [ll E. Binz, J. Sniatycki, H. Fisher: The Geometry of Classical Fields, North Holland, Amsterdam 1988. 121 F. Cantrijn, L. A. Ibort and M. de Lebn: On the Geometry of Multisymplectic Manifolds, J. Austral. Math. Sot. Ser. 66 (1999) 303-330. I31 J. F. Carifiena, M. Crampin and L. A. Ibort: On the multisymplectic formalism for first order field theories, Diff Geon. Appl. 1 (1991), 345-374. [41 A. Echeverria-Enrfquez and M. C. Mufloz-Lecanda: Variational calculus in several variables: a Hamiltonian approach, Ann. Inst. Henri Poincare’ 56(l) (1992), 27-47. [51 A. Echeverria-Enrfquez, M. C. Mufioz-Lecanda and N. Roman-Roy: Forts. Phys. 44 (1996) 235-280. E61 A. Echevern’a-Enriquez, M. C. Mmioz-Lecanda and N. Roman-Roy: “Geometry of Hamiltonian First-Order Classical Field Theories”. DMAT-UPC (2000) (in preparation). [71 P. L. Garcia: “The Poincar&Cartan invariant in the calculus of variations”, Symp. Math. 14, (Convegno di Geometria Simplettica e Fisica Matematica, INDAM, Rome, 1973) Acad. Press, London 1974, p. 219-246. [81 G. Giachetta, L. Mangiarotti and G. Sardanashvily: Constraint Hamiltonian Systems and Gauge Theories, Int. J. Theor. Phys. 34(12) (1995), 2353-2371. [91 G. Giachetta, L. Mangiarotti and G. Sardanashvily: New Lagrangian and Hamiltonian Methods in Field Theory, World Scientific, Singapore (1997). [lOI H. Goldschmidt and S. Stemberg: The Hamilton-Cartan formalism in the calculus of variations, Ann. Inst. Fourier Grenoble 23(l) (1973), 203-267. [ill M. .I. Gotay: “A Multisymplectic Framework for Classical Field Theory and the Calculus of Variations I: Covariant Hamiltonian formalism”, Mechanics, Analysis and Geometry: 200 Years after Lagrange, M. Francaviglia Ed., Elsevier, Amsterdam 1991, 203-235. [121 M. J. Gotay, J. Isenberg and J. E. Marsden: Momentum maps and classical relativistic fields, Part I: Covariant Field Theory, Preprint November 18, 1997 (66 pages). II31 X. Gracia: “Fibre derivatives: some applications to singular lagrangians”, Proc. V Fall Workshop: Geometry and Physics, 1996, J. F. CariIiena, E. Martinez, M. F. Raiiada, (Eds.) Anales de Fisica Monografias RSEF 4 Madrid (1997). 43-58. 1141 I. Kanatchikov: “Novel algebraic structures from the polysymplectic form in field theory”, GROUP21, Physical Applications and Mathematical Aspects of Geometry. Groups and Algebras, Vol. 2, H. D. Doebner, W Scherer, C. Schulte (Eds.), World Scientific, Singapore 1997, p. 894. [151 J. Kijowski: A Finite-dimensional Canonical Formalism in the Classical Field Theory, Common. Math. Phys. 30 (1973), 99-128. [161 J. Kijowski and W. Szczyrba, “Multisymplectic manifolds and the geometrical construction of the Poisson Brackets in the classical field theory”, Geometric Symplectique et Physique Mathektatique, Coll. Int. C.N.R.S. 237 (1975), 347-378. I171 J. Kijowski and W. M. Tulczyjew: A Symplectic Framework for Field Theories, Lect. Notes Phys. 170, Springer, Berlin 1979. [181 D. Krupka: Geometry of Lagrangean Structures, Supp. Rend. Circ. Mat. Palermo, ser. II, no. 14 (1987), 187-224.
ON THE MULTIMOMENTUM BUNDLES AND THE LEGENDRE MAPS IN FIELD THEORIES ](I5 [ 191 B. A. Kupershmidt: Geometry of Jet Bundles and the Structure of Lagrangian and Hamiltonian Formalisms, Lect. Notes in Math. 775, Springer, New York 1980, pp. 162-218. [20] M. de I.&n, J. Marfn-Solano and J. C. Marrero: “Ehresmann connections in classical field theories”. Proc. III Fall Workshop: Differential Geometry and its Applicafions, Anales de Fisica, Monografias 2 (1995), 73-89. [21] M. de Leon, J. Marfn-Solano and J. C. Marrero: “A geometrical approach to classical field theories: in Differential Geometry. A constraint algorithm for singular theories”, Proc. on New Developments L. Tamassi, J. Szenthe (Eds.), Kluwer, Dordrecht 1996, pp. 291-312. [22] M. de Leon, J. Marfn and J. Marrero: The constraint algorithm in the jet formalism, Difl Geom. Appl. 6 (1996), 275-300. [23] M. de Le6n and P. R. Rodrigues: Hamiltonian structures and Lagrangian field theories, Bol. Acad. Gabga de Ciencias VII (1988), 69-81. 1241 J. E. Marsden and S. Shkoller: “Multisymplectic geometry, covariant Hamiltonians and water waves”, Math. Proc. Camb. Phil. Sot. 1251 G. Sardanashvily: Generalized
125 (1999), 553-575. Hamiltonian Formalism
for Field Theory. Constraint Systems. World Scientific, Singapore 1995. 1261 G. Sardanashvily and 0. Zakharov: On application of the Hamilton formalism in fibred manifolds to field theory, Difl Geom. Appl. 3 (1993), 245-263. [27] D. J. Saunders: The Cartan form in Lagrangian field theories, J. Phys. A: Math. Gen. 20 (1987). 333-319. [28] D. J. Saunders, The Geometry of Jet Bundles, London Math. Sot. Lect. Notes Ser. 142, Cambridge University Press, Cambridge 1989.