Vilasi G. Hamiltonian Dynamics
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234
Symplectic Manifolds and Hamiltonian Systems
so
that
and Eq.
(9.4)
becomes
or
I
dH
dXj
w..X%
=
--.
31
Since det(wij(x))
#
0,
the last relation gives
Thus, the first order differential equations
Hamiltonian vector fields
X
have the following
.
.
dH
_-
-
AaJ-
dxi
dt
dxj
'
for the integral curves of the
form:
(9.5)
and they are very similar to the Eqs.
(2.24)
of Sec.
2.4.1.
trix, whose elements are
Aij,
satisfies the Jacobi identity
Equations
(2.24)
and
(9.5)
coincide, provided that the antisymmetric ma-
Actually, the Jacobi identity
is
satisfied because of the closure of the sym-
plectic form
w.
Indeed, in
a
coordinate basis, we may write
so
that
The reader can easily check that, if
Aihwhj
=
dj,
Eqs.
(9.6)
are equivalent
to
dwij
dwjk
dwki
-
+
-+
-
=o.
BX~
ax%
ax3
Hamiltonian
Flowa
235
9.3
Hamiltonian
Flows
What has been said in the previous section can be repeated, more geometri-
cally,
as
follows.
Let
us
consider
a
function
f
defined on the differentiable symplectic mani-
fold
(M,w).
Its differential
df,
at the point
p
E
M
belongs to
TM
dfp:7,M+!R,
VpEM.
The bi-vector field
A
associates to
df,
a tangent vector to
M
at the point
p
E
M
as
follows:
Xf(P)
=
A(df
(PI1
-1
*
With the vector field
X,(p),
a one-parameter group of diffeomorphisms is
associated
(Eq.
(5.13))
as
ut:M+M,
such that
The group
ot,
which is called
Hamiltonian
flow
with Hamilton function H,
preserves the symplectic structure, that is
Ut*W
=
w
(9.7)
where
at*
is the derivative
of
at.
More explicitly,
Eq.
(9.7)
can be written in the following form:
(at*4p(X,
y>
=
wut(,)(ff~,(X),~l,(Y))
=
%J(Xl
y>
9
(9.8)
where
X,
Y
E
7,M
and
ofp
:
7,M
+
7,t(,lM
is the derivative of
at
at the point
p.
The Lie derivative of the %form
w
along
Xf
is given by
where the relation
(9.8)
has been used.
236
Symplectic Manifolds and Hamiltonian Systems
Since
w
is closed, we may write
LX,W=O
H
dix,w=O.
Of course,
ix,w
is an exact differential 1-form, since
(9.9)
9.3.1
Lie algebras
of
Hamiltonian vector
fields
and
of
Hamilton functions
It is worth recalling that
a
Lie algebra is a vector space
A
supplied with
a
bracket
which is
0
bilinear
(9.10)
[X,Y1
=
-[Y,ZI
VGY
E
A;
(9.11)
0
antisymmetric
0
and satisfying the Jacobi identity
“Z,YI,Zl
+
“Y,Z1,4
+
“Z>ZI,Yl
=
0,
‘dX,Y,Z
E
A.
(9.12)
The Lie bracket
[X,
YI
=
LXY
,
(9.13)
which satisfies the relations (9.10), (9.11) and (9.12), provides the infinite-
dimensional vector space of differentiable vector fields, on
a
manifold
M,
with
a
Lie algebra structure.
Let
X
and
Y
be two vector fields and
ox
and
u+
be the corresponding flows,
respectively.
As
already said, such flows are diffeomorphisms defined over all
M,
if the manifold is compact. Otherwise,
ok
and
a+
are defined only in
open sets in
M
and
for
small values of the parameters
t
and
s.
However, this
suffices for
our
purposes.
Hamiltonian
Flows
237
An important property
of
the Lie bracket, of two vector fields, is that its
vanishing is
a
necessary and sufficient condition for the corresponding flows to
commute2:
[X,
Y]
=
0
*
.:a+
=
a+ax.
t
(9.14)
Let
(M,u)
be a symplectic manifold, and
f
and
g
two differentiable func-
The bracket
{
f,
g},
defined by
tions on
M.
(9.15)
where
a;
denotes the Hamiltonian flow corresponding to the Hamiltonian vec-
tor field
Xf
defined by
ix,w
=
df,
is called the
Poisson bracket
of
the functions
f
and
g.
From
definition (9.15), we have that the vanishing
of
the Poisson bracket
{
f,
g}
is
a
necessary and sufficient condition for the function
g
to be a first
integral
of
the flow
c;
with Hamilton function
f.
Became
of the isomorphism
(9.2)
between vector fields and differential
forms, the Poisson bracket
(9.15)
can be written in the following form:
{f,d(P)
=
Ap(df,dg)
=
ql(Xf,XB)
*
(9.16)
Indeed,
Vp
E
M
Exercise
9.3.1.
Prove,
by
using
(9.16),
that the Poisson Bracket
is
bilinear, antisymmetric, and satisfies the Jacobi identity
{{f,g},
h}
+
((91
hllf
1
+
{{hlf 1191
=
0
*
(9.17)
Thus, the Poisson bracket provides the set
T(M),
of differentiable functions
on
M,
with a Lie algebra structure. This Lie algebra, as
it
has already been
shown in Part
I,
modulo the constants, is isomorphic to the Lie algebra of
differentiable vector fields on
M.
238
Symplectic
Manifolds and
Hamiltonian Systems
Exercise
9.3.2.
Let w be a closed differential %form and
X
and
Y
be any
two vector fields on a manifold
M,
locally represented by
We have
Lxiyw
-
iyLxw
=
dixiyw
+
ixdiyw
-
iydixw
=
d(w(Y, X))
+
ixdiyw
-
iydixw
=
d(-wijXiY3)
+
ixd(wijYidd
-
wjjYjdxi)
-
iyd(wijXidxj
-
wijXjdxi)
=
wij[X, Y]adxj
-
wij[X, Y]jdxi
=
i[X,Y]W.
Prove the relation
i[x,y]w
=
Lxiyw
-
iyLxw,
(9.18)
without use
of
the coordinates.
Let
Xf
and
X,
be the Hamiltonian vector fields associated with the func-
tions
f
and
g,
respectively; i.e.
ix,
w
=
df
,
ix,~
=
dg
.
By using
Eq.
(9.18)
for
the Hamiltonian vector fields
Xj
and
Yj,
we
have
i[x,,x,lw
=
Lx,ix,w
=
dix,ix,w
+
ix,dix,w
=
dix,ix,w
=
d{f,g},
(9.19)
so
that
L[X,,X,]W
=
0.
(9.20)
Therefore,
[X,, X,]
is
a
globally Hamiltonian vector field with Hamilton
function given
by
H(P)
=
WP(Xf
,
X,)
=
{f,
SHP)
*
Thus, the set of globally Hamiltonian vector fields on
a
symplectic manifold
close
on
a
Lie subalgebra of all vector fields.
The
Cotangent
Bundle
and
Its
Symplectic Structure
239
Exercise
9.3.3.
Prove
that the
set
of
first
integrals
of
a Hamiltonian
flow
constitute a subalgebra
of
the Lie algebra
of
all differentiable functions.
Exercise
9.3.4.
Prove, by using
Eq.
(9.18),
that the Lie bracket
of
two locally
Hamiltonian vector fields,
X
and
Y,
is a globally Hamiltonian vector field, with
Hamiltonian function given by
H(p)
=
wp(Y,
X).
It
follows that the set
of
locally Hamiltonian vector fields constitute a sub-
algebra
of
the
Lie algebra
of
all vector fields too.
The considerations developed in Sec.
2.4.4 (Further generalizations
of
the
Jacobi-Poisson dynamics),
can be repeated, of course, also in this new context.
A useful reading on the theory of ordinary Jacobi-Poisson manifolds is given
by Vaisman’s book.54
9.4
The
Cotangent Bundle and Its Symplectic Structure
An example of symplectic manifold is given by the cotangent bundle
‘T*Q
of an n-dimensional manifold
Q.
An element
29
of
T*Q
is
a
differential
1-
form on
7,Q,
the tangent space to
Q
at
a
point
p.
In a coordinates basis
(q’,.
.
.
,
qn),
a differential 1-form
6
has components
PI,.
.
.
,pn
and the
2n
numbers
(PI,.
.
.
,pn,
q’,
.
.
.
,
qn)
can be taken
as
local coordinates
of
a
point
in
T’Q.
Thus, the cotangent bundle
M
=
T*Q
has a natural structure of a
2n-
dimensional differential manifold.
211
Moreover, it can be proven (see Appendix
E)
that
TQ
has
a
natural
symplectic structure
w,
which, in local coordinates, can be written
as
follows:
or
wC
=
d6,,
with
The differential forms
6,
and
w,
are called the
canonical differential
f-form
and the
canonical symplectic structure,
respectively.
240
Symplectic Manifolds and Hamiltonion Systems
But there
is
much more, in the sense that any symplectic manifold can be
locally considered
as
a cotangent bundle. This is guaranteed by the Darboux+
the~rem,~*'>~ according to which:
Theorem
29
(Darboux)
At
every point po
of
a 2n-dimensional symplectic
manifold
M,
there exists a chart
(U,po)
in
which the symplectic structure
w
assumes the
form
w=dxiAdxi+n,
i=l,
...,
n.
Such
a
chart
(U,
po)
is
called
a
Darboux chart.
In
a
Darboux chart, setting
(PI
=
x
1
,.
*.
,pn
=
x",ql
f
xn+l,.
.
.
,q"
=
P),
the symplectic structure
w
and the bivector field
A,
given by Eqs.
(9.3),
assume
the canonical forms
wc
=
dpi
A
dq'
and
respectively. Moreover, the Eq.
(9.3),
dxi
..aH
-
=
A"-,
dt
6x3
become the familiar Hamilton equations
(9.22)
An atlas
for M,
composed by Darboux charts, is called
a
Darboux atlas
or
a
symplectic atlas.
'Gaston Darboux, born in Nimes in
1842
and died in Paris in
1917,
has been
a
professor
at
the Sorbonne University for about
40
years. His work in four volumes on
Thdorie
des
Surfaces
is
considered a classic. Besides giving new and remarkable contributions to differential
geometry, he deeply influenced the development
of
the theory
of
differential equations and,
thanks to
a
deep
geometrical insight and a sagacious
use
of
algorithms, gave solutions to
relevant problems in calculus and mechanics.
Revisited
Analytical
Mechanics
241
At this point it is clear that the
Hamiltonian formulation
of
the dynamics,
described in Part
I
(Analytical Mechanics) is, at least
for
systems which do
not depend explicitly on time, the local version (i.e. in
a
Darboux chart) of the
theory
of
Hamiltonian vector fields on a symplectic manifold
M.
9.5
Revisited
Analytical
Mechanics
The reader can discover by himself the global version
of
many results obtained
in Part
I.
Indeed,
A system of particles has n-degrees
of
freedom if its configurations
define an n-dimensional differential manifold
Q.
The
state space
of
the system
is
the tangent bundle
TQ,
while the
phase space
is
the
cotangent bundle
T"Q.
0
A
Lagrangian function
L
is
a
differentiable map
from
TQ
into
8.
0
Lagrange's equations
can
be written in the intrinsic form,
Lad=
=
dL
,
(9.23)
or
iAdfiL:
=
-dEr:,
(9.24)
where
-
A
is the vector field given by
A
=
vha/aqh
+
LhkFk(q/v)a/avh;
-
Lhk
are the elements
of
the matrix
L-',
with
L
=
(a2L/avhvk);
-
19c
the differential l-form
on
TQ
defined by
dr
=
(aL/avh)dqh;
242
Symplectic Manafolds and Hamiltonian Systems
-
EL
is
the energy
EL
=
iAdL
-
C.
We notice that, if the Hessian determinant of the Lagrangian is not
vanishing, then
WL
=
ddL
is a symplectic structure on
‘TQ.
The intrinsic form of Lagrange’s equation allows
us
to introduce the Nother
theorem as follows.
Consider
a
complete vector field
X
on
‘TQ;
i.e. the generator
of
a
one-
parameter group
cp7
of diffeomorphisms on
TQ.
Let
us
calculate the
infinites-
imal transformation,
6.C
=
LxC,
which
X
induces on the Lagrangian function
C.
From Eq.
(9.23),
we have
612
E
LxC
=
ixdC
=
~XLAI~L
=
(LAI~L,
X)
=
LA(~L,
X)
-
(Gr,
LAX)
=
i[x,A]dL
-k
LAiXdL.
It follows that
LxL
=
0,
and
[X,
A]
=
0
=+
L~ixfi~:
=
0,
i.e.
Theorem
30
(Nother)
A
symmetry
X
of
both the Lagrangian
L:
and the
dynamics
A
gives rise
to
a
first
integral given by
LA(ixdL).
The translation of the previous geometrical formulation in coordinate language
gives back the original formulation by Emmy Nother.
Remark
17
In order
for
ixdL
to be a
first
integral,
it
sufices that
i(x,~]d~
-
LxC
vanishes, which
is
less
stringent than the separate vanishing
of
each term.
0
The Legendre transformation defines
a
vector bundle isomorphism be-
tween
‘TQ
and
PQ.
Indeed, the map
f
:
(q,v)
E
‘TQ+(Q,P)
E
7*Ql
with
ph
=
(a.C/awh)(q,
v),
induces the derivative map
f*
X
E
qq,w)
(TQ)
I--+
X*
=
f*X
E
qq,p)
(TQ)
*
Revisited
Analytical
Mechanics
243
The Legendre transformation is then defined by
(q,
v/Q, V)
-+
(q,
P/Q,
P>
,
where
Q,V
denote the sets of
“q”
and the
“v”
components of the
vector field
X,
respectively, and
Q,
P
the ones of the
“q”
and the
“p”
components of the vector field
X,.
ha
a
X=Q
-+Vh-
%lh
bvh
’
x,
=
Qh-
+
Ph-.
aqh
aph
a a
In matrix notation, setting
f*=(
ML’
I
0)
with
I
the
n
x
n
identity matrix, and
we have
where the
tilde
N
indicates that the velocities
v’s
must be expressed in
terms of the
q’s
and
p’s
by inverting the relations
ph
=
(aC/bvh)(q,
v).
If
the Lagrangian is degenerate; i.e. the Hessian determinant vanishes,
the Legendre map defines only a vector bundle homomorphism from
TQ
into
TQ.
The theory of constraints by Dirac and
just starts
from this observation.
A
geometrical analysis can be found in Refs.
41,
108, 177, 146
and
136.
0
An algebraic formulation of Lagrangian dynamics, suitable to be used
in a general context, including situations with no global Lagrangian
and/or fermionic variables, can be found in Ref.
76.