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\topmatter
\title
CONFORMALLY SYMPLECTIC DYNAMICS and SYMMETRY OF THE LYAPUNOV SPECTRUM
\endtitle
\rightheadtext{CONFORMALLY SYMPLECTIC DYNAMICS}
\author
Maciej P. Wojtkowski and Carlangelo Liverani
\endauthor
\affil University of Arizona, University of Rome {\sl Tor Vergata}
\endaffil
\address
Liverani Carlangelo,
Mathematics Department,
University of Rome ``Tor Vergata",
00133 Rome, Italy.
Maciej P. Wojtkowski,
Department of Mathematics,
University of Arizona,
Tuscon, Arizona 85 721
USA
\endaddress
\email
liverani@mat.utovrm.it
maciejw@math.arizona.edu
\endemail
\date
February 10, 1997
\enddate
\abstract
A generalization of the Hamiltonian formalism is studied
and the symmetry of the Lyapunov spectrum established
for the resulting systems.
The formalism is applied to the Gausssian isokinetic
dynamics of
interacting particles with hard core collisions and other systems.
\endabstract
\thanks
{\bf
We thank Dmitri Alexeevski, Federico Bonetto and Philippe Choquard
for many enlightening discussions during our stay at the ESI in
December of 1996. In addition, we thank David Ruelle for valuable comments.
We are also grateful for the opportunities provided by the
hospitality of the Erwin Schr\"odinger Institute in Vienna
where this work was done. M.P.Wojtkowski has been partially supported by NSF
Grant DMS-9404420.}
\endthanks
\endtopmatter
\document
\vskip.7cm
\subhead \S 0. Introduction \endsubhead
\vskip.4cm
We study the symmetry of the Lyapunov spectrum in systems more general
than Hamiltonian but closely related to the symplectic formalism. We
call these systems conformally Hamiltonian. They are determined by a
non-degenerate $2$-form $\Theta$ on the phase space and a function
$H$, called again a Hamiltonian. The form $\Theta$ is not assumed to
be closed but it satisfies the following basic condition $d\Theta =
\gamma\wedge\Theta$, for some closed $1$-form $\gamma$. This condition
guarantees that, at least locally, the form $\Theta$ can be multiplied
by a nonzero function to give a bona fide symplectic structure. The
skew-orthogonality of tangent vectors is preserved under
multiplication of the form by any nonzero function, hence the name
{\it conformally symplectic structure}. These ideas were known to
geometers for a long time, see for example the paper of Vaisman
\cite{V}.
The conformally Hamiltonian (with respect to the form $\Theta$)
vector field $\nabla_{\Theta}H$ is defined
by the usual relation
$$
\Theta\left(\cdot,\, \nabla_\Theta H\right) = dH(\cdot).
$$
The Hamiltonian function $H$ is again a first integral of the
system. In Section 2 we prove that for any conformally Hamiltonian
system restricted to a smooth level set of the Hamiltonian the
Lyapunov spectrum is symmetric, with symmetric exponents adding up to
a constant. More precisely the direction of the flow has to be
factored out. In Section 3 we extend this formalism to flows with
collisions.
In Section 1 we give an independent proof of the fact that for any
conformally symplectic cocycle we have the symmetry of the Lyapunov
spectrum. The first proof of this fact in the symplectic case goes
back to Benettin et al. \cite{B-G-G-S}. Our proof is based on an
alternative description of Lyapunov exponents and it borrows an idea
from \cite{W1} (Lemma 1.2).
In Section 4 we present examples which were recently the subject
of several papers. We show that the Gaussian isokinetic dynamics
can be viewed as a conformally Hamiltonian system, by which
we immediately recover the results of Dettmann and Morriss \cite{D-M 1},
\cite{D-M 2}, on the symmetry of the Lyapunov spectrum and the
Hamiltonian character of the dynamics. We extend these results to
systems with collisions, taking advantage of the fact that our
formalism works equally well for collisions as it does for flows. Such
systems were studied in the paper by Dellago, Posch and Hoover,
\cite{D-P-H}, and the symmetry of the Lyapunov spectra was
demonstrated numerically. Chernov et al, \cite{Ch-E-L-S} studied
rigorously the Lorentz gas of periodic scatterers with an electric
external field in dimension 2. Latz, van Beijeren and Dorfman,
\cite{L-B-D}, considered thermostated random Lorentz gas in
$3$-dimensions and found the symmetry there. Let us note that we
prove the symmetry of the Lyapunov spectrum for any invariant
(ergodic) measure and not only for the SRB measure, which is the
easiest to access numerically.
We also show that the Gaussian isokinetic dynamics on a Riemannian
manifold can be given the same treatment.
The last application is to Nos\'e--Hoover dynamics. We show that the
Hoover equations can be naturally viewed as a conformally Hamiltonian
system, thus giving another proof of the symmetry of Lyapunov spectra
for this system. It was originally proven by Dettmann and Morriss,
\cite{D-M 3}.
Finally let us note that our approach is one of several possible.
Recently Choquard, \cite{Ch}, showed that the isokinetic and
Nos\'e-Hoover dynamics can be considered as Lagrangian systems. Even
in the conformally symplectic framework one can keep the Hamiltonian
unchanged and modify the form $\Theta$, or keep the form unchanged and
modify the Hamiltonian, or keep both the form and the Hamiltonian
unchanged, but change time on a level set of the Hamiltonian. We
elaborate on that in Remark 2.1. We believe that our approach sheds
new light on these issues.
\vskip.7cm
\subhead \S 1. Conformally symplectic group \endsubhead
\vskip.4cm
Let $\omega = \sum_{i=1}^n dp_i \wedge dq_i$ be the standard linear
symplectic form
in $\Bbb R^n \times \Bbb R^n$.
\proclaim{Proposition 1.1}
For an invertible linear mapping $S$ acting on
$\Bbb R^{2n} = \Bbb R^n \times \Bbb R^n$
the following are equivalent
\roster
\item"(a)" $\omega(Su,Sv) = \beta \omega(u,v) \ \ \
\text{for some scalar} \ \ \ \beta \ \ \
\text{and all} \ \ \ u,v \in \Bbb R^{2n};$
\item"(b)" $\omega(Su,Sv) = 0 \ \ \ \ \ \text{if and only if}\ \ \ \ \
\omega(u,v) = 0,$\hfil\break
i.e., $S$ preserves skew-orthogonality of vectors;
\item"(c)" $S$ takes Lagrangian subspaces of $\Bbb R^{2n}$ into
Lagrangian subspaces.
\endroster
\endproclaim
\demo{Proof}
It is apparent that (a) implies (b) and that (b) is equivalent
to (c).
It remains to prove that (c) implies (a).
By composing $S$ with an appropriate linear symplectic map
we can assume without loss of generality that $S$ preserves
the Lagrangian subspaces $\Bbb R^n\times\{0\}$ and $\{0\}\times\Bbb R^n$,
i.e., $S$ is block diagonal.
Moreover, again by multiplying by an appropriate linear symplectic
map, we can assume that $S$ is equal to identity on $\Bbb R^n\times\{0\}$.
By (b) we conclude that $S$ is diagonal on $\{0\}\times\Bbb R^n$.
A simple calculation shows that to satisfy (b) this diagonal matrix
must be a multiple of identity, which gives us (a). \qed
\enddemo
We will call a linear map from $GL(\Bbb R^{2n})$ conformally symplectic
if it satisfies one of the properties in the Proposition 1.1.
The group of all conformally symplectic maps will be denoted by
$CSp(\Bbb R^{2n})$.
Let $X$ be a measurable space with probabilistic
measure $\mu$ and let $T: X \to X$ be an ergodic map.
Let further $A : X \to GL(\Bbb R^{2n})$ be a measurable map
such that
$$
\int_X\log_+\|A(x)\| d\mu(x) < +\infty.
\tag 1.1
$$
We define the matrix valued cocycle
$$
A^m(x) = A(T^{m-1}x)\dots A(x).
$$
By the Oseledets Multiplicative Ergodic Theorem, \cite{O}, which in
this generality was first proven by Ruelle, \cite{R}, there are numbers
$\lambda_1 < \dots < \lambda_s$, called
the Lyapunov exponents of the measurable
cocycle $A(x),\ x \in X$, and for almost all $x$ a flag of
subspaces
$$
\{0\} = \Bbb V_0 \subset \Bbb V_1(x) \subset \dots \subset\Bbb V_{s-1}(x)
\subset\Bbb V_s = \Bbb R^{2n},$$
such that
for all vectors $v\in \Bbb V_k(x) \setminus\Bbb V_{k-1}(x)$
$$
\lim_{m \to +\infty} \frac 1m\log\|A^m(x)v\|:=\lambda(v) = \lambda_k.
$$
In addition, denoting by $d_k$ the difference between the dimensions of
$\Bbb V_k$ and $\Bbb V_{k-1}$ ($d_k$ is called the multiplicity of
the $k$-th Lyapunov exponent), the following holds:
$$
\sum_{k=1}^s d_k\lambda_k=\int_X\log|\hbox{det}(A(x))|d\mu(x),
\tag 1.2
$$
i.e., the sum of all Lyapunov exponents is equal to the average exponential
rate of volume growth.
Given a measurable cocycle $A(x), \ x \in X$, satisfying \thetag{1.1}
and with values in the conformally symplectic group $CSp(\Bbb R^{2n})$
we obtain a measurable function $\beta = \beta(x)$ such that
$$
\omega(A(x)u,A(x)v) = \beta(x) \omega(u,v),
\tag 1.3
$$
for all vectors $u,v \in \Bbb R^{2n}$.
Let us define
$$
b:=\int_X\log|\beta(x)|d\mu(x).\tag{1.4}
$$
\proclaim{Lemma 1.1}If a measurable cocycle $A(x), \ x \in X$,
satisfies \thetag{1.1} and it has values in the conformally symplectic group
$CSp(\Bbb R^{2n})$, then
$$
\sum_{k=1}^sd_k\lambda_k=nb.
$$
\endproclaim
\demo{Proof}
Since $\omega^n$ is the volume form
it follows from \thetag{1.3} that the determinant of $A(x)$ is
$$
\det A(x) = \beta(x)^n.
$$
The lemma follows by applying (1.2).
\qed\enddemo
\proclaim{Lemma 1.2} Given a measurable cocycle $A(x), \ x \in X$,
satisfying \thetag{1.1} and with values in the conformally symplectic group
$CSp(\Bbb R^{2n})$, for each two non skew-orthogonal vectors
$u,v \in \Bbb R^{2n}$, i.e., $\omega(u,v) \neq 0$, we have
$$
\lambda(u)+\lambda(v)\geq b .
$$
\endproclaim
\demo{Proof}
For the standard Euclidean norm $\|\cdot\|$
we have $|\omega(u,v)| \leq \|u\| \|v\|$.
From \thetag{1.3} we obtain
$$
\omega(A^m(x)u, A^m(x)v) = \omega(u,v)\prod_{i = 0}^m\beta(T^{i}x).
$$
Therefore,
$$
\log|\omega(A^m(x)u, A^m(x)v)| = \log|\omega(u,v)|+\sum_{i = 0}^m
\log|\beta(T^{i}x)|,
$$
and
$$
\frac 1m \log|\omega(A^m(x)u, A^m(x)v)| \leq
\frac 1m \log\|A^m(x)u\| + \frac 1m \log\|A^m(x)v\|.
$$
Putting these relations together and using the Birkhoff Ergodic Theorem
we conclude that
$$
b = \int_X\log|\beta(x)| d\mu(x) \leq \lambda(u) + \lambda(v).
$$
\qed\enddemo
The following Lemma is obvious. We formulate it to streamline the
proof of Theorem 1.4 where it is used twice.
For a linear subspace
$X \subset \Bbb R^{2n}$ we denote by $X^\sk$ the skew-orthogonal
complement of $X$, i.e, $X^\sk \subset \Bbb R^{2n}$ is the
linear subspace containing all vectors $v$ such that
$\omega(u,\,v) = 0$ for all $u \in X$. Since $\omega$ is assumed to
be nondegenerate we have
$$
\dim X + \dim X^\sk = 2n \ \ \ \ \text{and} \ \ \ \
(X^\sk)^\sk = X.
$$
\proclaim{Lemma 1.3} Let $U,V \subset \Bbb R^{2n}$ be two linear
subspaces. If $\omega(u,\,v) = 0$ for all $u \in U$ and $v \in V$,
then
$$
U \subset V^\sk, \ \ V \subset U^\sk \ \ \ \ \text{and}
\ \ \ \ \dim U + \dim V \leq 2n.
$$
\endproclaim\qed
\proclaim{Theorem 1.4}
If a measurable cocycle $A(x), \ x \in X$, satisfies \thetag{1.1}
and it has values in the conformally symplectic group $CSp(\Bbb R^{2n})$
then we have the following symmetry of the Lyapunov spectrum:
$$
\lambda_k + \lambda_{s-k+1} = b,
$$
where $b$ is given by \thetag{1.4},
and the multiplicities of $\lambda_k$ and $ \lambda_{s-k+1}$
are equal, for $k = 1,2, \dots, s$.
Moreover the subspace $\Bbb V_{s-k}$ is the skew-orthogonal
complement of $\Bbb V_k$.
\endproclaim
\demo{Proof}
Let $\mu_1 \leq \mu_2 \leq \dots \leq \mu_{2n}$ be the Lyapunov
exponents taken with repetitions according to their multiplicities.
By Lemma 1.1 $\mu_1 + \mu_2 + \dots + \mu_{2n} = nb$.
We can choose a flag of subspaces
$$
\{0\} = \Bbb W_0 \subset \Bbb W_1(x) \subset \dots \subset\Bbb W_{2n-1}(x)
\subset\Bbb W_{2n} = \Bbb R^{2n},$$
such that $\dim \Bbb W_l =l$ and
for all vectors $v\in \Bbb W_l(x) \setminus\Bbb W_{l-1}(x)$ the
Lyapunov exponent $\lambda(v) = \mu_l$, for $l =1,2,\dots,2n$.
(Note that except in the case of all multiplicities equal to $1$
there is a continuum of such flags.)
Since for any $l \leq n$,
$\dim \Bbb W_l + \dim \Bbb W_{2n-l+1} = 2n+1$,
by Lemma 1.3 there are vectors $u \in \Bbb W_l$ and $v \in \Bbb W_{2n-l+1}$
such that $\omega(u,v)\neq 0$. By continuity there must be also
vectors $\tilde u \in \Bbb W_l \setminus \Bbb W_{l-1}$ and
$\tilde v \in \Bbb W_{2n-l+1} \setminus
\Bbb W_{2n-l}$ such that $\omega(\tilde u,\,\tilde v)\neq 0$.
It follows from Lemma 1.2 that
$$
\mu_l + \mu_{2n-l+1} \geq b,
$$
for $l =1,2,\dots, n$.
Adding these inequalities together, we get
$$
nb = \sum_{l=1}^n(\mu_l + \mu_{2n-l+1}) \geq nb,
$$
which shows that all the inequalities must be actually
equalities. It follows immediately that for any $k = 1,\dots, s,$
the multiplicities of $\lambda_k$ and $\lambda_{s-k+1}$ are equal
and $\lambda_k + \lambda_{s-k+1} = b$.
To show that the subspace
$\Bbb V_{s-k}$ is the skew-orthogonal complement
of the subspace $\Bbb V_k$
we observe that $\omega(u,\,v) = 0$ for any $u\in \Bbb V_k$
and $ v \in \Bbb V_{s-k}$. Indeed, if this is not the case we
could use Lemma 1.2 to claim that
$ \lambda_k + \lambda_{s-k} \geq b$, which leads to
the contradiction
$$
b = \lambda_k + \lambda_{s-k+1} > \lambda_k + \lambda_{s-k} \geq b.
$$
We can now apply Lemma 1.3 and we obtain $\Bbb V_{s-k} \subset \Bbb
V_k^\sk$. Since the dimensions of these subspaces are equal
we must have
$\Bbb V_{s -k} = \Bbb V_k^\sk $.
\qed\enddemo
\vskip.7cm
\subhead \S 2. Conformally symplectic manifolds
and conformal Hamiltonian flows \endsubhead
\vskip.4cm
Let $M$ be a smooth manifold of even dimension.
A conformally symplectic structure on $M$ is
a differentiable $2$-form $\Theta$ which is non-degenerate and
has the following basic property
$$
d\Theta = \gamma \wedge \Theta,\tag2.1
$$
for some closed $1$-form $\gamma$. A manifold with such a form $\Theta$
is called conformally symplectic. The origin of this name becomes
clear when one observes that locally $\gamma = dU$ for some smooth function
$U$ and
$$
d(e^{-U}\Theta) = 0,
$$
i.e. $e^{-U}\Theta$ defines a bona fide symplectic structure.
For a given function $H: M \to \Bbb R$, called a Hamiltonian,
let us consider a vector field $\nabla_{\Theta}H$ defined by the
usual relation
$$
\Theta(\cdot,\, \nabla_{\Theta}H ) = dH.\tag2.2
$$
We will call it the conformally
Hamiltonian vector field, or conformally symplectic,
or simply a Hamiltonian vector field when the
conformally symplectic structure is clearly chosen.
Note that our definition does not coincide
with the definition of a Hamiltonian vector field from \cite{V}.
Let $\Phi^t$ denote the flow defined by the vector field
$F = \nabla_{\Theta}H$. The Hamiltonian function $H$ is a first integral
of the system. Indeed we have
$$
\frac{d}{dt}H = dH(\nabla_\Theta H) = \Theta(\nabla_\Theta H,\,
\nabla_\Theta H) = 0.
$$
Let us consider the Lie derivative of the form $\Theta$ in the
direction of vector field $F$, i.e.,
$$
\left(L_F\Theta\right)(\xi,\eta) :=
\frac{d}{du}\Theta(D\Phi^u\xi,D\Phi^u\eta)_{|u = 0}.
$$
\proclaim{Theorem 2.1}
For a Hamiltonian vector field $F = \nabla_\Theta H$
we have
$$
L_F\Theta = \gamma(F)\Theta + \gamma\wedge dH.\tag2.3
$$
\endproclaim
\demo{Proof}
We will use the Cartan formula (\cite{A-M-R})
$$
L_F = i_Fd + di_F,
$$
where $i_F$ is the interior and $d$ the exterior derivative.
(For a differential $m$-form $\zeta$ the interior derivative
$i_F\zeta$ is the differential $(m-1)$-form obtained by substituting
$F$ as the first vector argument of $\zeta$.)
We have $i_F\Theta = -dH$ and we get immediately
$$
L_F\Theta = i_Fd\Theta -d^2H = i_F (\gamma\wedge \Theta) =
\gamma(F) \Theta +\gamma \wedge dH.
$$
\enddemo\qed
Let us restrict the flow $\Phi^t$ to one smooth level set of the
Hamiltonian, $M^c = \{z \in M| H(z) = c\}$. In particular we assume
that on $M^c$ the differential $dH$ and the vector field $F$ do not
vanish.
For two vectors $\xi, \eta$ from the tangent space $T_zM^c$,
we introduce
$$
w(t) = \Theta(D_z\Phi^t\xi,D_z\Phi^t\eta).
$$
By \thetag{2.3} we get
$$
\frac{d}{dt} w(t) = \gamma(F(\Phi^tz)) w(t),
$$
since $dH$ vanishes on the tangent space $T_zM^c$. We conclude that
$$
\Theta(D_z\Phi^t\xi,D_z\Phi^t\eta) = \beta(t)\Theta(\xi, \eta),\tag2.4
$$
for every $\xi, \eta,$ from $T_zM^c$ and
$$
\beta(t) = exp\left(\int_0^t\gamma(F(\Phi^uz))du\right) .
\tag 2.5
$$
{\bf Remark 2.1}
Let us note that under a non-degenerate time change a conformally
Hamiltonian vector field is still conformally
Hamiltonian with the same Hamiltonian function but with respect
to a modified conformally symplectic form.
More precisely if $F = \nabla_{\Theta} H$ is a Hamiltonian
vector field, then if the new time $\tau$ is related to the original
time $t$ by $$
\frac{d\tau}{dt} = f,
$$
for some function $f$ of the phase point, we get that
the vector field $\frac 1f F$ is conformally symplectic
with respect to the form $\widetilde\Theta = f\Theta$. Indeed
$$
d\widetilde\Theta = (d\ln f+\gamma)\wedge\widetilde\Theta \ \ \
\text{and} \ \ \ \widetilde\Theta(\cdot,\, \frac 1f F) =
dH.
$$
Alternatively we can keep
the same conformally symplectic form and modify the Hamiltonian
separately on each level set. Indeed we have
$$
\Theta(\cdot,\, \frac 1f F) = \frac 1f dH = d \left(\frac 1f(H-c)\right),
$$
where the last equality is valid only on the level set $\{H = c\}$.
Finally, let us consider the symplectic form $e^{-U}\Theta$.
On the level set $\{H = c\}$ we have
$$
d(e^{-U}(H-c)) = e^{-U}dH.
$$
It follows that on this level set
$$
e^{-U}\Theta(\cdot,\, F) = d\left(e^{-U}(H-c)\right),
$$
and, as a result, the vector field $F$ coincides
locally with the Hamiltonian vector field given by the Hamiltonian
$e^{-U}(H-c)$ (with respect to the symplectic form $e^{-U}\Theta$).
This observation provides an alternate way to derive \thetag{2.4} by using the
preservation of the symplectic form $e^{-U}\Theta$ by any Hamiltonian
flow (with respect to this symplectic form).
\bigskip
For a fixed level set $M^c$ we introduce the quotient of the tangent bundle
$TM^c$ of $M^c$ by the vector field $F = \nabla_\Theta H$,
i.e., by the one dimensional
subspace spanned by $F$. Let us denote the quotient bundle by $\widehat TM^c$.
The form $\Theta$ factors naturally from $TM^c$ to $\widehat TM^c$, in view of
\thetag{2.2}. The factor form defines in each of the quotient tangent spaces
$\widehat T_zM^c, z \in M^c$, a linear symplectic form.
The derivative of the flow preserves the vector field
$F$, i.e.,
$$
D_z\Phi^t(F(z)) = F(\Phi^tz).
$$
As a result the derivative can be also
factored on the quotient bundle and we call it the transversal
derivative cocyle and denote
it by
$$A^t(z): \widehat T_zM^c \to \widehat T_{\Phi^tz}M^c. $$
It follows immediately from \thetag{2.4}
that the transversal derivative cocycle
is conformally symplectic with respect to $\Theta$ (or more precisely the
linear symplectic form it defines in the quotient tangent spaces).
We fix an invariant probability measure $\mu_c$ on $M^c$ and assume that
$$
\int_{M^c}\|D_z\Phi^t\|d\mu_c(z) < +\infty.
$$
Under this assumption the derivative cocycle
has well defined Lyapunov exponents, cf. \cite{O},\cite{R}.
Then the transversal derivative cocycle
has also well defined Lyapunov exponents which coincide with the former
except that one zero Lyapunov exponent is skipped.
We can immediately apply Theorem 1.4 to the transversal derivative cocycle
and we get the following.
\proclaim{Theorem 2.2}
For a Hamiltonian flow $\Phi^t$, defined by the vector
field $F= \nabla_{\Theta}H$, restricted to one level set $M^c$
we have the following
symmetry of the Lyapunov spectrum of the transversal derivative
cocyle with respect to an invariant ergodic probability measure $\mu$.
Let
$$\{0\} \subset\Bbb V_0(z) \subset \Bbb V_1(z) \subset \dots \subset
\Bbb V_{s-1}(z) \subset \Bbb V_s = \widehat
T_zM^c$$
be the flag of subspaces at $z$ associated with the Lyapunov spectrum
$\lambda_1 < \lambda_2 < \dots < \lambda_{s-1} < \lambda_s$
of the transversal derivative cocycle $A^t(z), \ z \in M^c$. Then
the multiplicities of $\lambda_k$ and $\lambda_{s-k+1}$ are equal
and
$$
\lambda_k + \lambda_{s-k+1} = a , \ \ \ \text{for}\ \ \ k = 1,2, \dots, s,
$$
where $a = \int_{M^c}\gamma(F(z))d\mu_c(z)$.
Moreover the subspace $\Bbb V_{s-k}$ is the skew-orthogonal
complement of the subspace $\Bbb V_k$.
\endproclaim
\qed
Note that the invariant measure $\mu_c$ can be supported on a single periodic
orbit, so that Theorem 2.2 applies as well to the real parts of the
Floquet exponents.
To apply Theorem 1.4 it is enough to have linear symplectic forms in each
of the quotient tangent spaces (to the level set), not necessarily coming
from a conformally symplectic structure on the phase space. But then one
needs to check directly
how the transversal derivative cocycle acts on these forms, because we do
not have the advantage of Theorem 2.1.
This is essentially the line of argument in \cite{D-M 1} and \cite{D-M 3}.
\vskip.7cm
\subhead \S 3. Conformally symplectic flows with collisions\endsubhead
\vskip.4cm
Let $M$ be a smooth manifold with piecewise
smooth boundary $\partial M$. We assume that the manifold
$M$ is equipped with a conformally symplectic structure
$\Theta$, as defined in Section 2. Given a smooth function $H$ on $M$
with {\it non vanishing} differential we obtain the non vanishing
conformally Hamiltonian vector field $F = \nabla_{\Theta}H$ on $M$.
The vector field $F$ is tangent to the level sets of the Hamiltonian
$M^c = \{z\in M| H(z) = c\}$.
We distinguish in the boundary $\partial M$
the regular part, $\partial M_r$, consisting of
the points which do not belong to more than one smooth piece of the
boundary and where the vector field $F$ is transversal to the boundary.
The regular part of the boundary is further split into ``outgoing'' part,
$\partial M_-$, where the vector field $F$ points outside
the manifold $M$
and the ``incoming'' part, $\partial M_+$, where the vector field is
directed inside the manifold.
Suppose that additionally we have a piecewise smooth mapping
$\Gamma: \partial M_- \to \partial M_+$, called the collision map.
We assume that the mapping $\Gamma$ preserves the Hamiltonian,
$H\circ\Gamma = H$, and so it can be restricted to each level set of the
Hamiltonian.
We assume that all
the integral curves of the vector field $F$ that end (or begin)
in the singular part of the boundary lie in a codimension 1 submanifold of
$M$.
We can now define a flow $\Psi^t: M \to M$, called a flow with collisions,
which is a concatenation of the continuous time dynamics $\Phi^t$ given
by the vector field $F$, and the collision map $\Gamma$.
More precisely a trajectory of the flow with collisions,
$\Psi^t(x), \ x \in M$,
coincides with the trajectory of the flow $\Phi^t$ until it gets to the
boundary of $M$ at time $t_c(x)$, the collision time. If the point on the
boundary lies in the singular part then the flow is not defined for times
$t > t_c(x)$ (the trajectory ``dies'' there). Otherwise the trajectory is
continued at the point $\Gamma(\Psi^{t_c}x)$ until the next collision time,
i.e., for $0 \leq t \leq t_c\left(\Gamma(\Psi^{t_c(x)}x)\right)$
$$
\Psi^{t_c+t}x = \Phi^t\Gamma\Psi^{t_c}x.
$$
We define a flow with collisions to be conformally symplectic, if
for the collision map $\Gamma$ restricted to any level set $M^c$
of the Hamiltonian we have
$$\Gamma^*\Theta = \beta\Theta, \tag3.1$$
for some non vanishing function $\beta$ defined on the boundary.
More explicitly we assume that for every vectors $\xi$ and $\eta$
from the tangent space $T_z\partial M^c$
to the boundary of the level set $M^c$ we have
$$
\Theta(D_z\Gamma \xi, D_z\Gamma \eta) = \beta \Theta(\xi,\eta).
$$
We restrict the flow with collisions to one level set $M^c$ of the
Hamiltonian and we denote the resulting flow by $\Psi_c^t$.
This flow is very likely to be badly discontinuous
but we can expect that for a fixed time $t$ the mapping $\Psi_c^t$ is piecewise
smooth, so that the derivative $D\Psi_c^t$ is well
defined except for a finite union of codimension one submanifolds of $M^c$.
We will consider only such cases.
We choose an invariant measure in our system which satisfies the
condition that all the trajectories that begin (or end) in the singular part
of the boundary have measure zero. Usually there are many natural
invariant measures satisfying this property. For instance we get one by
taking a Lebesgue
measure $\nu$ in $M^c$ and averaging it over increasing time intervals
($\frac 1T\int_0^T\Psi_{c*}^t\nu dt$ as $T \to +\infty$).
Let us denote the chosen
invariant measure by $\mu_c$. This measure $\mu_c$ defines the
measure $\mu_{cb}$ on the boundary $\partial M^c$,
which is an invariant measure for the section of the flow (Poincar\'e
map of the flow). With respect to the measure $\mu_c$
the flow $\Psi_c^t$ is a measurable flow in the sense of the Ergodic
Theory and we obtain a measurable derivative cocycle $D\Psi_c^t : T_xM^c \to
T_{\Psi_c^tx}M^c$. We can
define Lyapunov exponents of the flow $\Psi_c^t$ with respect to the measure
$\mu_c$, if
we assume that
$$
\int_{M^c} \log_+||D_x\Psi_c^t||d\mu_c(x) < +\infty \ \ \ \text{and} \ \ \
\int_{\partial M^c_-} \log_+||D_y\Gamma||d\mu_{cb}(y) < +\infty
$$
(cf.\cite{O},\cite{R}).
The derivative of the flow with collisions can be also naturally
factored onto the quotient of the tangent bundle
$TM^c$ of $M^c$ by the vector field $F$, which we
denote by $\widehat TM^c$.
Note that for a point $z\in \partial M^c$ the tangent to the boundary at
$z$ can be naturally identified with the quotient space.
We will again denote the factor of the derivative cocycle by
$$A^t(x): \widehat T_xM^c \to \widehat T_{\Psi_c^tx}M^c. $$
We will call it the transversal derivative cocycle.
If the derivative cocycle
has well defined Lyapunov exponents then the transversal
derivative cocycle
has also well defined Lyapunov exponents which coincide with
the former ones except that one zero Lyapunov exponent is skipped.
For a conformally symplectic flow with collisions the factor
$A^t(x)$ of
the derivative cocycle on one level set changes the form $\Theta$
by a scalar, \thetag{2.3} and \thetag{3.1}, so that we can immediately apply
Theorem 1.4 and we get
\proclaim{Theorem 3.1}
For a conformally symplectic flow with collisions $\Psi_c^t$
we have the following
symmetry of the Lyapunov exponents for a given ergodic invariant probability
measure $\mu_c$. Let
$\{0\} \subset \Bbb V_0 \subset \Bbb V_1 \subset \dots \Bbb V_{s-1}
\subset \Bbb V_s = \widehat
T_xM^c$ be the flag of subspaces at $x$ associated with the Lyapunov spectrum
$\lambda_1(x) < \lambda_2(x) < \dots < \lambda_{s-1}(x) < \lambda_s(x)$
of the transversal derivative cocycle $A^t(x), \ x \in M^c$. Then the
multiplicities of $\lambda_k$ and $\lambda_{s-k+1}$ are equal and
$$
\lambda_k + \lambda_{s-k+1} = a + b, \ \ \ \text{for}\ \ \ k = 1,2, \dots, s,
$$
where $a = \int_{M^c}\gamma(F)d\mu_c$ and $b =
\frac{1}{\tau}\int_{\partial M^c_-} \log|\beta(y)|d\mu_{cb}(y)$. $\tau =
\int_{\partial M_-^c}t_c(y)d\mu_{cb}(y) $ is the average collision time
on the section of the flow.
Moreover the subspace $\Bbb V_{s-k}$ is the skew-orthogonal
complement to $\Bbb V_k$.
\endproclaim
\vskip.7cm
\subhead \S 4. Applications \endsubhead
\vskip.4cm
{\bf A. Gaussian isokinetic dynamics.}
The equations of the system are (cf. \cite{D-M 1})
$$
\aligned
\dot q &= p,\\
\dot p &= E - \alpha p, \ \ \text{where} \ \ \alpha =
\frac{\langle E,p\rangle}{\langle p,p\rangle}.
\endaligned
\tag{4.1}
$$
In these equations $q$ describes a point in the multidimensional
configuration space $\Bbb R^N$, $p$ is the momentum (velocity)
also in $\Bbb R^N$ and $\langle \cdot, \cdot \rangle$ is the arithmetic
scalar product in $\Bbb R^N$. The field of force $E = E(q)$
is assumed to be irrotational, i.e., it has locally
a potential function $U = U(q)$, $E = -\frac{\partial U}{\partial q}$.
Let us denote by $\kappa = \sum pdq$ the $1$-form which defines
the standard symplectic structure $\omega = d\kappa= \sum dp\wedge dq$.
We introduce the following $2$-form
$$
\Theta = \omega +
\frac{\langle E, dq\rangle}{\langle p,p\rangle}\wedge\kappa.
$$
We choose the Hamiltonian to be $H = \frac 12 \langle p,p\rangle$
and we denote the vector field defined by \thetag{4.1} by $F$.
We have
$$
\Theta(\cdot, F) = dH,
\tag{4.2}
$$
but the form $\Theta$ does not give us a conformally symplectic
structure because the relation \thetag{2.1} fails.
To correct this setback we fix one level set of the Hamiltonian
$M^c =\{H=\frac 12 \langle p,p\rangle = c\}$ and define another $2$-form
$$
\Theta_c = \omega +
\frac{\langle E, dq\rangle}{2c}\wedge\kappa.
$$
Now we get a conformally symplectic structure. Indeed
$$d\Theta_c = -\frac{\langle E, dq\rangle}{2c}\wedge\Theta_c, \ \
\text{and locally} \ \ \frac{\langle E, dq\rangle}{2c} =
d\left(\frac{-U}{2c}\right).
$$
Moreover on $M^c$ we still have $\Theta_c(\cdot, F) = dH$
so that the restriction of \thetag{4.1} to $M^c$ coincides with
a conformally Hamiltonian system with respect to the $2$-form $\Theta_c$
and with the Hamiltonian $H = \frac 12\langle p,p\rangle$.
We can immediately apply Theorem 2.1 and we obtain that
for any invariant ergodic probability measure $\mu_c$ on $M^c$
the Lyapunov exponents $ \lambda_1 < \dots < \lambda_s$
satisfy
$$
\lambda_k + \lambda_{s-k+1} = -\frac1{2c}\int_{M^c}\langle E, p\rangle d\mu_c
= \int_{M^c}\alpha d\mu_c.
$$
Note that if the vector field of force has a global potential,
$ E = - \frac{\partial U}{\partial q}$, then by the Birkhoff Ergodic Theorem
the integral
$-\frac1{2c}\int_{M^c}\langle E, p\rangle d\mu_c =
\frac1{2c}\int_{M^c}dU(F) d\mu_c$
is equal to the time average of $\frac{dU}{dt}$ and so it must vanish.
Another way to see it is that
$e^{-U}\Theta_c$ defines a global symplectic structure and on $M_c$
our flow is Hamiltonian with respect to this symplectic structure and
a modified Hamiltonian
$$\widetilde H = e^{-U}(\frac 12\langle p, p \rangle -c).$$
Indeed as discussed in the Remark 2.1 on $M^c$ we have
$$e^{-U}\Theta_c( \cdot, F) =d\widetilde H.$$
For a Hamiltonian flow the symmetric Lyapunov exponents must add up to zero.
\medskip
{\bf B. Gaussian isokinetic dynamics on a Riemannian manifold.}
For a given Riemannian manifold $N$ with the metric tensor
$ds^2 = \sum g_{ij}d q_i d q_j$ we can naturally
generalize the form $\Theta$ to the cotangent bundle $T^*N$.
Indeed the $1$-form $\kappa = \sum pdq$ is independent of the
coordinate system, cf. \cite{A},
and for a given closed $1$-form $\gamma$ we put
$$
\Theta_c = d\kappa - \frac 1c\gamma\wedge\kappa.
$$
We get $d\Theta_c = -\frac 1c\gamma\wedge\Theta_c$.
Taking $\gamma = dU$ for some potential function (single or multi-valued)
and the Hamiltonian $H = \frac 12\sum g^{ij} p_ip_j$
we obtain the Gaussian isokinetic dynamics, \cite{Ch}, on the level set
$H = c$ by the relation
\thetag{4.2}.
We can repeat the discussion in part A and
we conclude again that the Lyapunov exponents
must be symmetric and they add up
to zero, if the potential $U$ is single-valued.
\medskip
{\bf C. The Gaussian isokinetic dynamics with collisions.}
Let us consider $n$ spherical particles in a finite
box $B$ contained in $\Bbb R^d$ or the torus $\Bbb T^d$.
We assume that the particles interact with each other
by the potential $V(q_1,q_2,\dots,q_n)$ ($q_k \in B, \ k = 1, \dots, n$
denote the positions of the particles) and that they are subjected
to the external fields given by the potentials $V_k(q_k), \ k = 1, \dots,
n$.
Further we assume that the particles have the radii $r_1, \dots,r_n$, the
masses $m_1, \dots, m_n$, and that they collide elastically with each other
and the sides of the box, which can be flat or curved.
The last element in the description of the system is the Gaussian isokinetic
thermostat. As described in part A and B the Gaussian isokinetic
thermostat gives rise to a conformally Hamiltonian flow
with the Hamiltonian $H = \sum_{k=1}^n \frac{p_k^2}{2m_k}$
and an appropriate conformally symplectic structure. We will check
below that the collisions in this system preserve the form
$\Theta_c$ giving rise to a conformally symplectic flow with
collisions. Theorem 3.1 can be thus applied to our system
giving us the symmetry of the Lyapunov spectrum.
We introduce the canonical change of variables which bring
the kinetic energy into the standard form,
$$
\aligned
x_k &=\sqrt{m_k}q_k \\
v_k &=\frac{p_k}{\sqrt{m_k}}.
\endaligned
$$
The advantage of these coordinates is that although the collision
manifolds in the configuration space become less natural,
the collisions between particles
(and the walls of the box) are given by the billiard rule in the configuration
space.
The equations of motions in the $(x,v)$ coordinates are
$$
\aligned
\dot x &= v\\
\dot v &= - \frac {\partial U}{\partial x} + \alpha v,
\endaligned
\tag{4.4}
$$
where $U = U(x) = V+
\sum_{k=1}^n V_k$ is the total potential of the system
and $\alpha = -\frac{dU(v)}{\langle v,v\rangle}$.
We introduce the differential $2$-form
$$
\Theta_c = \sum dv\wedge dx - \frac1{2c}dU\wedge\langle v,dx\rangle
$$
As in part A we conclude that the form satisfies
\thetag{2.1} and the system \thetag{4.4} restricted to
$M^c$ coincides with the conformally Hamiltonian system
defined by this form and the Hamiltonian $H = \frac12 \langle v,v\rangle$.
\proclaim{Proposition 4.1}
The collision maps preserve the form $\Theta_c$.
\endproclaim
\demo{Proof}
A collision manifold is locally
given by an equation of the form $g(x) = 0$, where
$g$ is some differentiable $\Bbb R^{nd}$ valued map.
Note that the general form of the collision map is the same for
collisions of particles and the collisions with the sides of the
box.
Let $n(x)$, for $x \in \{x\in \Bbb R^{nd}| g(x) = 0\}$, denote the
unit normal vector to the collision manifold in the configuration space.
The collision map is defined as
$$
\aligned
x^+ &= x^-,\\
v^+ &= v^- - 2\langle v^-,n(x^-)\rangle n(x^-).
\endaligned
\tag{4.4}
$$
where the index $^+$ corresponds to the values of $x$ and $v$ after the
collision and the index $^-$ to the values before the collision.
As a result of these formulas we get immediately that
$$
\delta x^+ =\delta x^-.
\tag{4.5}
$$
It is well known, \cite{W1},\cite{W2}, that in an elastic collision
the symplectic form $\omega$ is preserved. It remains to show the
preservation of the second term in $\Theta_c$.
It follows immediately from
\thetag{4.4} and \thetag{4.5}, because
$$
\langle v^+, \delta x^+\rangle = \langle v^-, \delta x^-\rangle -
2\langle v^-,n(x^-)\rangle \langle n(x^-), \delta x^-\rangle,
$$
and the last term is zero since we only take the variations
$(\delta x^-, \delta v^-)$
tangent to the collision manifold, i.e., $\delta x^-$ is orthogonal to $n(x)$.
The Proposition is proven.
\enddemo\qed
It follows from Proposition 4.1 that also the form $e^{-U}\Theta_c$
is preserved under collisions. Hence, as remarked in parts A and B,
if the potential $U$ is singlevalued then the system restricted to one
energy level coincides with a globally Hamiltonian system
(with collisions) with respect
to the symplectic form $e^{-U}\Theta_c$ with the Hamiltonian function
equal to $\widetilde H = e^{-U}(\frac 12\langle p, p \rangle -c)$.
We conclude that the occurrence of dissipation in such systems
is related to the topology of the configuration space (the
multivaluedness of the potential $U$).
\medskip
{\bf D. Nos\'e-Hoover dynamics}
The Nos\'e Hamiltonian is, cf. \cite{D-M 3}
$$
H(q,s;\pi,p_s) = \sum_{i=1}^N\frac{\pi_i^2}{2m_is^2}+\varphi(q)+
\frac{p_s^2}{2} + C\ln s,
$$
with a non-physical time denoted by $\lambda$ and some constant $C$.
The symplectic form
is $\omega = \sum d\pi\wedge dq + dp_s\wedge ds$. Changing the variables
as $\pi = s p$ and $\sigma = \ln s$ the Hamiltonian becomes
$$
H(q,s;p,p_s) = \sum_{i=1}^N\frac{p_i^2}{2m_i}+\varphi(q)+
\frac{p_s^2}{2} + C\sigma,\tag{4.6}
$$
and the symplectic form is $\omega = e^\sigma\left(\sum_i dp_i \wedge dq_i +
dp_s \wedge d\sigma + d\sigma\wedge(\sum_i p_idq_i)\right).
$
Note that now in the Hamiltonian the thermostat $(\sigma,p_s)$
is decoupled from the system but the coupling is shifted to the symplectic
form.
We make finally the time change $\frac{d\lambda}{dt} = e^\sigma$.
We choose not to change the Hamiltonian but rather to modify the
$2$-form,
$$
e^{-\sigma}\omega (\cdot,\, e^\sigma\nabla_\omega H) = dH.
$$
We end up with the Hamiltonian \thetag{4.6} and the conformally
symplectic structure
$$
\Theta = e^{-\sigma}\omega = \sum_i dp_i \wedge dq_i +
dp_s \wedge d\sigma + d\sigma\wedge(\sum p_idq_i + p_sd\sigma).
$$
We have $d\Theta = d\sigma \wedge \Theta$.
Note the similarity of $\Theta$ with the form used in the discussion
of the isokinetic dynamics above.
This form and the Hamiltonian give us the Hoover equations
$$
\aligned
\dot q_i &= \frac{p_i}{m_i},\\
\dot p_i &= -\frac{\partial \varphi}{\partial q_i} - p_sp_i,\\
\dot \sigma &= p_s,\\
\dot p_s &= \sum_i\frac{p_i^2}{m_i} - C.
\endaligned
$$
On any level set we can drop the equation for $\sigma$
since $\sigma$ can be trivially obtained from other variables
using the constancy of the Hamiltonian.
By Theorem 2.1 we have the symmetry of the Lyapunov spectrum for
this system reduced to one level of the Hamiltonian.
Moreover the Lyapunov exponents add up to the time average
$\dot\sigma$. This average must be zero, unless $p_s$
grows linearly, which is unlikely.
Note that the Nos\'e--Hoover system is open in the sense that
arbitrarily large fluctuations of $p_s$ cannot be ruled
out.
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\enddocument