\input amstex
\documentstyle {amsppt}
\magnification \magstep1
\NoBlackBoxes
\pageno=1
\catcode`\@=11
\topmatter
\title
Magnetic flows and Gaussian thermostats \\
on manifolds of negative curvature
\endtitle
\rightheadtext{Magnetic flows and Gaussian thermostats}
\author
Maciej P. Wojtkowski
\endauthor
\affil University of Arizona
\endaffil
\address
Maciej P. Wojtkowski,
Department of Mathematics,
University of Arizona,
Tucson, Arizona 85 721
USA
\endaddress
\email
maciejw@math.arizona.edu
\endemail
\date
Jan 9, 1999
\enddate
\abstract
We consider a class of flows which
include both magnetic flows and Gaussian thermostats of
external fields.
We give sufficient conditions for such flows on manifolds of
negative sectional curvature to be Anosov.
\endabstract
\thanks
{\bf I thank Francois Ledrappier, Feliks Przytycki,
Marek Rychlik and Lai-Sang Young who generously
shared with me their knowledge of SRB measures.}
\endthanks
\endtopmatter
\document
\vskip.4cm
\subhead \S 0.Introduction \endsubhead
\vskip.3cm
The geodesic flow on a Riemannian manifold describes inertial
motion of a point particle confined to the manifold.
If the manifold has negative sectional curvature
we obtain an Anosov flow, a prime example of a dynamical
system with good statistical properties.
In the present paper we study flows generated by special
forces, magnetic flows and Gaussian thermostats.
Magnetic flows in this context were discussed already
30 years ago by Anosov and Sinai,
\cite{A-S}.
They were studied recently by Gouda, \cite{Go},
Grognet, \cite{Gr}, and M. and P. Paternain, \cite{P-P}.
Just like the geodesic flow the magnetic flow lives
naturally on the unit tangent bundle.
A different class of flows on the unit tangent bundle
was introduced recently in physics literature,
the Gaussian thermostat of an external field, \cite{H}.
We show (Section 1) that both classes of flows can be represented as
special cases of a general construction.
We define a generalized magnetic flow on the tangent bundle $TM$
(or the cotangent bundle $T^*M$)
by requiring that its velocity vector field $F$ satisfies
$$
i_F(\omega - \gamma) = -dH, \tag{0.1}
$$
where $\omega$ is the standard symplectic form, $H$ is a hamiltonian
and the 2-form $\gamma$ represents the generalized magnetic field.
This relation is classical for magnetic flows, where $\gamma$
is a closed 2-form on $M$.
In general, although it resembles Hamiltonian formalism,
it does not have the usual consequences. However, we show that for
any 2-form $\gamma$ on $M$ the resulting flow is volume preserving.
It seems that this fact was overlooked in the papers on
magnetic flows.
For the Gaussian thermostats the relation
\thetag{0.1} appeared in
\cite{W-L} and it was the basis for representing the flow as
a conformally symplectic system. In this paper we are
making the point that \thetag{0.1} holds because
the Gaussian thermostat can be viewed as a velocity dependent
magnetic field. We discuss these results in Section 2 and
compare them with the work of Dettmann and Morriss, \cite{D-M}.
In Section 3 we consider general flows with potential fields,
magnetic fields and Gaussian thermostats.
We show how such systems look like in Darboux coordinates
and we calculate the divergence of the resulting vector field.
It generalizes the coordinate free calculations of Sections 1 and 2.
We discuss the average rate of volume contraction which
was shown by Ruelle, \cite{R}, to represent the
average rate of entropy production in the
system. Its positivity (=dissipativity of the system)
is usually hard to establish in concrete
examples.
In the case of surfaces of constant negative curvature
Gaussian thermostats were studied by Bonetto, Gentile and
Mastropietro, \cite{B-G-M}, and they addressed the problem of
dissipativity. We show (Proposition 3.1) that if the flow is Anosov
and the external field has a local potential but no global potential
then the system is dissipative. We can allow magnetic fields
but no ``internal'' potential interactions. It would be interesting
to understand also this ``isoenergetic'' case.
The proof of Proposition 3.1 is deceptively simple because
it relies heavily on the deep results of Ledrappier and Young,
\cite{L-Y}.
In the rest of the paper we address the question: when is the general
flow on a manifold with negative sectional curvature Anosov.
Gouda, \cite{Go}, and Grognet,\cite{Gr}, obtained sufficient
conditions for magnetic flows. We obtain similar conditions
for the general flows, Theorem 4.1 and its corollaries.
The proof of Theorem 4.1(Section 6) is based on a criterion for Anosov
property, formulated and proven in Section 5, involving indefinte
quadratic forms. This criterion generalizes to Anosov flows
a result of Lewowicz, \cite{L}, on Anosov
diffeomorphisms (see also \cite{K-B} and \cite{W}).
Important technical advantage of this method is that we can do
calculations in the ambient space, although they relate to
a level set of the hamiltonian.
\vskip.4cm
\subhead \S 1. Potential flows, magnetic flows and Gaussian
thermostats
\endsubhead
\vskip.3cm
Let us consider a compact $n$-dimensional Riemannian manifold $M$ and its
tangent bundle $\pi : TM \to M$. For a smooth function
$W : M \to \Bbb R$ we introduce the potential flow on $TM$
defined by the equations
$$
\frac{dq}{dt} = v, \frac{Dv}{dt} = -\nabla W, \tag{1.1}
$$
where $q\in M$ and $\frac{D}{dt}$ denotes the covariant derivative.
Let $\omega$ be the canonical symplectic form transported from $T^*M$
to $TM$ by the natural isomorphism. The vector field $F$ on $TM$
defined by \thetag{1.1} is a Hamiltonian vector field with the
hamiltonian $H = \frac 12 v^2 +W$ and we have $i_F \omega = -dH$.
We are going to modify \thetag{1.1} by adding ``nonpotential
forces''. A 2-form $\gamma$ on $TM$ will
be called a generalized magnetic field (gmf), if there is an antisymmetric
operator $Y$ on the tangent spaces on $M$ (depending in general on the
point in $TM$) such that
$$
\langle Y d\pi\cdot, d\pi \cdot \rangle = \gamma(\cdot, \cdot),\tag{1.2}
$$
We consider the generalized magnetic flow on $TM$ defined by
$$
\frac{dq}{dt} = v, \frac{Dv}{dt} = -\nabla W + Yv. \tag{1.3}
$$
Now the vector field $F$ on $TM$ defined by \thetag{1.3}
satisfies
$$
i_F\Omega = -dH, \ \ \ \ \text{where} \ \ \ \ \Omega = \omega -
\gamma. \tag{1.4}
$$
To prove this let us introduce the identification of a tangent
space of $TM$ at $(q,v)$ with $T_qM \oplus T_qM$. For a tangent vector
defined by a parametrized curve $(q(u),v(u)),|u| < \epsilon,$
we use $(\xi,\eta) \in T_{q(0)}M \oplus T_{q(0)}M$
$$
\xi = \frac{dq}{du}, \eta = \frac{Dv}{du},
$$
as the linear coordinates. In these coordinates $F = (v,-\nabla W
+Yv)$ and
the symplectic form $\omega ((\xi_1,\eta_1),(\xi_2,\eta_2))=
\langle\xi_2,\eta_1\rangle - \langle \xi_1,\eta_2\rangle$.
We have further
$$
\aligned
\omega(F,(\xi,\eta)) =& -\langle v,\eta \rangle +
\langle -\nabla W + Yv,\xi \rangle,\\
\gamma(F,(\xi,\eta)) = &\langle Yv,\xi \rangle \\
dH((\xi,\eta)) = &\langle v,\eta \rangle +\langle \nabla W,\xi \rangle,
\endaligned
$$
which proves \thetag{1.4}.
$\Omega$ is always nondegenerate because, roughly speaking, $\gamma$
depends on $dq$ but not on $dv$, so it cannot destroy
the nondegeneracy of $\omega$. However $\Omega$ is closed only if $\gamma$
is closed. When $\Omega$ is not closed \thetag{1.4} does not amount to
much because it does not force the preservation of $\Omega$
by the flow, which is the cornerstone of Hamiltonian formalism.
Special cases of this construction include the magnetic flow
studied by Gouda, \cite{Go}, Grognet, \cite{Gr},
and M. and G. Paternain, \cite{P-P}. In this case the form $\gamma$ is the
pullback under the projection $\pi$ of a 2-form $\widehat \gamma$ on $M$.
In view of \thetag{1.4}, if $\widehat\gamma$ is closed
(which is a natural assumption for a magnetic field), the magnetic
flow is a Hamiltonian flow with the hamiltonian $H$
with respect to a modified symplectic structure.
If $\widehat\gamma$ is not
closed the resulting flow is not, in general, symplectic with respect to any
symplectic structure.
It turns out that
without any assumptions on $\widehat\gamma$ the magnetic flow
preserves the Lebegue measure on $TM$ and hence for
such flows we get smooth invariant
measures on the level sets of $H$ (energy shells).
\proclaim{Proposition 1.5}
If $\gamma$ is a pullback to $TM$ of a 2-form $\widehat\gamma$ on $M$
and a vector field $F$ on $TM$ satisfies
$$i_F\omega - i_F\gamma = -dH $$ for a smooth function $H$ on $TM$
then the flow defined by the vector field $F$ preserves the standard
Lebegue measure on $TM$ (the symplectic volume element).
\endproclaim
In the proof we will use the following observation
\proclaim{Lemma 1.6}
If a k-form $\zeta$ on $T^*M$
is the pullback of a k-form $\widehat\zeta$ defined on $M$
then
$$
\zeta \wedge \omega^{\wedge l} = 0 \ \ \ \ \text{for} \ \ \ \ l \geq n-k+1.
$$
\endproclaim
As a consequence we obtain that
$
(\omega - \gamma)^{\wedge n} = \omega^{\wedge n}.
$
We proceed with the calculation of the Lie derivative
$$
\aligned
\Cal L_F \omega^{\wedge n} &=
\Cal L_F (\omega - \gamma)^{\wedge n} = n d\left(i_F(\omega - \gamma) \wedge
(\omega - \gamma)^{\wedge (n-1)}\right)\\
& = - n d\left(dH \wedge
(\omega - \gamma)^{\wedge (n-1)}\right)
= - n(n-1) dH \wedge d\gamma\wedge (\omega - \gamma)^{\wedge (n-2)}.
\endaligned
$$
But $d\gamma\wedge (\omega - \gamma)^{\wedge (n-2)} = 0$ by Lemma 1.6.
Proposition 1.5 is proven. We will obtain a different, coordinate proof
in Section 3.
Another interesting example of a generalized magnetic field
is an external field with the Gaussian thermostat which we
are going to describe in detail. Let $E$ be a vector field on $M$,
for example the gradient vector field $E = -\nabla U$ for some
potential function $U$. We consider the flow in $TM$ given by
$$
\frac{dq}{dt} = v, \frac{Dv}{dt} = -\nabla W +E.
$$
The resulting flow does not preserve in general the energy function
$H = \frac 12 v^2 + W$.
We impose the preservation of $H$ via the Gauss least constraint
principle, \cite{H}. The resulting equations are
$$
\frac{dq}{dt} = v, \frac{Dv}{dt} = -\nabla W +
E- \frac{\langle E,v\rangle}{v^2}v. \tag{1.7}
$$
These systems appear in physics literature under the name of
``isoenergetic dynamics''.
They fit our formalism of generalized magnetic fields.
We introduce the 2-form $\gamma$ by
$$
\alpha((\xi,\eta)) = \langle v, \xi\rangle,\ \ \
\Upsilon((\xi,\eta)) = \langle E,\xi\rangle, \ \ \
\gamma = \frac 1{v^2} \alpha\wedge\Upsilon,
$$
Note that $d\alpha = \omega$ and in the special case of the
potential vector field, $E = -\nabla U$, the form $\Upsilon = -dU$
(strictly speaking $-\pi^*dU$, but we will allow this kind of
identification).
The antisymmetric operator
$$Y(\cdot) =
\frac 1{v^2} \left(\langle v,\cdot\rangle E - \langle E,\cdot\rangle v\right)
$$
satisfies \thetag{1.2} and the corresponding generalized magnetic flow
on $TM$ coincides with \thetag{1.7}.
\vskip.4cm
\subhead \S 2. Isokinetic dynamics and conformally symplectic
structures \endsubhead
\vskip.3cm
Let us look more carefully at the Gaussian thermostat
of an external field $E$ when $W \equiv 0$, i.e., $H = \frac 12 v^2$.
It is called the isokinetic dynamics.
The derivative $d\gamma$ does not vanish (except in trivial cases).
If the form $\Upsilon$ is closed (locally $\Upsilon = -dU$), then
for a fixed level set $v^2 = c$ we can modify
the form $\gamma$ to $\gamma_c = \frac 1{c}\alpha\wedge\Upsilon$,
without changing the gmf on the level set.
Now we have
$$
d\Omega_c = d(\omega - \gamma_c) = {1\over c}dU\wedge \omega =
{1\over c}dU\wedge \Omega_c.\tag{2.1}
$$
The significance of \thetag{2.1} is revealed when we calculate
$d\left(e^{-{U\over c}}\Omega_c\right) = 0$,
which shows that $ e^{-{U\over c}}\Omega_c$ defines a genuine
symplectic structure. In such a situation we say that
the form $\Omega_c$ defines a conformally
symplectic structure. It follows that when the field $E$ possesses
a global potential $U$ the resulting flow on the level set $v^2 = c$
is a Hamiltonian flow with respect to the symplectic structure
$ e^{-{U\over c}}\Omega_c$ and the hamiltonian $e^{-{U\over
c}}(\frac{v^2}2 -\frac c2)$. Indeed on the level set $v^2 = c$ we have
$$
i_F \left(e^{-{U\over c}}\Omega_c\right) = - d\left(e^{-{U\over
c}}\left(\frac{v^2}2 -\frac c2\right)\right).
$$
In particular when the field $E$ possesses a global potential
$U$ the flow has a smooth invariant measure. We will obtain the
density of this invariant measure at the end of this section
and then again, by coordinate calculations, in Section 3.
This geometric setup for the Gaussian thermostat of a locally potential
field $E$ was discovered in \cite{W-L}. It explains the symmetry of
Lyapunov spectrum which was first observed numerically by Evans, Cohen
and Morriss, \cite{E-C-M}, and then proved by Dettmann and Morris,
\cite{D-M}. In their proof they consider tangent subspaces transversal
to the flow, defined by the condition that $\xi$ be orthogonal to $v$.
This tnagent subbundle is not invariant under the flow. (We should not expect
in general any invariant geometrically defined transversal subbundle, the
case of a geodesic flow, a contact flow, is in this respect rather
special.) They consider the projection of the linearized flow onto
this subbundle and observe that the linear equations are
infinitesimally Hamiltonian up to addition of a multiple of identity.
This situation can be understood in our language in the following way.
Let us calculate the Lie derivative of $\Omega_c$ in the direction
of the vector field $F$ of our flow.
Using Cartan formula we obtain
$$
\Cal L_F\Omega_c = {1\over c}(dU(F) \Omega_c + dU\wedge dH).
$$
This is not a nice formula in general because it shows that such
flows do not preserve the conformally symplectic structure. That is
probably why in his geometric study of conformally symplectic dynamics
Vaisman \cite{V} defined Hamiltonian flows differently, and his
Hamiltonian flows do preserve the structure. However, if we restrict
the flow to the level set of the Hamiltonian, $v^2 = c$, (which we
have already done anyway) we obtain that
$$
\Cal L_F\Omega_c = {1\over c}dU(F) \Omega_c.\tag{2.2}
$$
Hence under the action of the flow the
restriction of the form $\Omega_c$ to the level set will be multiplied by
$e^{(U_1-U_0)/c}$ when we move from the value of the potential $U_0$
to the value $U_1$. The restriction of the form $\Omega_c$ factorizes
onto the quotient tangent space (quotient by the span of the velocity
vector field). When we restrict $\Omega_c$ further to the Dettmann -
Morris transversal subspace ($\xi$ orthogonal to $v$)
we see that the form $\alpha$ vanishes and
we obtain the restriction of the canonical symplectic form $\omega$.
Now \thetag{2.2} captures the essence of the Dettmann-Morriss proof.
Similarly to the symplectic case we can consider the k-th exterior
power of the form $\Omega_c$ and again its restriction to the
level set is preserved up to a scalar factor $e^{k(U_1-U_0)/c}$.
The $(n-1)$ exterior power defines a volume element on the
quotient transversal tangent subspace. This transversal volume
element can be further uniquely extended to the full volume
element $\Xi$ on the level set by the condition
$$
i_F \Xi = \Omega_c^{\wedge (n-1)}.
$$
A moment of reflection convinces us that because on the orthogonal
transversal subspace the form $\Omega_c$ coincides with the standard
symplectic form then
$\Xi$ is (up to a constant factor) the standard Lebesgue volume element
on the sphere bundle (the level set of $H$). Moreover this volume
element under the action of the flow will be multiplied by
the same coefficient as the form $\Omega_c^{\wedge (n-1)}$,
i.e., $e^{(n-1)(U_1-U_0)/c}$. It follows from a simple calculation
using Cartan formula and \thetag{2.2}. We will also obtain
a simple coordinate proof in the next section.
We conclude that if the field $E$ possesses a global potential $U$
then the resulting flow preserves the smooth invariant measure which has
the desity $e^{-(n-1)\frac Uc}$ with respect to the standard
volume element on the sphere bundle $v^2 = c$.
It was shown recently by Bonetto, Cohen and Pugh, \cite{B-C-P},
with a careful numerical study that
in general if $W \neq 0$, i.e., in the case
of isoenergetic dynamics, the Floquet exponents do not obey the
shifted symplectic symmetry.
Thus there is no conformally symplectic
structure preserved by the flow .
\vskip.4cm
\subhead \S 3. The general flow and entropy production\endsubhead
\vskip.3cm
Let us now combine a potential field,
a magnetic field and an external field with
the Gaussian thermostat, i.e., we consider equations
of the form
$$
\frac{dq}{dt} = v, \frac{Dv}{dt} = -\nabla W + Bv + E- \frac{\langle
E,v\rangle}{v^2}v. \tag{3.1}
$$
Now the vector field $F$ defined by \thetag{3.1} is the gmf
with respect to $\gamma = \beta + \frac 1{v^2}\alpha\wedge\Upsilon $,
where $B$ and $\beta$ describe the magnetic field.
In this case, even if $W \equiv 0$ and $\beta$ is closed,
we do not have in general a conformally symplectic
structure. We will calculate the divergence of the vector field to be
$-(n-1)\frac {1}{v^2}\langle E,v\rangle $,
equal to
$\frac {(n-1)}{v^2}\frac{dU}{dt}$ in the potential case.
It follows that in the isokinetic case $W \equiv 0$,
if the external field has a global
potential $U$ the flow
preserves the smooth invariant measure with the density
$e^{-\frac{(n-1)U}{v^2}}$
with respect to the standard volume element.
It seems that in the isoenergetic case $W \neq
0$, even if the external field has a global potential
the flow may have no absolutely continuous invariant measure,
because now $v^2$ is not constant. It would be interesting
to have an explicit example.
We will do the calculations in standard symplectic coordinates
in $T^*M$. Now $\omega = \sum dp\wedge dq$
denotes the standard symplectic form in $T^*M$ and the gmf
is defined by the form $\gamma =
\sum_{k,l} c_{kl}dq_k \otimes dq_l$ where the matrix of coefficients
$C=\{c_{kl}\}$ is antisymmetric. A vector field $F$ on $T^*M$ which
satisfies $i_F\omega - i_F\gamma = -dH $ for a smooth function $H$
on $T^*M$ generates the following differential equations
$$
\aligned
\dot q &= \frac{\partial H}{\partial p}\\
\dot p & = - \frac{\partial H}{\partial q} -C\frac{\partial
H}{\partial p}.
\endaligned
\tag{3.2}
$$
We can immediately calculate the divergence of the vector field
\thetag{3.2}. We obtain
$$
div F = -\sum_{k,l}\frac{\partial c_{kl}}{\partial p_k}
\frac{\partial H}{\partial p_l}.
$$
It follows that for magnetic flows the divergence is zero because
the entries of $C$ are functions on $M$.
This provides another proof of Proposition 1.5.
In the case of the flow \thetag{3.1} we obtain
$$
c_{kl} = \frac{1}{p^2}\left(p_k E_l
-p_l E_k\right),
$$
where $\Upsilon = \sum E_k dq_k$. Hence
$$
div F =\frac{1}{p^2} \sum_{k,l}c_{kl}\frac{\partial p^2}{\partial p_k}
\frac{\partial H}{\partial p_l} -\frac{1}{p^2}
\sum_{k}\sum_{l \neq k}E_l\frac{\partial H}{\partial p_l}.
$$
The first sum vanishes, if $H = \frac 12 p^2 + W(q)$ for some function
$W$ (the case of
isoenergetic dynamics), and the second term yields
$$
div F = -\frac{1}{p^2}\sum_{k}\left(\langle E,\dot q\rangle -
E_k \dot q_k\right) = \frac{-(n-1)}{p^2}
\langle E,\dot q\rangle.\tag{3.3}
$$
In the potential case $\Upsilon = -dU$ we get $\frac {(n-1)}{p^2}
\frac{dU}{dt}$.
If the external field $E$ is nonvanishing then
in the flow \thetag{3.1} we can expect that asymptotically
the velocity $v$ is aligned with the field $E$ and hence that
asymptotically the phase volume is contracted.
This intuition is supported by the following considerations.
Let us denote by $\phi^t = \phi^t_h$ our flow on the
energy shell $H = h$ (we assume that $h$ is large enough so that
$v^2$ does not vanish) and by $\lambda$
the normalized volume element on this energy shell.
Suppose that the flow has an asymptotic invariant measure
$\mu$ equal to the weak limit of the time averages
$ \frac 1T\int_0^T \phi^t_* \lambda dt $ as $T \to +\infty$
(possibly on a subsequence).
Now $e = -\int div F d\mu$ represents the average logarithmic
rate of volume contraction in this asymptotic state. As shown by
Ruelle, \cite{R}, this quantity can be interpreted as the average
rate of entropy production in the system and it is always nonnegative.
It is interesting to investigate conditions under which $e$ is
positive. One such case is a nonvanishing strong field $E$.
Indeed we can calculate
$$
\frac{d}{dt}\langle E, v \rangle =
\langle \nabla_v E, v \rangle -\langle E, \nabla W \rangle
+ \langle E, Bv \rangle + E^2 - \frac 1{c} \langle E, v \rangle^2,
$$
and we see that where $\langle E, v \rangle = 0$ we have
$\frac{d}{dt}\langle E, v \rangle > 0$, if only $E \neq 0$ and
we multiply it by a sufficiently large scalar.
It is not hard to prove that in such a case
the support of $\mu$ is contained in the domain $\langle E, v \rangle
>0$.
It follows from the fact that in the domain $\langle E, v \rangle
< 0$ the volume is expanded under the flow.
In the rest of this section we consider the isokinetic case $W \equiv
0$. If the external field $E$ has a global
potential then the system has a smooth invariant measure.
More interesting situation arises when the external field
has local potential but no global potential ($\Upsilon$ is
closed but not exact). Let us again consider
the asymptotic measure $\mu$ as introduced above.
Now the positivity of
$e$ can be also interpreted as the establishment of current
in the system since by \thetag{3.3}
$e = \frac{(n-1)}{v^2} \int \langle E, v \rangle d\mu$.
In the case of surfaces of constant negative curvature Gaussian
thermostats were studied by Bonetto, Gentile and Mastropietro,
\cite{B-G-M} and they addressed the question of positivity of $e$.
Let us consider more generally a Riemannian
manifold with negative sectional curvature. If the magnetic and
external fields are sufficiently small we obtain an Anosov flow
topologically conjugate to the geodesic flow, \cite{K-H}.
Moreover the topological conjugacy is homotopic to identity.
Such flows
have unique asymptotic measures, called SRB measures, \cite{R}.
We have
\proclaim{Proposition 3.1}
If the flow $\phi$ is an Anosov flow topologically conjugate
to the geodesic flow, the conjugacy is homotopic to
identity and the external field has local but no
global potential then the entropy production $e$ with respect to
the SRB measure $\mu$ is positive.
\endproclaim
\demo{Proof}
We begin by repeating the argument given by Ruelle in the case of
discrete time, \cite{R}.
Let us assume to the contrary that $e = -\int div F d\mu = 0$.
Then the sum of positive Lyapunov exponents (with respect to $\mu$)
is equal to the sum
of negative Lyapunov exponents and hence the metric entropy
of $\phi^{-1}$ satisfies the Pesin formula. It follows from the work of
Ledrappier and Young, \cite{L-Y}, that the measure
$\mu$ is absolutely continuous on both stable and unstable foliations
and hence it is absolutely continuous. Moreover the density with
respect to the volume element $\lambda$ must be positive and
continuous.
Such an invariant measure contradicts the multivaluedness of the
potential $U$. Indeed let us take a closed loop $l$ on $M$ such that
$\oint_l dU > 0$. There is a periodic geodesic $l'$ homotopic to
$l$. Further by the topological conjugacy
our flow has the periodic orbit $\zeta$ with the projection on $M$
homotopic to $l'$ and $l$. The time integral $\oint_\zeta div F dt$
of the divergence along the periodic orbit is equal to
$-\frac{n-1}{2h}\oint_l dU < 0$. On the other hand it is equal to
the logarithm of the Jacobian of $\phi^T$ on our $T$-periodic orbit.
Since the flow has an absolutely invariant measure with continuous
positive density the Jacobian must be equal to $1$.
\qed
\enddemo
This Proposition raises the question of explicit conditions
on the magnetic and external fields that imply the Anosov
property. The rest of the paper is devoted to this problem.
\vskip.4cm
\subhead \S 4. When is the general flow Anosov\endsubhead
\vskip.3cm
Gouda, \cite{G}, and Grognet, \cite{Gr}, obtained conditions on the
magnetic field on a manifold of negative sectional curvature which
imply that the flow is Anosov. We will extend their results by
including potential fields, external fields and Gaussian thermostats.
Hence we consider the general flow defined by the equations
$$
\frac{dq}{dt} = v, \ \ \ \frac{Dv}{dt} =
-\nabla W + Bv + E -\frac{\langle E,v\rangle}{v^2} v,
$$
where $W$ is a smooth function on $M$ (the potential energy of
internal interactions), $B$ is a magnetic field,
and $E$ is the external field. We restrict the system
to the level set
$ h = \frac 12 v^2 + W$ and we assume that $h$ is large enough so that
$v$ does not vanish. We denote by $R(\cdot,\cdot)$ the
curvature tensor.
\proclaim{Theorem 4.1}
If for every point $q\in M$ and $v \in T_qM$,
$\ \ v^2 = 2(h- W)$, the following quadratic form
in $\xi\in T_qM$
$$
\aligned
&-\langle R(v,\xi)v,\xi\rangle - \langle \nabla_\xi\nabla W,\xi \rangle
+ \langle \nabla_\xi Bv,\xi \rangle +\langle \nabla_\xi E,\xi
\rangle \\
&-\frac{\langle E,v\rangle^2}{4v^4}\xi^2 - \frac 14 (B\xi)^2
+\frac 1{4v^2}\langle \xi, -2\nabla W + Bv \rangle^2
-\frac 1{v^2}\langle \xi, -2\nabla W + Bv +E\rangle^2
\endaligned
$$
is positive definite on the subspace of $T_qM$ orthogonal to
$v$, then the system is Anosov.
\endproclaim
The condition in the theorem is clearly satisfied if the sectional
curvatures are negative and $h$ is sufficiently large.
It can be made less cumbersome by considering special cases.
Let the sectional curvatures of $M$ be bounded above by $-k^2 < 0$.
\proclaim {1. Pure potential flow, $ B = 0 , E = 0$}
If
$$
\sup_{q\in M} \left(\frac {\|\nabla^2W\|}{2(h- W)} +
3\left(\frac {\|\nabla W\|}{2(h-W)}\right)^2\right) < k^2
$$
then the system is Anosov.
\endproclaim
This criterion could be strengthened by the observation that where the
potential function is concave ($\langle \nabla_\xi\nabla W,\xi
\rangle$ negative definite), it adds to the dispersing
effect of the negative curvature.
\proclaim {2. Pure magnetic flow, $ W = 0, E = 0, v^2 = c$}
If $$\frac {\| \nabla B\|}{\sqrt{c}} + \left(\frac {\|
B\|}{\sqrt{c}}\right)^2 < k^2$$ then the system is Anosov.
\endproclaim
We obtained the condition of Gouda, \cite{Go}, and Grognet, \cite{Gr},
(Grognet does not get optimal
coefficients).
\proclaim {3. Pure Gaussian thermostat, $W = 0, B = 0, v^2 = c$}
If $$\frac {\| \nabla E\|}{c} + \left(\frac {\|
E\|}{c}\right)^2 < k^2$$ then the system is Anosov.
\endproclaim
\proclaim{4. Gaussian thermostat with the magnetic field, $W =0, v^2 = c$}
If
$$
\frac {\| \nabla B\|}{\sqrt{c}} + \frac {\| \nabla E\|}{c} +
\left(\frac{\|B\|}{\sqrt{c}} +\frac {\| E\|}{c} \right)^2 < k^2
$$
then the system is Anosov.
\endproclaim
\vskip.4cm
\subhead \S 5. A criterion for a flow to be Anosov \endsubhead
\vskip.3cm
Let $N$ be a smooth $k$-dimensional manifold with an auxiliary
Riemannian metric. Let further $F$ be a nonvanishing smooth vector
field on $N$. We consider the flow $\phi^t,\, t \in \Bbb R$, defined
by the vector field $F$.
The flow $\phi^t,\, t \in \Bbb R$ is Anosov (cf. \cite{K-H}),
if there is a $\phi^t$-invariant
splitting of the tangent bundle
$T_xM = E^0_x \oplus E^+_x \oplus E^-_x, x\in M$,
and two positive constants $a,b$ such that
\roster
\item $E^0_x = span\{F(x)\}, x\in M,$
\item $||D\phi^{\mp t} v|| \leq b e^{-ta}||v||
\ \ \ \ \text{for} \ \ \ \ v \in E^\pm_x
\ \ \ \ \text{and} \ \ \ \ t \geq 0 .$
\endroster
The linear subspaces of the splitting are called neutral, unstable
and stable, respectively.
In addition to the tangent bundle $TM$ we consider the
{\it quotient bundle}
$\widehat TM$ defined by $\widehat T_xM = T_xM/{span\{F(x)\}}$,
i.e., the linear space $\widehat T_xM$ is the linear quotient of
$T_xM$ by the onedimensional subspace spanned by the vector
$F(x)$. The quotient bundle inherits the scalar product
and the norm.
Since $D_x\phi^tF(x) = F(\phi^tx)$ we can factor the linear
map $$D_x\phi^t : T_xM \to T_{\phi^tx}M$$
to the quotient linear spaces
$\widehat T_xM $ and $\widehat T_{\phi^tx}M$. We denote the factor map
by
$$
A^t_x : \widehat T_xM \to \widehat T_{\phi^tx}M.
$$
The Anosov property can be reformulated in the language of the
quotient bundle.
\proclaim{Proposition 5.1}
The flow $\phi^t, t \in \Bbb R$, is Anosov if and only if
there is an $A^t$-invariant
splitting of the quotient bundle
$\widehat T_xM = \widehat E^+_x \oplus \widehat E^-_x, x\in M$,
and two positive constants $a,b$ such that
$$
||A^{\mp t}_x v|| \leq b e^{-ta}||v||
\ \ \ \ \text{for} \ \ \ \ v \in \widehat E^\pm_x
\ \ \ \ \text{and} \ \ \ \ t \geq 0.
$$
\endproclaim
\demo{Proof} The only thing that needs to be done is
the reconstruction of the un(stable) subspace from its
projection on the quotient space. We will do it for the stable
subspace $\widehat E^-_x$.
We identify the quotient tangent space $\widehat T_xM $ with the
orthogonal complement of $span\{F(x)\}$. Let
$\Pi : TM \to \widehat TM$ be the orthogonal projection
and let $\Cal L_F \Pi = \frac{d}{ds}D\phi^{-s}\Pi D\phi^s|_{s=0}$
be its Lie derivative.
Consider the linear functional $\lambda(x)$ defined on $\widehat E^-_x$ by
$$\lambda(x;v) = \int_0^{\infty} \frac{\langle F, (\Cal L_F
\Pi)A_x^sv\rangle}{\langle F, F \rangle} ds.$$
The integral converges thanks to the decay of $A_x^tv$
and we have for any vector $v \in \widehat E^-_x$
$$
D\phi^t(v+\lambda(x;v)F(x)) = A^tv + \lambda(\phi^tx;A^tv)F(\phi^tx).
$$
It follows that the stable subspace is given by
$$
E^-_x = \{ w = v + \lambda(x;v) F(x)\ | \ v \in \widehat E^-_x \}.
$$
\qed
\enddemo
Let $\Cal Q : TM \to \Bbb R$ be a continuous
quadratic form on the tangent bundle,
i.e., each
$\Cal Q_x = \Cal Q_{|T_xM}$ is a quadratic form on the tangent space $T_xM$ and
it depends continuously on $x$.
We assume further that the form can be factored onto the quotient bundle,
i.e.,
$\Cal Q_x(w + aF(x)) = \Cal Q_x(w) $, for any $w \in T_xM$ and any real $a$.
We will use the same notation for the factored form.
We assume that the factored form is {\it nondegenerate},
and hence being continuous it has constant signature $(l,m)$
($l+m = k-1$).
We assume that the Lie derivative
$ \Cal L_F \Cal Q(w) = \frac{d}{dt}\Cal Q(D\phi^tw)|_{t=0}$
of $\Cal Q$ in the direction of the
vector field $F$ is continuous. Clearly it can be also factored
onto the quotient space.
We say that the flow $\phi^t$ is {\it strictly monotone }
(with respect to a quadratic form $\Cal Q$), if
the factor of the Lie derivative
$ \Cal L_F \Cal Q$ onto the quotient bundle is positive definite.
The following theorem is a generalization of a result of
Lewowicz, \cite{L}, to flows.
\proclaim{Theorem 5.2}
If a flow is strictly monotone then it is Anosov.
\endproclaim
\demo{Proof}
We will work in the quotient bundle $\widehat TM$.
We define the bundle of positive cones
$C = \{v \in \widehat TM \ | \ \Cal Q(v) \geq 0\}$
associated with the form $\Cal Q$. By $C(x)$ we denote the
positive cone at $x \in M$.
By $C' = \{v \in \widehat TM \ | \ \Cal Q(v) \leq 0\}$
we denote the bundle of negative cones.
By the strict monotonicity we have that
$\Cal L_F \Cal Q(v) \geq c_1 \|v\|^2$ and also
(by compactness) $|\Cal Q(v)| \leq c_2 \|v\|^2$
for all $v \in \widehat TM$ and some positive constants
$c_1,c_2$. It follows that
$$
\frac{d}{dt}\Cal Q(A^tv) \geq c_1\|A^tv\|^2 \geq 2a |\Cal Q(A^tv)| \ \ \ \
\text{where} \ \ \ \ 2a = \frac{c_1}{c_2}.
$$
It follows that $A^tC \subset int C \cup \{0\}$ and
$A^{-t}C' \subset int C' \cup \{0\}$
for any $t > 0$. Moreover by integrating the last inequality
we obtain
$$
\frac {\Cal Q(A^tv)}{\Cal Q(v)} \geq e^{2at} \ \ \ \text{for} \ \ \
v \in int C \ \ \ \text{and} \ \ \
\frac {\Cal Q(A^tv)}{\Cal Q(v)} \leq e^{-2at} \ \ \ \text{for} \ \ \
A^tv \in int C'
$$
To compare the value of $\Cal Q(v)$ with $\|v\|^2$
let us introduce the bundles of narrow
cones $C_1 = A^1C$ and $C'_1 = A^{-1}C'$. We get by compactness that
for some positive constants $c_3,c_4$
$$
\Cal Q(v) \geq c_3 \|v\|^2 \ \ \ \text{for} \ \ \ v \in C_1 \ \ \
\text {and} \ \ \
-\Cal Q(v) \geq c_4 \|v\|^2 \ \ \ \text{for} \ \ \ v \in C'_1.
$$
We introduce the intersection $C_{\infty} = \bigcap_{t\ge\Cal Q 0} A^tC$.
In particular
$$
C_{\infty}(x) = \bigcap_{t\geq 0} A^t_{\phi^{-t}x}C(\phi^{-t}x).
$$
Our goal is to show that $C_{\infty}(x)$ is the unstable subspace.
Due to the monotonicity we have the invariance property $A^tC_\infty =
C_\infty$. Moreover since $\Cal Q$ has constant signature $(l,m)$
$C_{\infty}(x)$ must contain at least a subspace of dimension $l$.
Indeed the $l$-dimensional subspaces contained in
$A^t_{\phi^{-t}x}C(\phi^{-t}x)$
form compact subsets of the $l$-dimensional grassmanian, decreasing
as $t$ increases. Hence their intersection is nonempty.
It follows from the inequalities above and the inclusion
$C_\infty \subset C_1$ that
for any $v \in C_\infty, t > 0,$ we have
$$
c_2 \|v\|^2 \geq \Cal Q(v) = \Cal Q(A^tA^{-t}v) \geq e^{2at} \Cal Q(A^{-t}v) \geq
c_3e^{2at}\|A^{-t}v\|^2.
$$
i.e., $\|A^{-t}v\| \leq b
e^{-at}\|v\|$ for all $v \in C_\infty$ and some positive constant $b$.
This decay property holds also in the linear span of $C_\infty$, perhaps
with a different $b$.
Similarly starting with the negative cones $C'$ we can define
$C'_\infty$ which over every point contains at least an
$m$-dimensional
subspace and such that for any $v\in span C'_\infty, t > 0$ we have
$\|A^tv\| \leq b e^{-at}\|v\|$.
It remains to show that $C_\infty(x)$ is an $l$-dimensional subspace
and $C'_\infty(x)$ is an $m$-dimensional subspace. It follows from the
fact that the linear spans of $C_\infty(x)$ and $C'_\infty(x)$
must be transversal. Indeed for any nonzero vector in the intersection
we would have contradictory decay properties.
\qed\enddemo
Note that without the assumption that the factored form is
nondegenerate (and so it has constant signature) our stable and
unstable
subspaces could turn out to be lower dimensional, i.e., we would
obtain only partial hyperbolicity.
\vskip.4cm
\subhead \S 6. Proof of Theorem 4.1\endsubhead
\vskip.3cm
Let us introduce the linearized equations for the general flow \thetag{3.1}.
Let $q(t;u), |u| < \epsilon,$ be a family of trajectories
for the system. We introduce the Jacobi field
$\xi = \frac{dq}{du}$ and $\eta = \frac{Dv}{du} = \frac{D}{du}\frac{dq}{dt}$
We get the following equations for $(\xi, \eta)$ (the linearization
of the flow \thetag{3.1})
$$
\aligned
\frac{D\xi}{dt} = \eta,
\frac{D\eta}{dt} + R(v,\xi)v
= &-\nabla_\xi\nabla W + (\nabla_\xi B)v + B\eta +
\nabla_\xi E -\frac{\langle E,v\rangle}{v^2}\eta
\\&-\frac 1{v^2}\left(\langle
\nabla _\xi E,v\rangle + \langle E,\eta\rangle
- \frac{2\langle E,v\rangle
\langle v,\eta\rangle}{v^2}\right) v.
\endaligned
\tag{6.1}
$$
We introduce a quadratic form $\Cal H$ on the tangent spaces of $TM$
by $$
\Cal H(\xi,\eta) = \frac 12 \left(\xi^2 - \frac{\langle \xi,
v\rangle^2}{v^2}\right).
$$
The form $\Cal H$ factors naturally to the quotient bundle
(quotient by the span of the vector field \thetag{3.1}).
We define $\Cal Q$ as the Lie derivative of $\Cal H$
in the direction of the vector field \thetag{3.1}.
We get
$$
\Cal Q(\xi,\eta) = \langle \xi, \eta \rangle -
\frac{\langle \xi, v \rangle}{v^2}\left(\langle\eta, v \rangle -
\langle \xi,\nabla W \rangle +
\langle \xi, Bv \rangle + \langle \xi, E \rangle\right) +
\frac{\langle \xi, v \rangle^2}{v^2}
\left(\langle E, v \rangle -
\frac{\langle \nabla W, v \rangle }{v^2}\right).
$$
To apply Theorem 5.2 we need to calculate the Lie derivative of $\Cal Q$.
This cumbersome task is simplified by restricting the resulting
quadratic form to
the subspace defined by
$$\langle \eta, v \rangle = -\langle \xi , \nabla W
\rangle, \ \langle \xi, v \rangle = 0. \tag{6.2}$$
The first equation corresponds to the restriction of the energy
function $H = \frac 12 v^2 + W$, the second equation gives a
representation of the factor space. (Note that we can use the first
equation before differentiating the form but the second equation can
be used only after the differentiation.)
Thanks to this restriction
a considerable number of terms in $\frac{d}{dt}\Cal Q$ vanishes.
In particular we do not have to differentiate the terms in the
first bracket and the terms in the second bracket give no contribution
whatsoever to the restricted derivative. We obtain
$$
\aligned
\frac{d}{dt}\Cal Q = &\eta^2 - \langle R(v,\xi)v,\xi\rangle -
\langle \xi ,\nabla_\xi\nabla W\rangle +\langle \xi,
(\nabla_\xi B)v\rangle
+ \langle \xi,B\eta\rangle + \langle \xi,\nabla_\xi E\rangle
- \frac{\langle E,v\rangle}{v^2}\langle \xi,\eta\rangle
\\&-\frac 1{v^2}\langle \xi, -2\nabla W + Bv +E\rangle^2 =
\left(\eta - \frac{\langle \eta,v\rangle}{v^2}v -\frac{\langle E,v
\rangle}{2v^2}\xi -\frac 12 B\xi + \frac{\langle
B\xi,v\rangle}{2v^2}v\right)^2 \\&-
\langle R(v,\xi)v,\xi\rangle -
\langle \xi ,\nabla_\xi\nabla W\rangle +\langle \xi,
(\nabla_\xi B)v\rangle
+ \langle \xi,\nabla_\xi E\rangle \\&-
\frac{\langle E,v\rangle^2}{4v^4}\xi^2 - \frac 14 (B\xi)^2
+\frac 1{4v^2}\langle \xi, -2\nabla W + Bv \rangle^2
-\frac 1{v^2}\langle \xi, -2\nabla W + Bv +E\rangle^2.
\endaligned
$$
To end the proof of Theorem 4.1 we only need to check
that the factored form $\Cal Q$ is nondegenerate.
We can represent the factored form by the restriction on the
subspace \thetag{6.2}. We obtain the form $\langle \xi, \eta \rangle$.
It is straightforward to check that it is nondegenerate on the
subspace \thetag{6.2}.
\newpage
\Refs
\widestnumber\key{B-C-P}
\ref\key{A-S}\by D.V. Anosov, Ya.G. Sinai \paper
Certain smooth ergodic systems
\jour Russ. Math. Syrv. \vol 22 \pages 103 - 167 \yr 1967
\endref
\ref\key{B-C-P} \by F. Bonetto, E.G.D. Cohen, C. Pugh
\paper On the validity of the conjugate pairing rule for Lyapunov
exponents
\paperinfo preprint \yr 1998
\endref
\ref\key{B-G-M} \by F. Bonetto, G. Gentile, V. Mastropietro
\paper Electric fields on a surface of constant negative curvature
\paperinfo preprint \yr 1997
\endref
\ref\key{D-M}\by C.P. Dettmann, G.P. Morriss \paper Proof of
Lyapunov exponent pairing for systems at constant kinetic energy
\jour Phys. Rev. E \vol 53 \pages R5541 \yr 1996
\endref
\ref\key{E-C-M}\by D.J. Evans, E.G.D. Cohen, G.P. Morriss \paper
Viscosity of a simple fluid from its maximal Lyapunov exponents
\jour Physical Review\vol 42A \pages 5990 - 5997 \yr 1990
\endref
\ref\key{Go} \by N. Gouda \paper Magnetic flows of Anosov type
\jour Tohoku Math. J.\vol 49 \yr 1997 \pages 165 -- 183 \endref
\ref\key{Gr} \by S. Grognet \paper Flots magnetiques en courbure
negative
\paperinfo to appear in Erg. Th. Dyn. Sys. \endref
\ref\key{H}\by W.G. Hoover \book
Molecular dynamics, Lecture Notes in Physics 258
\publ Springer \yr 1986
\endref
\ref \key{K-B} \by A. Katok in collaboration with K. Burns
\paper Infinitesimal Lyapunov functions, invariant cone families
and stochastic properties of smooth
dynamical systems
\jour Erg. Th. Dyn. Sys. \yr 1994
\vol 14
\pages 757 -- 786
\endref
\ref\key{K-H}\by A. Katok, B. Hasselblatt \book
Introduction to the modern theory of dynamical systems
\publ Cambridge UP \yr 1995
\endref
\ref\key{L-Y}\by F. Ledrappier,L.-S. Young.
\paper The metric entropy of diffeomorphisms:
I. Characterization of measures satisfying Pesin's formula,
II. Relations between entropy, exponents and dimension
\jour Ann. of Math.
\vol 122
\yr 1985 \pages 509 -- 539 , 540 -- 574
\endref
\ref\key{L}\by J. Lewowicz \paper Lyapunov functions and topological
stability
\jour J.Diff.Eq. \vol 38 \yr 1980 \pages 192 -- 209
\endref
\ref\key{P-P} \by G.P. Paternain, M. Paternain
\paper Anosov geodesic flows and twisted symplectic structures
\jour Proc. Conf. Dyn, Systems, Montevideo 1995
\pages 132 - 145
\paperinfo Pitman Res.Notes Math.Ser. 362
\yr 1996
\endref
\ref\key{R}\by D. Ruelle \paper Positivity of entropy
production in nonequilibrium statistical mechanics
\jour J. Stat.Phys. \vol 85 \pages 1-25 \yr 1996
\endref
\ref\key{V}\by I. Vaisman \paper Locally conformal symplectic
manifolds \jour Int. J. Math.--Math.Sci.\vol 8
\yr 1985\pages 521--536
\endref
\ref\key{W-L} \by M.P. Wojtkowski, C. Liverani
\paper Conformally symplectic dynamics and
symmetry of the Lyapunov spectrum
\jour Commun.Math.Phys. \vol 194 \yr 1998 \pages 47-60 \endref
\ref \key{W} \by
M.P. Wojtkowski \paper Systems of classical interacting particles
with nonvanishing Lyapunov exponents \pages 243 -- 262
\yr 1991 \jour Lecture Notes in Math. 1486,
%Springer-Verlag \paperinfo Lyapunov Exponents, Proceedings,
%Oberwolfach 1990, L. Arnold, H. Crauel, J.-P. Eckmann (Eds)
\endref
\endRefs
\enddocument