Canonical Description of the Tachyon Kink in General Background and
Hamiltonian for Dp-brane-anti-Dp-brane system

J. Klusoň 111Email addresses<EMAIL_ADDRESS>(J. Klusoň)

Department of Theoretical Physics and Astrophysics, Faculty of Science,
Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic

In this short note we present analysis of the tachyon kink solution in the
canonical formalism. We also find Hamiltonian for simplified system of Dp-
brane and anti-Dp-brane.

##  1 Introduction and Summary

Unstable objects are very important parts of string theory. Their conformal
field theory description and their significance was discovered in remarkable
series of papers by A. Sen [1, 2, 3], for review and extensive list of
references, see [4]. They also led to renewed interest in the advanced topics
of string theory, as for example open string field theory [5, 6], background
independent string theory [9, 11, 10] and so on 222For review, see for example
[7, 8]..

It is remarkable that many phenomena related to unstable D-branes can be
captured by low energy effective action despite of the fact that the mass of
the tachyon is of the string scale [12, 13, 14, 15]. For example, it was shown
that the spatial dependent tachyon condensation leads to lower dimensional Dp-
brane with agreement with the conformal field theory and string field theory
description [16, 17, 18]. It was also shown that at the tachyon vacuum
unstable Dp-brane disappears and the dynamics reduces to the gas of
fundamental strings [26, 27, 28]. Moreover, it was also shown that tachyon
condensation in case of D-brane anti-D-brane system leads to emergence of
codimension two stable D-brane again with agreement with conformal field
theory description [16].

The goal of this paper is to continue the study of the tachyon condensation on
the world-volume of Dp-brane in general background when we focus on the
description of this problem using canonical formalism. In more details, we
find Hamiltonian for covariant form of unstable D-brane when we show that it
is given as sum of $p+1$ first class constraints where $p+1$ is number of
world-volume dimensions. Then we consider tachyon kink solution with the same
dependence as was given in [16], see also [17]. We also specify that all
remaining world-volume fields depend on remaining $p-$ coordinates. Due to the
fact that we study this problem in the canonical formalism we also propose an
ansatz for conjugate momenta that have more complicated form than conjugate
coordinates. Then we insert this proposed ansatz to the canonical equations of
motion and we show that these equations of motion are obeyed on conditions
when the fluctuation fields obey the equations of motion that arise from the
action for stable D(p-1)-brane. This result shows that the tachyon kink
solution reproduces lower dimensional D(p-1)-brane even in the canonical
formalism. We also discover the presence of the electric flux that is
transverse to the kink and that has meaning of the fundamental string
transverse to stable D(p-1)-brane. However this flux is localized in the
transverse region of the size of order $a^{-1}$ where $a$ i the parameter that
defines tachyon kink solution and that goes to $\infty$ in the end. As a
result this electric flux goes to zero in the limit $a\rightarrow\infty$ with
agreement with the observation given in [25].

As the second goal of this paper we focus on the canonical analysis of Dp-
brane anti-Dp-brane action. The first proposal of the action for this system
was given in [16]. Then an alternative derivation of this action was presented
in [24, 23] where this action was derived by specific projection from non-
abelian action for two non-BPS Dp-branes. However this proposed action was
discussed in [21] where it was shown that it does not reproduce solution of
the boundary field theory that describes this system. As a result this system
is not well understood yet. Despite of this fact we try to find canonical form
of the original form of the action [16]. However even in this case we are not
able to find corresponding Hamiltonian in the full generality simply from the
fact that it is not possible to find exact form of the dependence of tachyon
on canonical variables. This problem is a consequence of the fact that the
effective action for this coupled system is the sum of two square root terms.
For that reason we restrict ourselves to the case of the zero gauge fields on
both worldvolumes of branes and also to the case when their separation is
small. Then we will be able to find closed form of the Hamiltonian for this
system and discuss its properties. In particular, we analyse firstly the case
when brane and anti-brane coincide so that the tachyon is pure time dependent
and corresponds to rolling tachyon solution. The second solution corresponds
to the situation when the tachyon is in its vacuum state and we study dynamics
of relative coordinate when we show that it linearly depends on time which
however means that it quickly breaks our approximation of the small separation
of brane and anti-brane.

Let us outline our results and suggest possible extension of this work. We
found canonical action for non-BPS Dp-brane. Then we studied tachyon kink
solution in the canonical formalism and we showed that this solution
corresponds to stable D(p-1)-brane. Then we analysed action for Dp-brane anti-
Dp-brane system and we found Hamiltonian for its simplified form. We mean that
the detailed study of D-brane anti-Dbrane system should be goal of the future
project. In particular, we would like to see whether it would be possible to
find covariant form of the action suggested in [21] and how it is related to
the actions proposed in [16] and [24, 23]. This problem is currently under
study.

The structure of this paper is as follows. In the next section (2) we find
canonical form of the action for non-BPS Dp-brane and study tachyon kink
solution. Then in section (3) we analyse Dp-brane anti-Dp-brane system in the
canonical formalism.

##  2 Tachyon Kink on non-BPS Dp-Brane at Canonical Formalism

We start with the action for non-BPS Dp-brane that has the form [12, 14, 15]

$S=-\tilde{\tau}_{p}\int
d^{p+1}e^{-\phi}\mathcal{V}(T)\sqrt{-\det(g_{\alpha\beta}+l_{s}^{2}F_{\alpha\beta}+\partial_{\alpha}T\partial_{\beta}T)}$
(1)

where

$g_{\alpha\beta}=g_{\mu\nu}\partial_{\alpha}x^{\mu}\partial_{\beta}x^{\nu}\
,\quad F_{\alpha\beta}=\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}\
.$ (2)

Further, $T$ is the tachyonic field defined on the world-volume of non-BPS Dp-
brane, $\mathcal{V}(T)$ is its potential when we presume that $\mathcal{V}$ is
even function with three extremes, where $T_{0}=0$ is unstable maximum with
$\mathcal{V}(T_{0})=0$ while $\pm T_{min}$ are global minima of the potential
with $\mathcal{V}(\pm T_{min})=0$. Further,
$x^{\mu}(\xi^{\alpha}),\mu,\nu=0,1,\dots,9$ are scalar fields on the world-
volume of non-BPS Dp-brane that describe embedding of Dp-brane into target
space-time and $A_{\alpha},\alpha=0,1,\dots,p$ are gauge fields living on the
world-volume of non-BPS Dp-brane. Note that the $p+1$-dimensional world-volume
is labelled with coordinates $\xi^{\alpha}$. Finally $g_{\mu\nu}(x)$ is target
space metric and $\phi$ is dilaton field that generally depends on $x$. For
simplicity we will presume that $\phi$ is constant equal to $\phi_{0}$. Note
also that $\tilde{\tau}_{p}$ is non-BPS Dp-brane tension proportional to
$l_{s}^{p+1}$ where $l_{s}$ is string length.

We would like to study tachyon kink solution in the canonical formalism so
that we should firstly determine corresponding Hamiltonian. From (1) we find
following conjugate momenta

$\displaystyle p_{\mu}=\frac{\partial
S}{\partial(\partial_{0}x^{\mu})}=-\tilde{\tau}_{p}\mathcal{V}e^{-\phi_{0}}g_{\mu\nu}\partial_{\alpha}x^{\nu}\left(\mathbf{A}^{-1}\right)_{S}^{\alpha
0}\sqrt{-\det\mathbf{A}}\ ,$ $\displaystyle\pi^{\alpha}=\frac{\partial
S}{\partial(\partial_{0}A_{\alpha})}=-\tilde{\tau}_{p}\mathcal{V}e^{-\phi_{0}}\left(\mathbf{A}^{-1}\right)^{\alpha
0}_{A}\sqrt{-\det\mathbf{A}}\ ,$ $\displaystyle p_{T}=\frac{\partial
S}{\partial(\partial_{0}T)}=-\tilde{\tau}_{p}\mathcal{V}e^{-\phi_{0}}\partial_{\alpha}T\left(\mathbf{A}^{-1}\right)^{\alpha
0}_{S}\sqrt{-\det\mathbf{A}}\ ,$

where

$\mathbf{A}_{\alpha\beta}=g_{\alpha\beta}+\partial_{\alpha}T\partial_{\beta}T+l_{s}^{2}F_{\alpha\beta}\
,$ (4)

and where $\left(\mathbf{A}^{-1}\right)^{\alpha\beta}$ is matrix inverse to
$\mathbf{A}_{\alpha\beta}$ so that
$\mathbf{A}_{\alpha\beta}\left(\mathbf{A}^{-1}\right)^{\beta\gamma}=\delta_{\alpha}^{\gamma}$
and we also defined

$\left(\mathbf{A}^{-1}\right)^{\alpha\beta}_{S}=\frac{1}{2}(\left(\mathbf{A}^{-1}\right)^{\alpha\beta}+\left(\mathbf{A}^{-1}\right)^{\beta\alpha})\
,\quad\left(\mathbf{A}^{-1}\right)^{\alpha\beta}_{A}=\frac{1}{2}(\left(\mathbf{A}^{-1}\right)^{\alpha\beta}-\left(\mathbf{A}^{-1}\right)^{\beta\alpha})\
.$ (5)

Note also that the definition of the canonical momenta $\pi^{\alpha}$ implies
following primary constraint

$\pi^{0}\approx 0\ .$ (6)

Using definition of the canonical momenta given in (2) we easily find that the
bare Hamiltonian density is equal to

$\mathcal{H}_{B}=p_{T}\partial_{0}T+\pi^{\alpha}\partial_{0}A_{\alpha}+p_{\mu}\partial_{0}x^{\mu}-\mathcal{L}=\pi^{i}\partial_{i}A_{0}$
(7)

while this theory possesses $p+1$ primary constraints

$\displaystyle\mathcal{H}_{i}=p_{\mu}\partial_{i}x^{\mu}+p_{T}\partial_{i}T+F_{ij}\pi^{j}\
,\quad$
$\displaystyle\mathcal{H}_{\tau}=p_{\mu}g^{\mu\nu}p_{\nu}+p_{T}^{2}+\pi^{i}(g_{ij}+\partial_{i}T\partial_{j}T)\pi^{j}+\tilde{\tau}_{p}^{2}\mathcal{V}^{2}e^{-2\phi_{0}}\det(g_{ij}+l_{s}^{2}F_{ij}+\partial_{i}T\partial_{j}T)\
.$

Then we can also write the action (1) in the canonical form

$\displaystyle S=\int
d^{p+1}\xi(p_{\mu}\partial_{0}x^{\mu}+\pi^{i}\partial_{0}A_{i}+p_{T}\partial_{0}T-$
$\displaystyle-N(p_{\mu}g^{\mu\nu}p_{\nu}+p_{T}^{2}+\pi^{i}(g_{ij}+\partial_{i}T\partial_{j}T)\pi^{j}+\tilde{\tau}_{p}^{2}\mathcal{V}^{2}e^{-2\phi_{0}}\det(g_{ij}+l_{s}^{2}F_{ij}+\partial_{i}T\partial_{j}T))-$
$\displaystyle-N^{i}(p_{\mu}\partial_{i}x^{\mu}+F_{ij}\pi^{j}+p_{T}\partial_{i}T)-\pi^{i}\partial_{i}A_{0})\
,$

where $N$ and $N^{i}$ are Lagrange multipliers corresponding to the
constraints $\mathcal{H}_{i},\mathcal{H}_{\tau}$. Then performing variation of
the action (2) we determine corresponding equations of motion

$\displaystyle\partial_{0}x^{\mu}=2Ng^{\mu\nu}p_{\nu}+N^{i}\partial_{i}x^{\mu}\
,$
$\displaystyle\partial_{0}A_{i}=2N\mathbf{A}_{ij}^{S}\pi^{j}+N^{j}F_{ji}+\partial_{i}A_{0}\
,$
$\displaystyle\partial_{0}p_{\mu}=2\partial_{i}[N\pi^{i}g_{\mu\nu}\partial_{j}x^{\nu}\pi^{j}]-N\pi^{i}\partial_{i}x^{\rho}\partial_{\mu}g_{\rho\sigma}\partial_{j}x^{\sigma}\pi^{j}+$
$\displaystyle+2\tau_{p}^{2}\partial_{i}[Ne^{-2\phi_{0}}\mathcal{V}^{2}h_{\mu\nu}\partial_{j}x^{\nu}\left(\mathbf{A}^{-1}\right)^{ji}_{S}\det\mathbf{A}]+\partial_{i}[N^{i}p_{\mu}]\
,$ $\displaystyle\partial_{0}T=2Np_{T}+N^{i}\partial_{i}T\ ,$
$\displaystyle\partial_{0}p_{T}=2\partial_{i}[N\pi^{i}\partial_{j}T\pi^{j}]-2\tau_{p}^{2}Ne^{-2\phi_{0}}\mathcal{V}\frac{d\mathcal{V}}{dT}\det\mathbf{A}+$
$\displaystyle+2\tau_{p}^{2}\partial_{i}[Ne^{-2\phi_{0}}\mathcal{V}^{2}\partial_{j}T\left(\mathbf{A}^{-1}\right)^{ji}_{S}\det\mathbf{A}]+\partial_{i}[N^{i}p_{T}]\
,$
$\displaystyle\partial_{0}\pi^{i}=2\tau_{p}^{2}\partial_{j}[Ne^{-2\phi_{0}}\mathcal{V}^{2}\left(\mathbf{A}^{-1}\right)^{ij}_{A}\det\mathbf{A}]+\partial_{k}[N^{k}\pi^{i}]-\partial_{k}[N^{i}\pi^{k}]\
.$

Further, variation of the action (2) with respect to $N$ and $N^{i}$ where we
treat them as independent variables imply constraints $\mathcal{H}_{i}\approx
0\ ,\mathcal{H}_{\tau}\approx 0$. In the same way variation of the action (2)
with respect to $A_{0}$ implies following Gauss law constraint

$\mathcal{G}\equiv\partial_{i}\pi^{i}\approx 0\ .$ (11)

Now we are ready to proceed to the search of the tachyon kink solution in the
canonical formalism. Analysis performed in the Lagrangian formalism shows that
this kink solution should correspond to stable D(p-1)-brane [16], see also
[17, 18]. Following the same logic we will presume that the tachyon field has
following dependence of the world-volume coordinates

$T(\xi)=f(a(y-t(\xi^{\hat{\alpha}})))\ ,$ (12)

where $y\equiv\xi^{p}$ and where $\xi^{\hat{\alpha}}\
,\hat{\alpha}=0,1,\dots,p-1$. Then as in [16] we presume that $f(u)$ is
arbitrary function of $u$ with the property that $f^{\prime}(u)>0,\forall u$
and $f(\pm\infty)=\pm\infty$. Finally $a$ is free parameter that we can send
to $\infty$ in the end.

Let us further presume that $A_{\alpha}$ and $x^{\mu}$ do not depend on $y$ so
that

$x^{\mu}(\xi^{\alpha})=\tilde{x}^{\mu}(x^{\hat{\alpha}})\ ,\quad
A_{\hat{\alpha}}(\xi^{\alpha})=\tilde{a}_{\hat{\alpha}}(\xi^{\hat{\alpha}})\
,\hat{i}=1,\dots,p-1\ $ (13)

and without lost of generality we set $A_{y}=0$. The ansatz for conjugate
momenta has more complicated form

$p_{T}=af^{\prime}\mathcal{V}\tilde{p}_{T}\ ,\quad
p_{\mu}=\frac{\tau_{p}}{T_{p-1}}f^{\prime}\mathcal{V}\tilde{p}_{\mu}\
,\quad\pi^{\hat{\alpha}}=\frac{\tau_{p}}{T_{p-1}}af^{\prime}\mathcal{V}\tilde{\pi}^{\hat{\alpha}}\
,\quad$ (14)

where $T_{p-1}$ is tension of stable D(p-1)-brane and the factor
$\frac{\tilde{\tau}_{p}}{T_{p-1}}$ was chosen through dimensional reason since
$p_{\mu}$ is $p+1$-dimensional density while $\tilde{p}_{\mu}$ is
$p-$dimensional density. Further, $\pi^{y}$ will be determined bellow. Finally
we presume following ansatz for $N$ and $N^{i}$

$N=\frac{T_{p-1}}{\tau_{p}}\frac{1}{af^{\prime}\mathcal{V}}\tilde{n}(\xi^{\hat{\alpha}})\
,\quad N^{\hat{i}}=\tilde{n}^{\hat{i}}(\xi^{\hat{\alpha}})\ $ (15)

while $N^{y}$ will be determined by equations of motion.

With this ansatz we find that the matrix $\mathbf{A}_{ij}$ has the form

$\displaystyle\mathbf{A}_{ij}=\left(\begin{array}[]{cc}\
\mathbf{a}_{\hat{i}\hat{j}}+a^{2}f^{\prime
2}\partial_{\hat{i}}t\partial_{\hat{j}}t&-a^{2}f^{\prime
2}\partial_{\hat{i}}t\\\ -a^{2}f^{\prime 2}\partial_{\hat{j}}t&a^{2}f^{\prime
2}\\\ \end{array}\right)\ ,\quad\det\mathbf{A}_{ij}=a^{2}f^{\prime
2}\det\mathbf{a}_{\hat{i}\hat{j}}\ ,$ (18)

where

$\mathbf{a}_{\hat{i}\hat{j}}=g_{\mu\nu}\partial_{\hat{i}}\tilde{x}^{\mu}\partial_{\hat{j}}\tilde{x}^{\nu}+l_{s}^{2}f_{\hat{i}\hat{j}}\
,\quad
f_{\hat{i}\hat{j}}=\partial_{\hat{i}}\tilde{a}_{\hat{j}}-\partial_{\hat{j}}\tilde{a}_{\hat{i}}\
.$ (20)

For further purposes we determine matrix inverse to $\mathbf{A}_{ij}$ that has
the form

$\mathbf{A}^{ij}=\left(\begin{array}[]{cc}\tilde{\mathbf{a}}^{\hat{i}\hat{j}}&\tilde{\mathbf{a}}^{\hat{i}\hat{k}}\partial_{\hat{k}}t\\\
\partial_{\hat{k}}t\tilde{\mathbf{a}}^{\hat{k}\hat{j}}&\frac{1}{a^{2}f^{\prime
2}}+\partial_{\hat{i}}t\tilde{\mathbf{a}}^{\hat{i}\hat{j}}\partial_{\hat{j}}t\\\
\end{array}\right)\ ,$ (21)

where $\tilde{\mathbf{a}}^{\hat{i}\hat{j}}$ is matrix inverse to
$\mathbf{a}_{\hat{i}\hat{j}}$ so that

$\mathbf{a}_{\hat{i}\hat{j}}\tilde{\mathbf{a}}^{\hat{j}\hat{k}}=\delta_{\hat{i}}^{\hat{j}}\
.$ (22)

Let u start with the equation of motion for $T$ that for (12),(13) and (14)
takes the form

$\displaystyle-
af^{\prime}\partial_{0}t=af^{\prime}\mathcal{V}\tilde{p}_{T}+N^{y}af^{\prime}-\tilde{n}^{\hat{i}}\partial_{\hat{i}}taf^{\prime}\
.$

To proceed further note that the equation of motion for $N^{y}$ implies

$p_{T}\partial_{y}T+\partial_{y}x^{\mu}p_{\mu}+F_{y\hat{i}}\pi^{\hat{i}}=0$
(24)

that, since $\partial_{y}T\neq 0$ and since
$\partial_{y}x^{\mu}=F_{y\hat{i}}=0$ implies that $p_{T}=0$. Note that
$F_{y\hat{i}}=\partial_{y}A_{\hat{i}}-\partial_{\hat{i}}A_{y}=0$ due to the
fact that $A_{\hat{i}}$ does not depend on $y$ and since $A_{y}=0$. Now since
$p_{T}=0$ the equation (2) allows us to express $N^{y}$ as

$N^{y}=\tilde{n}^{\hat{i}}\partial_{\hat{i}}t-\partial_{0}t\ .$ (25)

As the next step we proceed to the equation of motion for $\pi^{\hat{i}}$.
Inserting (12),(13) and (14) into right side of this equation we get

$\displaystyle
2\tilde{\tau}_{p}^{2}\partial_{j}[Ne^{-2\phi_{0}}\mathcal{V}^{2}\left(\mathbf{A}^{-1}\right)^{\hat{i}j}\det\mathbf{A}]+\partial_{k}[N^{k}\pi^{y}]-\partial_{k}[N^{\hat{i}}\pi^{k}]=$
$\displaystyle
af^{\prime}\tilde{\tau}_{p}\mathcal{V}[2T_{p-1}\partial_{\hat{j}}[e^{-2\phi_{0}}\tilde{n}\tilde{\mathbf{a}}^{\hat{i}\hat{j}}_{A}\det\mathbf{a}]+\frac{1}{T_{p-1}}\partial_{\hat{k}}[\tilde{n}^{\hat{k}}\tilde{\pi}^{\hat{i}}]-\frac{1}{T_{p-1}}\partial_{\hat{k}}[\tilde{n}^{\hat{i}}\tilde{\pi}^{\hat{k}}]]+$
$\displaystyle+\frac{\tilde{\tau}_{p}}{T_{p-1}}\partial_{y}[af^{\prime}\mathcal{V}](-\partial_{\hat{k}}t\tilde{n}^{\hat{k}}\tilde{\pi}^{\hat{i}}+\tilde{n}^{\hat{k}}\partial_{\hat{k}}t\tilde{\pi}^{\hat{i}}-\partial_{0}t\tilde{\pi}^{\hat{i}}+\partial_{\hat{k}}t\tilde{n}^{\hat{i}}\tilde{\pi}^{\hat{k}})-\tilde{n}^{\hat{i}}\partial_{y}\pi^{y}$

while on the left-side we have

$\partial_{0}\pi^{\hat{i}}=-\frac{\tilde{\tau}_{p}}{T_{p-1}}\partial_{0}t\partial_{y}[af^{\prime}\mathcal{V}]\tilde{\pi}^{\hat{i}}+\frac{\tilde{\tau}_{p}}{T_{p-1}}af^{\prime}\mathcal{V}\partial_{0}\tilde{\pi}^{\hat{i}}$
(27)

using the fact that

$\partial_{\hat{\alpha}}[af^{\prime}\mathcal{V}]=-\partial_{y}[af^{\prime}\mathcal{V}]\partial_{\hat{\alpha}}t\
.$ (28)

Then collecting (2) together with (27) we obtain

$\displaystyle
af\mathcal{V}\partial_{0}\tilde{\pi}^{i}=af^{\prime}\mathcal{V}\left(2T_{p-1}^{2}\partial_{\hat{j}}[\tilde{n}e^{-2\phi_{0}}\tilde{\mathbf{a}}^{\hat{i}\hat{j}}_{A}\det\mathbf{a}]+\partial_{\hat{k}}[\tilde{n}^{\hat{k}}\tilde{\pi}^{\hat{i}}]-\partial_{\hat{k}}[\tilde{n}^{\hat{i}}\tilde{\pi}^{\hat{k}}]\right)+$
$\displaystyle+\partial_{y}[af^{\prime}V]\partial_{\hat{k}}t\tilde{n}^{\hat{i}}\tilde{\pi}^{\hat{k}}-\frac{T_{p-1}}{\tilde{\tau}_{p}}\tilde{n}^{\hat{i}}\partial_{y}\pi^{y}$

We see that the expression on the second line vanishes when we presume that
$\pi^{y}$ is equal to

$\pi^{y}=\frac{\tilde{\tau}_{p}}{T_{p-1}}af^{\prime}\mathcal{V}\partial_{\hat{i}}t\tilde{\pi}^{\hat{i}}\
.$ (30)

Then (2) has final form

$af^{\prime}\mathcal{V}(\partial_{0}\tilde{\pi}^{\hat{i}}-2T_{p-1}^{2}\partial_{\hat{j}}[\tilde{n}e^{-2\phi_{0}}\tilde{\mathbf{a}}^{\hat{i}\hat{j}}_{A}\det\mathbf{a}]-\partial_{\hat{k}}[\tilde{n}^{\hat{k}}\tilde{\pi}^{\hat{i}}]+\partial_{\hat{k}}[\tilde{n}^{\hat{i}}\tilde{\pi}^{\hat{k}}])=0\
.$ (31)

Now this expression is equal to zero in the limit $a\rightarrow\infty$ for
$y\neq t(\xi^{\hat{\alpha}})$. On the other hand we find that the expression
$af^{\prime}\mathcal{V}$ is finite for $y=t(\xi^{\hat{\alpha}})$ and hence in
order to obey equations of motion for $\pi^{\hat{i}}$ at the core of the kink
defined as $y^{p}=t(\xi^{\hat{\alpha}})$ we should demand that the expressions
in the bracket is zero. However they are precisely the equations of motion for
$\tilde{\pi}^{\hat{i}}$ that follow from an action for D(p-1)-brane. In more
details, let us consider canonical form of the action for stable D(p-1)-brane

$S=-T_{p-1}\int
d^{p}\xi(\tilde{p}_{\mu}\partial_{0}\tilde{x}^{\mu}+\tilde{\pi}^{\hat{i}}\partial_{0}\tilde{a}_{\hat{i}}-\tilde{n}\tilde{\mathcal{H}}-\tilde{n}^{\hat{i}}\tilde{\mathcal{H}}_{\hat{i}}-\tilde{\pi}^{\hat{i}}\partial_{\hat{i}}\tilde{a}_{0})\
,$ (32)

where

$\displaystyle\tilde{\mathcal{H}}=\tilde{p}_{\mu}g^{\mu\nu}\tilde{p}_{\nu}+\tilde{\pi}^{\hat{i}}g_{\hat{i}\hat{j}}\tilde{\pi}^{\hat{j}}+T_{p-1}^{2}e^{-2\phi_{0}}\det(g_{\hat{i}\hat{j}}+l_{s}^{2}f_{\hat{i}\hat{j}})\
,$
$\displaystyle\tilde{\mathcal{H}}_{\hat{i}}=\tilde{p}_{\mu}\partial_{\hat{i}}\tilde{x}^{\mu}+f_{\hat{i}\hat{j}}\tilde{\pi}^{\hat{j}}\
,\quad
g_{\hat{i}\hat{j}}=g_{\mu\nu}\partial_{\hat{i}}\tilde{x}^{\mu}\partial_{\hat{j}}\tilde{x}^{\nu}\
,\quad
f_{\hat{i}\hat{j}}=\partial_{\hat{i}}\tilde{a}_{\hat{j}}-\partial_{\hat{j}}\tilde{a}_{\hat{i}}\
.$ (33)

Then from the action (32) we obtain following equations of motion for
$\tilde{x}^{\mu}$

$\partial_{0}\tilde{x}^{\mu}=2\tilde{n}g^{\mu\nu}\tilde{p}_{\nu}+\tilde{n}^{\hat{i}}\partial_{\hat{i}}\tilde{x}^{\mu}\
$ (34)

and for $\tilde{a}_{\hat{i}}$

$\partial_{0}\tilde{a}_{\hat{i}}=2\tilde{n}g_{\hat{i}\hat{j}}\tilde{\pi}^{\hat{j}}+\tilde{n}^{\hat{j}}f_{\hat{j}\hat{i}}+\partial_{\hat{i}}\tilde{a}_{0}\
,$ (35)

together with equations of motion for canonically conjugate momenta
$\tilde{p}_{\mu}$

$\displaystyle\partial_{0}\tilde{p}_{\mu}=2\partial_{\hat{i}}[\tilde{n}\tilde{\pi}^{\hat{i}}g_{\mu\nu}\partial_{\hat{j}}\tilde{x}^{\nu}\tilde{\pi}^{\hat{j}}]-\tilde{n}\tilde{\pi}^{\hat{i}}\partial_{\hat{i}}\tilde{x}^{\rho}\partial_{\mu}g_{\rho\sigma}\partial_{\hat{j}}\tilde{x}^{\sigma}\tilde{\pi}^{\hat{j}}+$
$\displaystyle+T_{p-1}^{2}e^{-2\phi_{0}}\partial_{\hat{i}}\tilde{x}^{\rho}\partial_{\mu}g_{\rho\sigma}\partial_{\hat{j}}\tilde{x}^{\nu}\tilde{\mathbf{a}}^{\hat{j}\hat{i}}\det\mathbf{a}+$
$\displaystyle+2T_{p-1}^{2}\partial_{\hat{i}}[\tilde{n}e^{-2\phi_{0}}g_{\mu\nu}\partial_{\hat{j}}\tilde{x}^{\nu}\tilde{\mathbf{a}}^{ji}_{S}\det\mathbf{a}]+\partial_{\hat{i}}[\tilde{n}^{\hat{i}}p_{\mu}]\
$

and $\tilde{\pi}^{\hat{i}}$

$\displaystyle\partial_{0}\tilde{\pi}^{\hat{i}}=2T_{p-1}^{2}\partial_{\hat{j}}[\tilde{n}e^{-2\phi}\tilde{\mathbf{a}}^{\hat{i}\hat{j}}_{A}\det\mathbf{a}]+\partial_{\hat{k}}[\tilde{n}^{\hat{k}}\tilde{\pi}^{\hat{i}}]-\partial_{\hat{k}}[\tilde{n}^{\hat{i}}\tilde{\pi}^{\hat{k}}]\
.$ (37)

Now we return back to non-BPS Dp-brane and consider equation of motion for
$\pi^{y}$

$\partial_{0}\pi^{y}=2\tilde{\tau}_{p}^{2}\partial_{j}[Ne^{-2\phi_{0}}\mathcal{V}^{2}\left(\mathbf{A}^{-1}\right)^{yj}_{A}\det\mathbf{A}]+\partial_{k}[N^{k}\pi^{y}]-\partial_{k}[N^{y}\pi^{k}]\
.$ (38)

Inserting ansatz (12),(13) and (14) into it we obtain after some calculations
following result

$\displaystyle
af^{\prime}\mathcal{V}\partial_{\hat{i}}t[\partial_{0}\tilde{\pi}^{\hat{i}}-T_{p-1}^{2}\partial_{\hat{k}}[\tilde{n}e^{-2\phi_{0}}\tilde{\mathbf{a}}_{A}^{\hat{i}\hat{k}}\det\mathbf{a}]+\partial_{\hat{k}}[\tilde{n}^{\hat{i}}\tilde{\pi}^{\hat{k}}]-\partial_{\hat{k}}[\tilde{n}^{\hat{k}}\tilde{\pi}^{\hat{i}}]]=0\
,$

where we also used the fact that

$\left(\mathbf{A}^{-1}\right)^{y\hat{i}}_{A}=\frac{1}{2}(\partial_{\hat{k}}t\tilde{\mathbf{a}}^{k\hat{i}}-\tilde{\mathbf{a}}^{\hat{i}\hat{k}}\partial_{\hat{k}}t)=\partial_{\hat{k}}t\tilde{\mathbf{a}}_{A}^{\hat{k}\hat{i}}\
.$ (40)

It is clear that (2) is obeyed when the equations of motion (37) hold. This is
again nice consistency check.

As the next step we proceed to the equation of motion for $p_{T}$. Note that
the left side is equal to zero since $p_{T}=0$ while the right side is equal
to

$\displaystyle
2\partial_{i}[N\pi^{i}\partial_{j}T\pi^{j}]-\tilde{\tau}_{p}^{2}Ne^{-2\phi_{0}}\mathcal{V}\frac{d\mathcal{V}}{dT}\det\mathbf{A}+$
$\displaystyle+\tilde{\tau}_{p}^{2}\partial_{i}[Ne^{-2\phi_{0}}\mathcal{V}^{2}\partial_{j}T\left(\mathbf{A}^{-1}\right)^{ji}_{S}\det\mathbf{A}]+\partial_{i}[N^{i}p_{T}]\
.$

Let us consider the first term on the first line in (2). Inserting (12),(13)
and (14) into it we obtain that it identically vanishes since

$\displaystyle\partial_{i}[N\pi^{i}\partial_{j}T\pi^{j}]=\frac{\tilde{\tau}_{p}}{T_{p-1}}\left(\partial_{y}[a^{2}f^{\prime
2}\mathcal{V}]\tilde{\pi}^{\hat{i}}\partial_{\hat{i}}t\tilde{\pi}^{\hat{k}}\partial_{\hat{k}}t-\partial_{y}[a^{2}f^{\prime
2}\mathcal{V}]\tilde{\pi}^{\hat{i}}\partial_{\hat{i}}t\partial_{\hat{k}}t\tilde{\pi}^{\hat{k}}\right.+$
$\displaystyle+\left.+\partial_{\hat{i}}[a^{2}f^{\prime
2}\mathcal{V}\tilde{\pi}^{\hat{i}}\tilde{\pi}^{\hat{k}}\partial_{\hat{k}}t]-\partial_{\hat{i}}[a^{2}f^{\prime
2}\mathcal{V}\tilde{\pi}^{\hat{i}}\partial_{\hat{j}}t\tilde{\pi}^{\hat{j}}]\right)=0\
.$

Let us now insert (12),(13) and (14) into the first term on the second line in
(2) and we obtain

$\displaystyle\partial_{i}[Ne^{-2\phi_{0}}\mathcal{V}^{2}\partial_{j}T\left(\mathbf{A}^{-1}\right)^{ji}_{S}\det\mathbf{A}]=$
$\displaystyle=\tilde{\tau}_{p}T_{p-1}\partial_{y}[\tilde{n}a^{2}f^{\prime
2}e^{-2\phi_{0}}\mathcal{V}\det\mathbf{a}(\left(\mathbf{A}^{-1}\right)^{yy}-\partial_{\hat{j}}t\left(\mathbf{A}^{-1}\right)^{jy}_{S})]+$
$\displaystyle+\tilde{\tau}_{p}T_{p-1}\partial_{\hat{i}}[\tilde{n}a^{2}f^{\prime
2}e^{-2\phi_{0}}\mathcal{V}\det\mathbf{a}(\left(\mathbf{A}^{-1}\right)^{y\hat{i}}_{S}-\partial_{\hat{j}}t\left(\mathbf{A}^{-1}\right)^{\hat{j}\hat{i}}_{S})]=$
$\displaystyle=\tilde{\tau}_{p}T_{p-1}\tilde{n}e^{-2\phi_{0}}af^{\prime}\frac{d\mathcal{V}}{dT}\det\mathbf{a}$

which exactly cancel the second term on the first line in (2). As a result we
obtain that the equation of motion for $p_{T}$ is identically obeyed for the
ansatz (12),(13) and (14).

As the next step we consider equation of motion for $A_{\alpha}$

$\partial_{0}A_{\hat{i}}=2N\mathbf{A}_{\hat{i}j}^{S}\pi^{j}+N^{j}F_{ji}+\partial_{\hat{i}}A_{0}$
(44)

that for the ansatz (12),(13) and (14) reduces into

$af^{\prime}\mathcal{V}\left(\partial_{0}\tilde{a}_{\hat{i}}-\partial_{\hat{i}}\tilde{a}_{0}-2\tilde{n}\tilde{\mathbf{a}}_{\hat{i}\hat{j}}^{S}\tilde{\pi}^{\hat{j}}-\tilde{n}^{\hat{j}}f_{\hat{j}\hat{i}}\right)=0$
(45)

and we see that this equation is obeyed at the core of the kink on condition
when the expression in the bracket vanishes. In fact, this expression is the
equation of motion for the gauge field $\tilde{a}_{\hat{i}}$ given in (35).

In the same way we can proceed with the equation of motion for $x^{\mu}$ and
we obtain

$\displaystyle
af^{\prime}\mathcal{V}\left(\partial_{0}\tilde{x}^{\mu}-2\tilde{n}g^{\mu\nu}\tilde{p}_{\nu}-\tilde{n}^{\hat{i}}\partial_{\hat{i}}\tilde{x}^{\mu}\right)=0$

and we see that this equation of motion is obeyed at the core of the kink when
the expression in the bracket vanishes. Again, this expression has the same
form as the equation of motion for $\tilde{x}^{\mu}$ (34).

The situation is more involved in case of the equation of motion for
$p_{\mu}$. Inserting the ansatz (14)) to the left side of this equation we
obtain

$\partial_{0}p_{\mu}=\frac{\tilde{\tau}_{p}}{T_{p-1}}(-\partial_{y}(af^{\prime}\mathcal{V})\partial_{0}t\tilde{p}_{\mu}+af^{\prime}\mathcal{V}\partial_{0}\tilde{p}_{\mu})\
.$ (47)

On the other hand when we insert (12),(13) and (14) to the right side of the
equation of motion for $p_{\mu}$ we obtain

$\displaystyle
2\partial_{i}[N\pi^{i}g_{\mu\nu}\partial_{j}x^{\nu}\pi^{j}]-N\pi^{i}\partial_{i}x^{\rho}\partial_{\mu}g_{\rho\sigma}\partial_{j}x^{\sigma}\pi^{j}-\tilde{\tau}_{p}^{2}Ne^{-2\phi_{0}}\partial_{i}x^{\rho}\partial_{\mu}g_{\rho\sigma}\partial_{j}x^{\sigma}\left(\mathbf{A}^{-1}\right)^{ji}\det\mathbf{A}+$
$\displaystyle+2\tilde{\tau}_{p}^{2}\partial_{i}[Ne^{-2\phi_{0}}\mathcal{V}^{2}g_{\mu\nu}\partial_{j}x^{\nu}\left(\mathbf{A}^{-1}\right)^{ji}_{S}\det\mathbf{A}]+\partial_{i}[N^{i}p_{\mu}]=$
$\displaystyle
2\frac{\tilde{\tau}_{p}}{T_{p-1}}af^{\prime}\mathcal{V}\partial_{\hat{i}}[\tilde{n}e^{-2\phi_{0}}g_{\mu\nu}\partial_{\hat{j}}\tilde{x}^{\nu}\tilde{\pi}^{\hat{j}}]-\frac{\tilde{\tau}_{p}}{T_{p-1}}af^{\prime}V\tilde{\pi}^{\hat{i}}\partial_{\hat{i}}\tilde{x}^{\rho}\partial_{\mu}g_{\rho\sigma}\partial_{\hat{j}}\tilde{x}^{\sigma}\tilde{\pi}^{\hat{j}}+$
$\displaystyle+2af^{\prime}\mathcal{V}\tilde{\tau}_{p}T_{p-1}\partial_{\hat{i}}[\tilde{n}e^{-2\phi_{0}}g_{\mu\nu}\partial_{\hat{j}}\tilde{x}^{\nu}\tilde{\mathbf{a}}^{\hat{j}\hat{i}}_{S}\det\mathbf{a}]-$
$\displaystyle-\frac{\tilde{\tau}_{p}}{T_{p-1}}\partial_{y}(af^{\prime}\mathcal{V})\partial_{\hat{i}}t\tilde{n}^{\hat{i}}\tilde{p}_{\mu}+\frac{\tilde{\tau}_{p}}{T_{p-1}}af^{\prime}\mathcal{V}\partial_{\hat{i}}[\tilde{n}^{\hat{i}}\tilde{p}_{\mu}]+\frac{\tilde{\tau}_{p}}{T_{p-1}}\partial_{y}(af^{\prime}\mathcal{V})(\tilde{n}^{\hat{i}}\partial_{\hat{i}}t-\partial_{0}t)\tilde{p}_{\mu}\
,$

where we used

$\displaystyle\partial_{i}[N\pi^{i}g_{\mu\nu}\partial_{j}x^{\nu}\pi^{j}]=\frac{\tilde{\tau}_{p}}{T_{p-1}}af^{\prime}\mathcal{V}\partial_{\hat{i}}[\tilde{n}g_{\mu\nu}\partial_{\hat{j}}\tilde{x}^{\nu}\tilde{\pi}^{\hat{j}}]$

and also

$\displaystyle\tilde{\tau}_{p}^{2}\partial_{i}[Ng_{\mu\nu}\partial_{j}x^{\nu}\left(\mathbf{A}^{-1}\right)^{ji}_{S}\det\mathbf{A}]=af^{\prime}\mathcal{V}\tilde{\tau}_{p}T_{p-1}\partial_{\hat{i}}[\tilde{n}g_{\mu\nu}\partial_{\hat{j}}\tilde{x}^{\nu}\tilde{\mathbf{a}}^{\hat{j}\hat{i}}_{S}\det\mathbf{a}]\
.$

Taking (47) together with (2) we obtain

$\displaystyle
af^{\prime}\mathcal{V}\left(-\partial_{0}\tilde{p}_{\mu}+\partial_{\hat{i}}[\tilde{n}g_{\mu\nu}\partial_{\hat{j}}\tilde{x}^{\nu}\tilde{\pi}^{\hat{j}}]-\tilde{\pi}^{\hat{i}}\partial_{\hat{i}}\tilde{x}^{\rho}\partial_{\mu}g_{\rho\sigma}\partial_{\hat{j}}\tilde{x}^{\sigma}\tilde{\pi}^{\hat{j}}+\right.$
$\displaystyle+\left.T_{p-1}^{2}e^{-2\phi_{0}}\partial_{\hat{i}}\tilde{x}^{\rho}\partial_{\mu}g_{\rho\sigma}\partial_{\hat{j}}\tilde{x}^{\nu}\tilde{\mathbf{a}}^{\hat{j}\hat{i}}\det\mathbf{a}+2T^{2}_{p-1}\partial_{\hat{i}}[\tilde{n}e^{-2\phi_{0}}g_{\mu\nu}\partial_{\hat{j}}\tilde{x}^{\nu}\tilde{\mathbf{a}}^{\hat{j}\hat{i}}_{S}\det\mathbf{a}]+\partial_{\hat{i}}[\tilde{n}^{\hat{i}}\tilde{p}_{\mu}]\right)=0$

and we see that the equation of motion for $p_{\mu}$ is obeyed at the core of
the kink on condition when the fluctuation fields around the kink solution
obey the equations of motion (2).

In summary, we have shown that the tachyon kink solution describes
D(p-1)-brane at the core of the kink when the kink is localized along
$y-$direction at the point $t(\xi)$. Note that $t(\xi^{\alpha})$ can be
completely arbitrary which is manifestation of the diffeomorphism invariance
of the world-volume theory. On the other hand non-trivial profile $t(\xi)$
rises a puzzle since it induces, according to the analysis presented above,
non-zero electric field equal to

$\pi^{y}=\frac{\tilde{\tau}_{p}}{T_{p-1}}af^{\prime}\mathcal{V}\tilde{\pi}^{\hat{i}}\partial_{\hat{i}}t\
.$ (52)

It is remarkable that when $t(\xi^{\hat{\alpha}})$ is constant then the
electric flux along $y-$ direction is zero. In fact, this electric flux is
localized around the core of the kink due to the presence of the factor
$af^{\prime}\mathcal{V}$ which is in agreement with the result derived in
[25].

##  3 Canonical Analysis $Dp-\overline{D}p$-brane action

In this section we perform canonical analysis of the action for Dp-brane anti-
Dp-brane system at least in its simplified form.

The first proposal for the action that describes this system was given by A.
Sen in [16]. It was suggested there that such an action has the form 333For
related works, see [19, 20, 21, 22, 23, 24].

$S=-\int d^{p+1}\xi
V(T,X_{(1)}-X_{(2)})(\sqrt{-\det\mathbf{A}_{(1)}}+\sqrt{-\det\mathbf{A}_{(2)}})\
,$ (53)

where

$\displaystyle\mathbf{A}_{(i)\mu\nu}=\eta_{\mu\nu}+\partial_{\mu}X^{I}_{(i)}\partial_{\nu}X^{I}_{(i)}+F_{\mu\nu}^{(i)}+\frac{1}{2}(D_{\mu}T)^{*}D_{\nu}T+\frac{1}{2}(D_{\nu}T)^{*}D_{\mu}T\
,$ $\displaystyle
F_{\mu\nu}^{(i)}=\partial_{\mu}A_{\nu}^{(i)}-\partial_{\nu}A_{\mu}^{(i)}\
,\quad D_{\mu}T=(\partial_{\mu}-iA_{\mu}^{(1)}+iA_{\mu}^{(2)})T\ ,$

where $(i)=(1),(2)$ where $(1)$ corresponds to the first Dp-brane while $(2)$
corresponds to the second one. Further, A. Sen suggested that the tachyon
potential has the form

$V(T,X_{(1)}-X_{(2)})=T_{p+1}\left[1+\frac{1}{2}\sum_{I}\frac{(X_{(1)}^{I}-X_{(2)}^{I})^{2}}{l_{s}^{2}})T^{2}+O(T^{4})\right]$
(55)

It is important to stress that these Dp-brane actions were formulated in the
static gauge when

$\xi^{\alpha}=X^{\alpha}_{(1)}=X^{\alpha}_{(2)}\ .$ (56)

Then it is natural to generalize this action to the manifestly diffeomorphism
invariant form when we write it as

$\displaystyle\mathbf{A}_{(i)\alpha\beta}=\partial_{\alpha}X^{M}_{(i)}\partial_{\beta}X^{N}_{(i)}\eta_{MN}+l_{s}^{2}F_{\alpha\beta}^{(i)}+\frac{1}{2}(D_{\alpha}T)^{*}D_{\beta}T+\frac{1}{2}(D_{\beta}T)^{*}D_{\alpha}T\
,$ $\displaystyle
V(T,X_{(1)}-X_{(2)})=T_{p}\left[1+\frac{1}{2}\sum_{M}\frac{(X^{M}_{(1)}-X^{M}_{(2)})^{2}}{l_{s}^{2}})T^{2}+O(T^{4})\right]\
,$

where
$(X^{M}_{(1)}-X^{M}_{(2)})^{2}=(X^{M}_{(1)}-X^{M}_{(2)})\eta_{MN}(X^{N}_{(1)}-X^{N}_{(2)})$
that in the static gauge (56) the potential given above reduces into (55).

It is very difficult to find canonical form of this action in the full
generality due to the fact that both determinants $\det\mathbf{A}_{(1)}$ and
$\det\mathbf{A}_{(2)}$ contain kinetic term for tachyon and we were not able
to find Hamiltonian constraint. For that reason we restrict ourselves to
simpler case when the gauge fields $A_{\alpha}^{(1)}$ and $A_{\alpha}^{(2)}$
vanish. We further presume that the separation of D-brane anti-D-brane pair is
small so that when we write $X^{M}_{(1)}=Y^{M}+\frac{l^{M}}{2}\
,X^{M}_{(2)}=Y^{M}-\frac{l^{M}}{2}$ we obtain

$\displaystyle\sqrt{-\det\mathbf{A}_{1}}+\sqrt{-\det\mathbf{A}_{(2)}}=\sqrt{-\det\mathbf{Y}}(2+\frac{1}{4}\mathbf{Y}^{\alpha\beta}\partial_{\alpha}l^{M}\partial_{\beta}l_{M})\
,$ (58)

where

$\mathbf{Y}_{\alpha\beta}=g_{\alpha\beta}+\partial_{\alpha}\rho\partial_{\beta}\rho+\rho^{2}\partial_{\alpha}\phi\partial_{\beta}\phi\
,$ (59)

and where we introduced following parametrization of $T$

$T=\rho e^{i\phi}\ .$ (60)

Note that the potential $V$ is now function of $\rho$ only.

To proceed further we introduce common notation $Z^{P}=(Y^{M},\rho,\phi)$ and
matrix $H_{PQ}=\mathrm{diag}(\eta_{MN},1,\rho^{2})$ so that the matrix
$\mathbf{Y}_{\alpha\beta}$ has the form
$\mathbf{Y}_{\alpha\beta}=H_{PQ}\partial_{\alpha}Z^{P}\partial_{\beta}Z^{Q}$.
In order to find canonical formulation of this theory we use ADM like
decomposition of the matrix $\mathbf{Y}_{\alpha\beta}$ in the form

$\mathbf{Y}_{\alpha\beta}=\left(\begin{array}[]{cc}-M^{2}+M_{i}m^{ij}M_{j}&M_{j}\\\
M_{i}&m_{ij}\\\ \end{array}\right)\
,\quad\mathbf{Y}^{\alpha\beta}=\left(\begin{array}[]{cc}-\frac{1}{M^{2}}&\frac{M^{j}}{M^{2}}\\\
\frac{M^{i}}{M^{2}}&m^{ij}-\frac{M^{i}M^{j}}{M^{2}}\\\ \end{array}\right)\ ,$
(61)

where $m^{ij}$ is matrix inverse to $m_{ij}$ so that
$m_{ij}m^{jk}=\delta_{i}^{k}$. Let us compare this notation with the
definition
$\mathbf{Y}_{\alpha\beta}=\partial_{\alpha}Z^{P}H_{PQ}\partial_{\beta}Z^{Q}$
and we obtain

$\displaystyle m_{ij}=\partial_{i}Z^{P}H_{PQ}\partial_{j}Z^{Q}\ ,\quad
M_{i}=\partial_{i}Z^{P}H_{PQ}\partial_{0}Z^{Q}\ ,$ $\displaystyle
M^{2}=-\partial_{0}Z^{P}V_{PR}\partial_{0}Z^{R}\ ,\quad
V_{PR}=H_{PR}-H_{PQ}\partial_{i}Z^{Q}m^{ij}\partial_{j}Z^{S}H_{SR}\ .$

With this notation the action has the form

$\displaystyle S=-\int d^{p+1}\xi VM\sqrt{\det
m_{ij}}(2-\frac{1}{4}\nabla_{m}l^{M}\nabla_{m}l_{M}+\frac{1}{4}m^{ij}\partial_{i}l^{M}\partial_{j}l_{M})\
,$ $\displaystyle\nabla_{m}l=\frac{1}{M}(\partial_{0}l-M^{i}\partial_{i}l)\ .$

Now we are ready to find conjugate momenta. From (3) we obtain momentum
conjugate to $l^{M}$ in the form

$p_{M}^{l}=\frac{\delta S}{\delta\partial_{0}l^{M}}=\frac{1}{2}V\sqrt{\det
m_{ij}}\eta_{MN}\nabla_{m}l^{N}\ .$ (64)

In case of the fields $Z^{P}$ we should be more careful since $M$ and $M_{i}$
depends on $\partial_{0}Z^{P}$. Explicitly, from (3) we obtain

$\frac{\partial
M}{\partial\partial_{0}Z^{P}}=-\frac{V_{PQ}\partial_{0}Z^{Q}}{N}\
,\quad\frac{\partial
M_{i}}{\partial\partial_{0}Z^{P}}=H_{PQ}\partial_{i}Z^{Q}\ .$ (65)

Then, after some calculations, we get

$\displaystyle\Pi_{P}\equiv(K_{P}+H_{PQ}\partial_{i}Z^{Q}m^{ij}\partial_{j}l^{M}p_{M}^{l})=$
$\displaystyle=V\frac{V_{PQ}\partial_{0}Z^{Q}}{M}\sqrt{\det
m_{ij}}(2-\frac{1}{V^{2}\det
m_{ij}}p^{l}_{M}p^{l}_{N}\eta^{MN}+\frac{1}{4}m^{ij}\partial_{i}l^{M}\partial_{j}l^{N}\eta_{MN})+$
$\displaystyle 2\frac{V}{M}V_{PQ}\partial_{0}Z^{Q}\frac{1}{V^{2}\det
m_{ij}}p_{M}^{l}p^{l}_{N}\eta^{MN}\ ,$

where $K_{P}$ is momentum conjugate to $Z^{P}$. To proceed further we use the
fact that

$\partial_{i}Z^{P}V_{PQ}=0\ ,\quad V_{PQ}H^{QR}V_{RT}=V_{RT}\ .$ (67)

Using the first relation given above we obtain

$\partial_{i}Z^{P}K_{P}=-\partial_{i}l^{M}p_{M}^{l}$ (68)

that implies following $p-$primary constraints

$\mathcal{H}_{i}\equiv\partial_{i}Z^{P}K_{P}+\partial_{i}l^{M}p_{M}^{l}\approx
0\ .$ (69)

Further, the second relation in (67) implies

$\displaystyle\Pi_{P}H^{PQ}\Pi_{Q}=-V^{2}\det
m_{ij}[4+m^{ij}\partial_{i}l^{M}\partial_{j}l_{N}]-4p_{M}^{l}p_{N}^{l}\eta^{MN}$

that implies an existence of Hamiltonian constraint

$\displaystyle\mathcal{H}_{\tau}=\Pi_{P}H^{PQ}\Pi_{Q}+4p^{M}_{l}p^{N}_{l}\eta_{MN}+V^{2}\det
m_{ij}[4+m^{ij}\partial_{i}l^{M}\partial_{j}l_{M}]\approx 0$

or explicitly

$\displaystyle\mathcal{H}_{\tau}=p_{M}\eta^{MN}p_{N}+p_{\rho}^{2}+\frac{1}{\rho^{2}}p_{\phi}^{2}+4p_{M}^{l}p_{N}^{l}\eta^{MN}+V^{2}\det
m_{ij}[4+m^{ij}\partial_{i}l^{M}\partial_{j}l_{M}]\approx 0\ ,$

where we used the fact that
$\partial_{i}Z_{P}K^{P}=\mathcal{H}_{i}-p_{M}^{l}\partial_{i}l^{M}$ and then
we neglected terms of the fourth order in $p_{M}^{l}$ and $l_{M}^{l}$. In
summary, we get Hamiltonian as sum of $p+1$ primary constraints

$H=\int d^{p}\xi\mathcal{H}\
,\quad\mathcal{H}=N\mathcal{H}_{\tau}+N^{i}\mathcal{H}_{i}\ .$ (73)

Using this Hamiltonian we can study dynamics of Dp-brane anti-Dp-brane system.
The canonical equations of motion for $Y^{M}$ and $p_{M}$ have the form

$\displaystyle\partial_{0}Y^{M}=\left\\{Y^{M},H\right\\}=2N\eta^{MN}p_{N}+N^{i}\partial_{i}Y^{M}\
,$
$\displaystyle\partial_{0}p_{M}=\left\\{p_{M},H\right\\}=2\partial_{i}[NV^{2}\eta_{MN}\partial_{j}Y^{N}m^{ji}\det
m_{mn}(4+m^{kl}\partial_{k}l^{M}\partial_{l}l_{M})]-$
$\displaystyle-2\partial_{k}[NV^{2}\det
m_{mn}m^{ik}\eta_{MN}\partial_{l}Y^{N}m^{lj}\partial_{i}l^{K}\partial_{j}l_{K}]+\partial_{i}(N^{i}p_{M})\
.$

In case of $\phi$ and $p_{\phi}$ we get

$\displaystyle\partial_{0}\phi=\left\\{\phi,H\right\\}=2\frac{N}{\rho^{2}}p_{\phi}+N^{i}\partial_{i}\phi\
,$
$\displaystyle\partial_{0}p_{\phi}=\left\\{p_{\phi},H\right\\}=\partial_{i}(N^{i}p_{\phi})+$
$\displaystyle+2\partial_{i}[NV^{2}\partial_{j}\phi m^{ji}\det
m_{mn}(4+m^{kl}\partial_{k}l^{M}\partial_{l}l_{M})]-$
$\displaystyle-2\partial_{k}[NV^{2}\det m_{mn}m^{ik}\partial_{l}\phi
m^{lj}\partial_{i}l^{K}\partial_{j}l_{K}]\ .$

The situation is slightly more involved in case of $\rho$ and $p_{\rho}$ when
we have

$\displaystyle\partial_{0}\rho=\left\\{\rho,H\right\\}=2Np_{\rho}+N^{i}\partial_{i}\rho\
,$
$\displaystyle\partial_{0}p_{\rho}=\left\\{p_{\rho},H\right\\}=4\frac{N}{\rho^{3}}p_{\phi}^{2}-2V\frac{dV}{d\rho}\det
m_{ij}[4+m^{ij}\partial_{i}l^{M}\partial_{j}l_{M}]+$
$\displaystyle+2\partial_{i}[NV^{2}\partial_{j}p_{\rho}m^{ji}\det
m_{mn}(4+m^{kl}\partial_{k}l^{M}\partial_{l}l_{M})]-$
$\displaystyle-2\partial_{k}[NV^{2}\det m_{mn}m^{ik}\partial_{l}\rho
m^{lj}\partial_{i}l^{K}\partial_{j}l_{K}]+\partial_{i}(N^{i}p_{M})-$
$\displaystyle-2NV^{2}\det m_{mn}\partial_{i}\phi\partial_{j}\phi
m^{ji}(4+m^{kl}\partial_{k}l^{M}\partial_{l}l_{M})+$
$\displaystyle+2NV^{2}\det m_{mn}m^{ik}\rho\partial_{k}\phi\partial_{l}\phi
m^{lj}\partial_{i}l^{M}\partial_{j}l_{M}+\partial_{i}(N^{i}p_{\rho})\ .$

Finally the equations of motion for $l^{M}$ and $p_{M}^{l}$ have the form

$\displaystyle\partial_{\tau}l^{M}=\left\\{l^{M},H\right\\}=8N\eta^{MN}p_{N}^{l}+N^{i}\partial_{i}l^{M}\
,$
$\displaystyle\partial_{\tau}p_{M}^{l}=\left\\{p_{M}^{l},H\right\\}=-2V\frac{\partial
V}{\partial l^{M}}\det
m_{ij}[4+m^{ij}\partial_{i}l^{M}\partial_{j}l^{M}\eta_{MN}]+$
$\displaystyle+2\partial_{i}[V^{2}\det
m_{mn}m^{ij}\eta_{MN}\partial_{j}l^{N}]+\partial_{i}(N^{i}p_{M}^{l})\ .$

Let us now solve these equations. First of all we impose static gauge

$Y^{\alpha}=\xi^{\alpha}\ ,\quad l^{\alpha}=0\ ,\quad\alpha=0,1,\dots,p\ .$
(78)

Let us also presume that remaining fields depend on $\xi^{0}$ only. Then the
spatial diffeomorphism constraints imply

$\mathcal{H}_{i}=p_{i}=0\ ,i=1,\dots,p\ .$ (79)

Further, (78) and the fact that $\rho$ and $\phi$ depend on $\xi^{0}$ only
implies that $m_{ij}$ is $p\times p$ diagonal matrix

$m_{ij}=\partial_{i}Y^{M}\eta_{MN}\partial_{j}Y^{N}=\delta_{ij}\ .$ (80)

Let us now consider equation of motion for $Y^{0}$

$\partial_{0}Y^{0}=-2Np_{0}$ (81)

that using (78) fixes $N$ to be equal to $N=-\frac{1}{2p_{0}}$. Further,
inserting (78) into equation of motion for $Y^{i}$ and using $p_{i}=0$ we
obtain that $N^{i}=0$. Finally, equations of motion for $p_{0}$ implies

$\displaystyle\partial_{0}p_{0}=0\ $ (82)

and hence $p_{0}$ is constant.

The equations of motion for $Y^{K},K=p+1,\dots,9$ and $p_{K}$ have the form

$\displaystyle\partial_{0}Y^{K}=\left\\{Y^{K},H\right\\}=2N\delta^{KL}p_{L}=-\frac{1}{p_{0}}\delta^{KL}p_{L}\
,$ $\displaystyle\partial_{0}p_{K}=\left\\{p_{K},H\right\\}=0\ ,$

due to the fact that space-time metric does not depend on $Y^{K}$. Solving (3)
we get

$Y^{K}=-\frac{1}{2p_{0}}\delta^{KL}p_{L}^{c}\xi^{0}+Y^{K}_{0}\ ,\quad
p_{K}^{c}=\mathrm{const}\ $ (84)

and this solution corresponds to the free motion of the the center of mass
coordinates $Y^{K}$ in the transverse space.

Following [24] we write the potential $V(T,X)$ as

$V(T,l)=\mathcal{V}(\rho)\sqrt{1+\frac{\rho^{2}}{l_{s}^{2}}l^{M}l^{N}\eta_{MN}}\
,$ (85)

where $\mathcal{V}(\rho)$ has the same form as potential on the world-volume
of unstable Dp-brane.

In case of $\phi$ and $p_{\phi}$ we obtain

$\displaystyle\partial_{\tau}\phi=\frac{2N}{\rho^{2}}p_{\phi}\
,\quad\partial_{\tau}p_{\phi}=0\ .$

This equation shows that $p_{\phi}$ is constant that we choose to be equal to
zero. Then $\phi=\mathrm{const}$ too.

Finally we consider equations of motion for $l^{M}$. We see that an
interesting physics is hidden in the dynamics of $l^{M}$ and $\rho$ due to the
fact that the potential is function of these variables. In case of $l^{M}$ we
obtain

$\displaystyle\partial_{0}l^{I}=8N\delta^{IJ}p_{J}^{l}\
,\quad\partial_{0}p^{l}_{I}=-8\frac{N}{l_{s}^{2}}\mathcal{V}^{2}\rho^{2}\delta_{IJ}l^{J}$

that together give following differential equation for $l_{I}$

$\partial_{0}^{2}l_{I}=\frac{16}{p^{2}_{0}}\mathcal{V}^{2}\rho^{2}\delta_{IJ}l^{J}\
.$ (88)

Further, the equation of motion for $\rho$ and $p_{\rho}$ have the from

$\displaystyle\partial_{0}\rho=2Np_{\rho}\ ,$
$\displaystyle\partial_{0}p_{\rho}=-8N\mathcal{V}\frac{d\mathcal{V}}{d\rho}(1+\frac{\rho^{2}}{l_{s}^{2}}l^{I}l_{I})-8N\mathcal{V}^{2}\frac{\rho}{l_{s}^{2}}l^{I}l_{I}\
.$ (89)

Clearly these equations of motion have solution when $l^{I}=0$ so that the
tachyon obeys the equation

$\partial^{2}_{0}\rho=-\frac{4}{p_{0}^{2}}\mathcal{V}\frac{d\mathcal{V}}{d\rho}\
.$ (90)

that corresponds to standard rolling tachyon solution. Another interesting
situation occurs when we consider the tachyon at the minimum of the potential
$\mathcal{V}(\rho)$. Since, according to [24] $\mathcal{V}(\rho)$ has minimum
at $\rho_{min}\pm\infty$ and $\mathcal{V}(\rho)\sim e^{-\rho}$ for large
$\rho$ we get that $\lim_{\rho\rightarrow\rho_{min}}V\rho=0$. Then the
equations of motion for $l^{I}$ have the form

$\partial_{0}^{2}l_{I}=0$ (91)

that has the solution $l_{I}=k_{I}\xi^{0}+l_{I}^{0}$ which of course is valid
for small values of $\xi^{0}$ since in the opposite case we would get out of
the region of small $l^{I}$ where our approximation is valid. In fact, more
general analysis of the dynamics of D-brane anti D-brane system can be found
in [24].

In summary, we performed canonical analysis of the Dp-brane anti-Dp-brane
systems in the case of the vanishing gauge fields and in the approximation of
the small separation of these objects. It would be certainly consider more
general case and also it would be interesting to find covariant form of the
action for Dp-brane anti Dp-brane system that was suggested in [21]. This
problem is currently under investigation.

Acknowledgement:
This work is supported by the grant “Integrable Deformations” (GA20-04800S)
from the Czech Science Foundation (GACR).

## References

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  * [2] A. Sen, _“NonBPS states and Branes in string theory,”_ [arXiv:hep-th/9904207 [hep-th]].
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# Dynamics of a Rotated Orthogonal Gravitational Wedge Billiard

K. D. Anderson

###### Abstract

We investigate a rotated, orthogonal gravitational wedge billiard—a special
case of the asymmetric gravitational wedge billiard—in which the dynamics are
integrable. We derive equations and conditions under which periodic orbits may
be constructed for this model, and show that any other trajectory will be
dense in the configuration space.

##  1 Introduction

Based on previous work on the asymmetric wedge billiard [1, 2], we now
investigate a special case of the asymmetric wedge billiard which leads to
integrable dynamics.

For this special case, we set $\theta_{1}=\theta$ and
$\theta_{2}=\pi/2-\theta$ in the asymmetric wedge billiard [1, 2]. This
corresponds to an orthogonal wedge which is rotated by an angle $\theta$ from
the vertical, see Figure 1.

Figure 1: The rotated orthogonal wedge billiard.

This work generalises the work done by Lehtihet and Miller [3], Richter et.
al. [4], and Szeredi [5, 6] on the symmetric orthogonal gravitational wedge
billiard. For that specific model, one can show [1] the existence of a
period-1 orbit in the billiard, illustrated in Figure 2, which corresponds to
fixed-point solutions of the collision maps.

(a)

(b)

Figure 2: Periodic orbits of the symmetric orthogonal gravitational wedge
billiard, represented in configuration space $(x,y)$.

##  2 Model

###  2.1 Geometry

We consider the motion of a point particle of mass $m$ within a constant
gravitational field $\bm{g}$, its motion restricted to the regions

$\displaystyle\mathcal{Q}_{A}$
$\displaystyle=\left\\{(x,y)\in\mathbb{R}^{2}:x\geq 0,\;y\geq
x\cot(\theta)\right\\},$ $\displaystyle\mathcal{Q}_{B}$
$\displaystyle=\left\\{(x,y)\in\mathbb{R}^{2}:x<0,\;y\geq
x\tan(\theta)\right\\},$

with corresponding boundaries

$\displaystyle\partial\mathcal{Q}_{A}$
$\displaystyle=\left\\{(x,y)\in\mathbb{R}^{2}:x\geq
0,\;y=x\cot(\theta)\right\\},$ $\displaystyle\partial\mathcal{Q}_{B}$
$\displaystyle=\left\\{(x,y)\in\mathbb{R}^{2}:x\leq
0,\;y=x\tan(\theta)\right\\},$

as illustrated in Figure 3.

Figure 3: Regions of motion and boundaries for the rotated orthogonal wedge
billiard.

The real-valued variable $\theta$ represents the angle measured clockwise from
the vertical and may take values on the interval $(0,\pi/2)$.

We shall call the set
$\partial\mathcal{Q}\coloneqq\partial\mathcal{Q}_{A}\cup\partial\mathcal{Q}_{B}$
the _rotated orthogonal wedge_. The boundary $\partial\mathcal{Q}_{A}$ shall
be called the _right-hand slope_ or _right-hand wall_ of the wedge; similarly,
the boundary $\partial\mathcal{Q}_{B}$ shall be called the _left-hand slope_
or _left-hand wall_ of the wedge. The intersection of the two boundaries
$\partial\mathcal{Q}_{A}\cap\partial\mathcal{Q}_{B}$ shall be called the
_wedge vertex_. The region
$\mathcal{Q}\coloneqq\mathcal{Q}_{A}\cup\mathcal{Q}_{B}$ is called the _region
of allowed motion_.

We introduce an inertial Cartesian reference system such that the origin $O$
is fixed at the wedge vertex, the reference axes with unit vectors
$\bm{e}_{x}=\begin{bmatrix}1&0\end{bmatrix}^{T}$ and
$\bm{e}_{y}=\begin{bmatrix}0&1\end{bmatrix}^{T}$ are orthogonal to each other
and directed along the horizontal and vertical, respectively, as illustrated
in Figure 4. We shall denote by $\mathcal{B}_{C}$ the set
$\\{\bm{e}_{x},\bm{e}_{y}\\}$.

Figure 4: Reference systems used in our study of the rotated orthogonal wedge
billiard. (Colour online)

Of import in later sections, will be the additional reference systems
$\mathcal{B}_{R}=\left\\{\bm{\bar{e}}_{r},\bm{\bar{e}}_{\varphi}\right\\}$
with origin coinciding with the particle, and
$\mathcal{B}_{W}=\left\\{\bm{\tilde{e}}_{r},\bm{\tilde{e}}_{\theta}\right\\}$
with origin coinciding with that of $\mathcal{B}_{C}$, as illustrated in
Figure 4. The former, introduced by Lehtihet and Miller [3], is dynamic and
changes during the motion of the particle; while the latter, introduced by
Szeredi [6], is fixed with its reference vectors coinciding with the wedge
walls.

###  2.2 Mechanics

Let $t$ be the variable representing time and let
$\bm{q}\coloneqq\bm{q}(t)\in\mathcal{Q}$ represent the position vector at some
time $t$, and let $\bm{p}\coloneqq\bm{p}(t)\in\mathbb{R}^{2}$ represent the
momentum vector of the particle at some time $t$. In the Cartesian reference
system, under a transformation to dimensionless quantities [1], we have the
coordinate vectors $[\bm{g}]_{C}=\begin{bmatrix}0&-1\end{bmatrix}^{T}$,
$[\bm{q}]_{C}=\begin{bmatrix}x&y\end{bmatrix}^{T}$ and
$[\bm{p}]_{C}=\begin{bmatrix}u&w\end{bmatrix}^{T}$. The corresponding
Hamiltonian function is

$H(x,y,u,w)=\frac{u^{2}+w^{2}}{2}+y.$ (1)

Integrating the Hamiltonian equations of motion, derived from the Hamiltonian
(1), we find that the particle moves along a parabolic path until it collides
with either of the two walls $\partial\mathcal{Q}_{j}$ ($j=\\{A,B\\}$). These
collisions are assumed to be elastic and obey the reflection law. Since the
motion between collisions is completely determined, we focus only on the
collision points themselves. The collision maps simplify remarkably in the
$\mathcal{B}_{R}$ reference system. For successive collisions on
$\partial\mathcal{Q}_{A}$, we have the map
$F_{A}:\partial\mathcal{Q}_{A}\to\partial\mathcal{Q}_{A}$ given by

$\displaystyle\bar{u}_{j+1}$
$\displaystyle=\bar{u}_{j}-2\bar{w}_{j}\cot(\theta),$ (2a)
$\displaystyle\bar{w}_{j+1}^{2}$ $\displaystyle=\bar{w}_{j}^{2}.$ (2b)

Similarly, for successive collisions on $\partial\mathcal{Q}_{B}$, we have the
map $G_{A}:\partial\mathcal{Q}_{B}\to\partial\mathcal{Q}_{B}$ given by

$\displaystyle\bar{u}_{j+1}$
$\displaystyle=\bar{u}_{j}+2\bar{w}_{j}\tan(\theta),$ (3a)
$\displaystyle\bar{w}_{j+1}^{2}$ $\displaystyle=\bar{w}_{j}^{2}.$ (3b)

For a collision with the opposite wall $\partial\mathcal{Q}_{B}$ for the
particle starting on $\partial\mathcal{Q}_{A}$, we have the map
$F_{B}:\partial\mathcal{Q}_{A}\to\partial\mathcal{Q}_{B}$ given by

$\displaystyle\bar{u}_{j+1}$
$\displaystyle=\bar{w}_{j}-\left(\bar{u}_{j}+\bar{w}_{j+1}\right)\tan(\theta),$
(4a) $\displaystyle\bar{w}_{j+1}^{2}$ $\displaystyle=2E-\bar{w}_{j}^{2}.$ (4b)

Similarly, for a collision with $\partial\mathcal{Q}_{A}$ for the particle
starting on $\partial\mathcal{Q}_{B}$, we have the map
$G_{B}:\partial\mathcal{Q}_{B}\to\partial\mathcal{Q}_{A}$ given by

$\displaystyle\bar{u}_{j+1}$
$\displaystyle=-\bar{w}_{j}-\left(\bar{u}_{j}-\bar{w}_{j+1}\right)\cot(\theta)$
(5a) $\displaystyle\bar{w}_{j+1}^{2}$ $\displaystyle=2E-\bar{w}_{j}^{2}.$ (5b)

###  2.3 One-dimensional approximation of motion

We now consider the transformation of the Hamiltonian from the Cartesian
reference system $\mathcal{B}_{C}$ to the reference system $\mathcal{B}_{W}$.
(The transformations between the different reference systems is detailed in
Appendix A.)

Let $\bm{q}=\begin{bmatrix}x&y\end{bmatrix}^{T}$ and
$\bm{p}=\begin{bmatrix}u&w\end{bmatrix}^{T}$ denote the coordinate vectors in
$\mathcal{B}_{C}$, as previously mentioned. We shall denote the corresponding
position and momentum coordinate vectors in $\mathcal{B}_{W}$ by
$\bm{\tilde{q}}=\begin{bmatrix}\tilde{x}&\tilde{y}\end{bmatrix}^{T}$ and
$\bm{\tilde{p}}=\begin{bmatrix}\tilde{u}&\tilde{w}\end{bmatrix}^{T}$.

Rewriting the Hamilton function (1) in terms of the $\mathcal{B}_{W}$
coordinates yields

$\tilde{H}=H(\tilde{x},\tilde{y},\tilde{u},\tilde{w})=\frac{\tilde{u}^{2}}{2}+\tilde{x}\cos(\theta)+\frac{\tilde{w}^{2}}{2}+\tilde{y}\sin(\theta).$
(6)

Define

$\tilde{H}_{\tilde{x}}\coloneqq\frac{\tilde{u}^{2}}{2}+\tilde{x}\cos(\theta),\quad\tilde{H}_{\tilde{y}}\coloneqq\frac{\tilde{w}^{2}}{2}+\tilde{y}\sin(\theta)$
(7)

and note that both $H_{\tilde{x}}$ and $H_{\tilde{y}}$ are Hamiltonians
associated with the one dimensional motion of a particle in a (rotated)
gravitational field. Thus, if we suppose that $\tilde{y}$, $\tilde{w}$ is
small enough, which corresponds to particle motion very close to
$\partial\mathcal{Q}_{A}$, then $H_{\tilde{y}}\to 0$ and $\tilde{H}\approx
H_{\tilde{x}}$. A similar argument holds for $H_{\tilde{y}}$ for very small
$\tilde{x},\ \tilde{u}$.

The equations of motion for the particle very close to
$\partial\mathcal{Q}_{A}$ are

$\displaystyle\tilde{x}(t)$
$\displaystyle=t\tilde{u}_{0}-\frac{t^{2}\cos(\theta)}{2},$ (8a)
$\displaystyle\tilde{u}(t)$ $\displaystyle=\tilde{u}_{0}-t\cos(\varphi)$ (8b)

and for the particle very close to $\partial\mathcal{Q}_{B}$ are

$\displaystyle\tilde{y}(t)$
$\displaystyle=t\tilde{w}_{0}-\frac{t^{2}\sin(\theta)}{2},$ (9a)
$\displaystyle\tilde{w}(t)$ $\displaystyle=\tilde{w}_{0}-t\sin(\theta).$ (9b)

From the Hamiltonians (7) we may derive bounds on the trajectories in the
configuration space:

$0\leq\tilde{x}(t)\leq\frac{E}{\cos(\theta)},\quad
0\leq\tilde{y}(t)\leq\frac{E}{\sin(\theta)},$ (10)

where $E$ is the constant energy fixed at the start of the particle’s motion.

##  3 Dynamics

###  3.1 Fixed points of the collision maps

The maps $F_{A}$ and $G_{A}$ have the family of fixed points
$(\bar{u}^{*},\bar{w}^{*})=(c,0)$ where $c\in\mathbb{R}$ is a constant. This
corresponds to the particle sliding up ($c>0$) or down ($c<0$) either
$\partial\mathcal{Q}_{A}$ or $\partial\mathcal{Q}_{B}$

For the maps $F_{B}$ and $G_{B}$, we obtain

$\bar{u}^{*}=\sqrt{E}\left(\frac{1-\tan(\theta)}{1+\tan(\theta)}\right),\qquad\bar{w}^{*}=\sqrt{E},$
(11)

and

$\bar{u}^{*}=\sqrt{E}\left(\frac{\cot(\theta)-1}{\cot(\theta)+1}\right),\qquad\bar{w}^{*}=\sqrt{E},$
(12)

respectively. In the symmetric wedge billiard, these two fixed points are
identical and correspond to a period one trajectory, as illustrated by Figure
2a. To obtain a similar period one trajectory for the rotated orthogonal wedge
billiard, we would need to reflect the momentum component across both
$\bar{\bm{e}}_{r}$ and $\bar{\bm{e}}_{\varphi}$ at the collision point, and
set it equal to (11), which results in
$\begin{bmatrix}\bar{u}^{\prime}&\bar{w}^{\prime}\end{bmatrix}^{T}=\begin{bmatrix}-\bar{u}&-\bar{w}\end{bmatrix}^{T}.$
However, such a reflection is only possible when $\theta=\pi/4$.

###  3.2 Periodic orbits

Using equations (8b) and (9b), we may define the following time maps for
collisions with the wedge walls in the one-dimensional approximation:

$t_{j+1}^{A}=\frac{2\tilde{u}(t_{j})}{\cos(\theta)}=2jT_{A},\quad
T_{A}\coloneqq\frac{\tilde{u}_{0}}{\cos(\theta)}$ (13)

and

$t_{k+1}^{B}=\frac{2\bar{w}(t_{k})}{\sin(\theta)}=2kT_{B},\quad
T_{B}\coloneqq\frac{\tilde{w}_{0}}{\sin(\theta)},$ (14)

with $j,k\in\left\\{0,1,2,\dotsc\right\\}$. The trajectories of the particle’s
motion will be dense if $T_{A}/T_{B}$ is irrational, illustrated in Figure 5.

(a) $\theta=50^{\circ}$

(b) $\theta=60^{\circ}$

(c) $\theta=70^{\circ}$

Figure 5: Dense trajectories in the rotated orthogonal wedge billiard for
various $\theta$, $\phi=90^{\circ}$, $\bar{u}_{0}=0$ and $\bar{w}_{0}=1$,
after $50$ collisions.

The ratio $T_{A}/T_{B}$ is rational for

$\theta^{*}=\arctan\left(\frac{p}{q}\right),$ (15)

where $p,q\in\mathbb{N}$ such that $p>0$, $q>0$ and $\gcd(p,q)=1$. To prove
this, note that

$\frac{T_{A}}{T_{B}}=\frac{\tilde{u}_{0}}{\tilde{w}_{0}}\tan(\theta)$

which may be simplified to

$\frac{T_{A}}{T_{B}}=\tan(\theta)$ (16)

if we make the subsitutions $\tilde{u}_{0}=\bar{u}_{0}$ and
$\tilde{w}_{0}=\bar{u}_{0}$ for the particle close to
$\partial\mathcal{Q}_{A}$ and $\partial\mathcal{Q}_{B}$ respectively. If we
substitute $\theta^{*}$ into equation (16), then

$\frac{T_{A}}{T_{B}}=\tan(\theta^{*})=\tan\left(\arctan\left(\frac{p}{q}\right)\right)=\frac{p}{q}$

and $T_{A}/T_{B}\in\mathbb{Q}$. The restriction that $p$ and $q$ be positive
follows from the restriction on the allowed values for $\theta$, that is,
$\theta\in(0,\pi/2)$.

If $p=q$, then $\theta^{*}=\pi/4$ which corresponds to the symmetric
orthogonal wedge billiard. The fixed points of the collision maps (4) and (5)
become identically $(\bar{u}_{*},\bar{w}_{*}^{2})=(0,E)$, which correspond to
the period-$1$ orbit in the symmetric orthogonal wedge billiard, as previously
mentioned.

For $p\neq q$, we either have $p>q$, from which follows that $p/q>1$ and
$\theta^{*}>\pi/4$, or $p<q$, from which follows that $p/q<1$ and
$\theta^{*}<\pi/4$.

Substituting $\theta^{*}$ into the fixed point equations (11) and (12) yields
the identical expression

$\bar{u}^{*}=\sqrt{E}\left(\frac{q-p}{q+p}\right),\qquad\bar{w}^{*}=\sqrt{E}.$
(17)

For a particle starting on $\partial\mathcal{Q}_{A}$, using (17) will yield a
periodic orbit of period $p+q$ with $p$ collisions on
$\partial\mathcal{Q}_{A}$ and $q$ collisions on $\partial\mathcal{Q}_{B}$. The
number of collisions per side is determined from

$\frac{T_{A}}{T_{B}}=\frac{p}{q}=\frac{p\bar{u}_{0}}{\cos(\theta^{*})}\,\frac{\sin(\theta^{*})}{q\bar{u}_{0}}=\frac{p\tilde{u}_{0}}{\cos(\theta^{*})}\,\frac{\sin(\theta^{*})}{q\tilde{w}_{0}}$

which leads to

$t_{p+1}^{A}=2pT_{A},\quad t_{q+1}^{B}=2qT_{B}$

in equations (13) and (14). Some examples of periodic orbits are illustrated
in Figure 6.

(a)  $p=1$, $q=2$

(b)  $p=1$, $q=3$

(c)  $p=3$, $q=1$

(d)  $p=2$, $q=3$

(e)  $p=2$, $q=5$

Figure 6: Periodic trajectories for a particle launced from
$\partial\mathcal{Q}_{A}$.

Numeric simulations show that these periodic orbits are sensitive with respect
to the initial conditions (17)—a small perturbation $\bar{u}^{*}+\varepsilon$
leads to dense orbits once more.

We also plotted the values of $\theta$ versus $\bar{u}$ for various $p$, $q$
in Figure 7. Figure 7a indicate a rotational symmetry in $(\theta,\bar{u})$
space, which allows us to focus on $\theta$-values in $(0,\pi/4]$ generated
from suitable $p$, $q$ values. Of interest are the “windows” which appear for
various critical initial $\theta$, we were unable to determine any special
relationship between these different $\theta$ values with respect each other.

(a) $p,q\in\\{1,2,3,\dotsc,25\\}$

(b) $p\in\\{1,2,3,\dotsc,25\\},\ q\in\\{p+1,\dotsc,25\\}$

Figure 7: A plot of the $\bar{u}$ component of a periodic trajectory versus
$\theta$.

##  4 Conclusion

We studied a special case of the asymmetric gravitational wedge billiard
previously introduced by the author [1, 2]. This extends work on the symmetric
orthogonal gravitational wedge billiard previously studied by Lehtihet and
Miller [3], Richter et. al. [4], and Szeredi [5, 6].

We derived the conditions under which periodic orbits as well as dense, non-
periodic orbits are found. For periodic orbits, we derived an explicit formula
for the wedge angle and initial components of the momentum which guarantees a
periodic orbit of certain period and known number of collisions on each wall
of the wedge. However, numeric simulations show that these periodic orbits are
unstable with respect to initial conditions, a small perturbation
$\bar{u}+\varepsilon$ leads to dense orbits once more. We were also unable to
reproduce Figure 2b using the formula derived for periodic trajectories. The
question is whether this is a degenerate case of the orbit presented in Figure
2a or not.

## Acknowledgements

This work extends work done in the author’s doctoral thesis [1]. The author
wishes to thank the University of Johannesburg for the financial assistance
and opportunity afforded to pursue the doctorate.

##  Appendix A Coordinate transformations

In this section, we present the technical details of transformations between
the various reference systems introduced in section 2.1. Note the time-
dependence of the angle $\varphi$ in Figure 4—this angle changes as the
particle moves, whereas $\theta$ remains fixed during the motion. Consider the
rotation matrix

$R_{\varphi}=\begin{bmatrix}\cos(\varphi)&-\sin(\varphi)\\\
\sin(\varphi)&\cos(\varphi)\end{bmatrix}$

and note that

$\displaystyle\begin{bmatrix}\bm{\bar{e}}_{r}&\bm{\bar{e}}_{\varphi}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\bm{e}_{x}&\bm{e}_{y}\end{bmatrix}\begin{bmatrix}\cos(\varphi)&-\sin(\varphi)\\\
\sin(\varphi)&\cos(\varphi)\end{bmatrix}=\begin{bmatrix}\bm{e}_{x}&\bm{e}_{y}\end{bmatrix}R_{\varphi},$
$\displaystyle\begin{bmatrix}\bm{\tilde{e}}_{r}&\bm{\tilde{e}}_{\theta}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\bm{e}_{x}&\bm{e}_{y}\end{bmatrix}\begin{bmatrix}\sin(\theta)&-\cos(\theta)\\\
\cos(\theta)&\sin(\theta)\end{bmatrix}=\begin{bmatrix}\bm{e}_{x}&\bm{e}_{y}\end{bmatrix}R_{\pi/2-\theta},$
$\displaystyle\begin{bmatrix}\bm{\tilde{e}}_{r}&\bm{\tilde{e}}_{\theta}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\bm{\bar{e}}_{r}&\bm{\bar{e}}_{\varphi}\end{bmatrix}\begin{bmatrix}\sin(\theta+\varphi)&-\cos(\theta+\varphi)\\\
\cos(\theta+\varphi)&\sin(\theta+\varphi)\end{bmatrix}=\begin{bmatrix}\bm{\bar{e}}_{r}&\bm{\bar{e}}_{\varphi}\end{bmatrix}R_{\varphi}^{T}R_{\pi/2-\theta}$

If we denote the (dimensionless) components of the position and momentum
vectors in the different reference systems analogously, that is,

$\displaystyle[\bm{q}]_{C}=\begin{bmatrix}x\\\
y\end{bmatrix},\quad[\bm{p}]_{C}=\begin{bmatrix}u\\\
w\end{bmatrix},\quad[\bm{q}]_{R}=\begin{bmatrix}\bar{x}\\\
\bar{y}\end{bmatrix},\quad[\bm{p}]_{R}=\begin{bmatrix}\bar{u}\\\
\bar{w}\end{bmatrix},$ $\displaystyle[\bm{q}]_{W}=\begin{bmatrix}\tilde{x}\\\
\tilde{y}\end{bmatrix},\quad[\bm{p}]_{W}=\begin{bmatrix}\tilde{u}\\\
\tilde{y}\end{bmatrix},$

then the transformations of the components between the various reference
systems are

$\displaystyle[\bm{q}]_{R}$
$\displaystyle=R_{\varphi}^{T}[\bm{q}]_{C}=R^{T}_{\varphi}R_{\pi/2-\theta}[\bm{q}]_{W},$
$\displaystyle[\bm{p}]_{R}$
$\displaystyle=R_{\varphi}^{T}[\bm{p}]_{C}=R^{T}_{\varphi}R_{\pi/2-\theta}[\bm{p}]_{W},$
$\displaystyle[\bm{q}]_{W}$
$\displaystyle=R_{\pi/2-\theta}^{T}[\bm{q}]_{C}=\mathbb{R}^{T}_{\pi/2-\theta}R_{\varphi}[\bm{q}]_{R}$
$\displaystyle[\bm{p}]_{R}$
$\displaystyle=R_{\pi/2-\theta}^{T}[\bm{p}]_{C}=R^{T}_{\pi/2-\theta}R_{\varphi}[\bm{p}]_{R}.$

Also note that

$[\bm{g}]_{R}=-\begin{bmatrix}\sin(\varphi)\\\
\cos(\varphi)\end{bmatrix},\quad[\bm{g}]_{W}=-\begin{bmatrix}\cos(\theta)\\\
\sin(\theta)\end{bmatrix}.$

For the particle on $\partial\mathcal{Q}_{A}$, $\varphi=\pi/2-\theta$ and

$\begin{bmatrix}\bm{\tilde{e}}_{r}&\bm{\tilde{e}}_{\theta}\end{bmatrix}=\begin{bmatrix}\bm{\bar{e}}_{r}&\bm{\bar{e}}_{\varphi}\end{bmatrix}\begin{bmatrix}1&0\\\
0&1\end{bmatrix}=\begin{bmatrix}\bm{\bar{e}}_{r}&\bm{\bar{e}}_{\varphi}\end{bmatrix},\qquad\begin{bmatrix}\tilde{u}\\\
\tilde{w}\end{bmatrix}=\begin{bmatrix}\bar{u}\\\ \bar{w}\end{bmatrix}.$

For the particle on $\partial\mathcal{Q}_{B}$, $\varphi=\pi-\theta$ and

$\begin{bmatrix}\bm{\tilde{e}}_{r}&\bm{\tilde{e}}_{\theta}\end{bmatrix}=\begin{bmatrix}\bm{\bar{e}}_{r}&\bm{\bar{e}}_{\varphi}\end{bmatrix}\begin{bmatrix}0&1\\\
-1&0\end{bmatrix}=\begin{bmatrix}-\bm{\bar{e}}_{\varphi}&\bm{\bar{e}}_{r}\end{bmatrix},\qquad\begin{bmatrix}\tilde{u}\\\
\tilde{w}\end{bmatrix}=\begin{bmatrix}-\tilde{w}\\\ \tilde{u}\end{bmatrix}.$

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