# A Curious Use of Extra Dimension in Classical Mechanics

Trung V. Phan<EMAIL_ADDRESS>Department of Physics, Princeton
University, Princeton, NJ 08544, USA  Anh Doan  Department of Astronomy and
Astrophysics, Pennsylvania State University, University Park, PA 16802, USA

###### Abstract

Extra dimensions can be utilized to simplify problems in classical mechanics,
offering new insights. Here we show a simple example of how the motion of a
test particle under the influence of an 1D inverse-quadratic potential is
equivalent to that of another test particle moving freely in 2D Euclidean
space and 3D Minkowskian space.

##  I Introduction

Broadly speaking, physics questions are often defined by or complicated by
their dimensions. On the surface, there seems to be a trend that physical
questions become more difficult as the dimensionality increases. For example,
collisions in two dimensions are harder to deal with than in one dimension,
because the velocity becomes a two-dimensional vector not a scalar. landau ;
morin_mechanics  Rigid body rotation in three dimensions is much more
complicated than in two dimensions, since the angular velocity becomes a
three-dimensional vector not a scalar. landau ; morin_mechanics  In quantum
mechanics, any potential-well in one dimension and two dimensions has at least
one bound state, but that claim is no longer correct in three dimensions.
yang_1D_2D ; chadan_1D_2D  However, many findings in modern theoretical
physics indicate that it is also possible to simplify problems by adding more
dimensions. In the theory of general relativity, electromagnetism and gravity
can be unified by adding an extra compact dimension. kaluza_unify ;
klein_unify  In condensed matter physics, quasicrystals can be treated as
projections of a higher-dimensional lattice. kramer_periodic ; levine_new ;
kalugin_6d  Finally, in string theory, a strongly interacting system can be
more easily understood by considering a gravitational system in one more
dimension hydrodynamic ; condensed_matter  via gauge/string duality. ads_cft

Though typically used in more advanced topics, the addition of extra
dimensions can help to simplify problems in classical physics. For example,
the electrostatic problem of finding the charge distribution on a thin
conducting circular disk, can be easily solved by an orthogonal projection of
the charge distribution of a conducting sphere onto an equatorial plane.
thomson_paper  But, those are rare. In this note, we will concretely show how
adding extra dimensions simplifies the classical mechanics problem of motion
under the influence of an inverse-quadratic potential. To the best of our
knowledge, this curious example has not been demonstrated elsewhere.

##  II From 1D to 2D Euclidean Space

Figure 1: The 1D/2D duality. A repulsive inverse-quadratic potential in an 1D
space is dual to no potential in a 2D space.

Consider a point particle of mass $m$ in one-dimensional space moving under
the influence of an inverse-quadratic potential $V_{1}(x)=\alpha/x^{2}$. This
potential appears in experimental atomic physics levy_1r2 ; francoise_1r2  and
also is of many theoretical interests because it gives a scale-invariant
Schrodinger’s equation. nicholson_bound ; coon_anomaly ; quantum_1x2 ;
quantum_alphax2  For now, let us focus on a repulsive potential with
$\alpha>0$. Initially, when $t=0$, the particle is at position $x_{0}$ with no
velocity. The motion of the particle can be described by applying conservation
of energy to arrive at the following integral:

$\begin{split}&m(dx/dt)^{2}/2+V_{1}(x)=V_{1}(x_{0})\\\ \Rightarrow\ \
&dx/dt=\Big{(}2\big{(}V_{1}(x_{0})-V_{1}(x)\big{)}\big{/}m\Big{)}^{1/2}\\\
\Rightarrow\ \
&\int^{x(t)}_{x_{0}}dx\big{(}2\alpha(x_{0}^{-2}-x^{-2})/m\big{)}^{-1/2}=t\ \
.\ \ \end{split}$ (1)

However, doing this integration is non-trivial. The solution requires changing
variables to $y=(x_{0}^{2}-x^{2})^{1/2}$, at which point the integral becomes
$\int dyy^{-1/2}$ up to a multiplication factor. We can get the equation of
motion:

$\begin{split}&\Big{(}mx_{0}^{2}\big{(}x^{2}(t)-x_{0}^{2}\big{)}\big{/}2\alpha\Big{)}^{1/2}=t\\\
\Rightarrow\ \ &x(t)=(x_{0}^{2}+2\alpha t^{2}/mx_{0}^{2})^{1/2}\ \ .\ \
\end{split}$ (2)

While this solution is tractable, there exists another way to describe the
motion of the particle without the need of calculus. A “magic” from an extra
dimension.

Consider a general central potential $V_{2}$ in a two-dimensional space. In
the polar coordinates $\vec{r}=(r,\theta)$ where the origin is the center of
the potential, we have rotational symmetry. The kinetic energy $K_{\theta}$
stored in the compact angular dimension depends on the angular momentum
$p_{\theta}$ and the moment of inertia $m_{\theta}=mr^{2}$:

$K_{\theta}=p_{\theta}^{2}/2m_{\theta}=p_{\theta}^{2}/2mr^{2}=K_{\theta}(r)\ \
.\ \ $ (3)

The effective potential in the radial dimension landau ; morin_mechanics  is
just the sum of the central potential $V_{2}(r)$ and the kinetic energy
$K_{\theta}(r)$. It should be noted that $K_{\theta}(r)$ and $V_{1}(x)$ are
both inverse-quadratic functions.

Figure 2: The 1D/3D duality. An inverse-quadratic potential in an 1D space is
dual to no potential in a 3D space with $(+,+,-)$ metric signature. Different
signs of the potential correspond to different planes of motion: $(+,+)$ plane
when the potential is repulsive and $(+,-)$ plane when the potential is
attractive.

We note now an exact correspondence between this 2D scenario and the 1D
problem considered above. The motion of the point particle in one-dimensional
space under $V_{1}$ potential is dual to the radial motion of its counterpart
moving freely in two-dimensional space (no potential $V_{2}=0$), given that
the angular momentum is exactly $p_{\theta}=(2m\alpha)^{1/2}$:

$x(t)\Big{|}_{V_{1}=\alpha/x^{2}}\ \ \leftrightarrow\ \
r(t)\Big{|}_{V_{2}=0,p_{\theta}=(2m\alpha)^{1/2}}\ \ .\ \ $ (4)

In other words, with an extra angular dimension we can eliminate the
potential.

Using $x(0)=x_{0}$ and $dx/dt(0)=0$, we have the corresponding radial position
$r(0)=x_{0}$ and radial velocity $dr/dt(0)=0$. The tangent velocity is given
by:

$v_{\theta}\equiv rd\theta/dt(0)=p_{\theta}/mr(0)=(2\alpha/mx_{0}^{2})^{1/2}\
\ .\ \ $ (5)

See Fig. 1 for the detail of this 1D/2D duality. We can arrive at the same
answer (2) with the Pythagorean theorem:

$r(t)=\big{(}r^{2}(0)+(v_{\theta}t)^{2}\big{)}^{1/2}=(x_{0}^{2}+2\alpha
t^{2}/mx_{0}^{2})^{1/2}\ \ .\ \ $ (6)

While we arrived at the same answer, the solution this time is purely
geometric and does not involve any calculus.

##  III From 1D to 3D Minkowskian Space

The tools developed here can also be used for an arbitrary inverse-quadratic
potential. However, it is more complicated and requires generalization to
three dimensions. For an attractive potential, we can directly use
$\alpha=-|\alpha|<0$ to get the equation of motion $x(t)$ and also the
lifetime $\tau$ until the point particle meets the singularity at position
$x=0$:

$\begin{split}&x(t)=\big{(}x_{0}^{2}-2|\alpha|t^{2}/mx_{0}^{2}\big{)}^{1/2}\ \
,\ \ \\\ &x(\tau)=0\ \ \Rightarrow\ \
\tau=\big{(}mx_{0}^{4}/2|\alpha|\big{)}^{1/2}\ \ .\ \ \end{split}$ (7)

However, as we re-examine the problem from a two-dimensional perspective as
explained above, this indicates an imaginary value of angular momentum
$p_{\theta}=i\big{(}2m|\alpha|\big{)}^{1/2}$. To generalize this extra-
dimensional trick for all real values of $\alpha$, we need to complexify the
angular dimension $\theta=\theta_{R}+i\theta_{I}$ (with $\theta_{R}$ and
$\theta_{I}$ are real). Thus the corresponding space will be three-dimensional
with $(+,+,-)$ metric signature. landau_vol2  Note that there are now two
extra dimensions instead of one: while $\theta_{R}$ is a compact dimension,
$\theta_{I}$ is an open one. The particle moves in the $(+,+)$ Euclidean plane
when $\alpha>0$, and in the $(+,-)$ Minkowskian plane when $\alpha<0$. See
Fig. 2 for the detail of this 1D/3D duality.

###### Acknowledgements.

This note comes from our final projects for MIT’s course 8.223 in 2012 and
2013. We thank Duy V. Nguyen and the xPhO club for their support to share this
finding to a wider audience. We thank Qiantan Hong and Leonid Levitov for
examples in physics using extra dimensions. We also thank Oak Nelson and Todd
Springer for many useful comments to make this note more approachable for
general readers.

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