# Relativistic formulation of curl force, relativistic Kapitza equation and
trapping

Partha Guha111E-mail<EMAIL_ADDRESS>
Department of Mathematics
Khalifa University of Science and Technology
Main Campus, P.O. Box -127788, Abu Dhabi
United Arab Emirates
Sudip Garai222E-mail<EMAIL_ADDRESS>
Department of Physics
Diamond Harbour Women’s University
D. H. Road, Sarisha, West-Bengal 743368, India

###### Abstract

In this present communication the relativistic formulation of the curl forces
with saddle potentials has been performed. In particular, we formulated the
relativistic version of the Kapitza equation. The dynamics and trapping
phenomena of this equation have been studied both theoretically and
numerically. The numerical results show interesting characteristics of the
charged particles associated with the particle trapping and escaping in the
relativistic domain. In addition, the relativistic generalization of the
Kapitza equation associated with the monkey saddle has also been discussed.

##### PACS numbers:

05.45.-a, 45.20.-d, 45.20.Dd, 45.50.Jf

##### 2020 Mathematics Subject Classification:

01A75, 34A05, 70J25, 70H14

##### Keywords:

Curl forces; Kapitza equation; Relativistic Lagrangian; Higher-order saddle
potentials; Trapping and escaping

##  1 Introduction

A Newtonian dynamics associated with a non-zero curl force has been a topic of
interest to many of the researchers round the globe in the recent past. These
forces are known as curl forces which has been introduced to the research
community by Berry and Shukla[1, 2]. These forces, in general, are velocity
independent in order to satisfy a non-zero curl _viz._ force $F$ depends only
on position ${\bf r}$ and is independent of velocity ${\bf v}$. The
interesting fact is that these curl forces can not be derivable from a scalar
potential. The direct applications about the understanding of the system
characteristics associated with these curl forces span to many compulsive
fields like optics, laser physics, noncentral forces, anisotropic Kepler
problem etc.[3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. The generalization of these
curl forces have been made with the introduction of dissipation factor present
in the system and gyroscopic terms. The application to this directs us to ion
trapping, two level atomic state, gain-loss mechanisms etc.

The curl force preserves the volume in phase space domain _viz._ $({\bf
r},{\bf v})$ without any attractors and this can be visualized as follows. Let
us take the dynamical equation for a unit mass as $\ddot{\bf r}=F({\bf r})$
with a non zero curl i.e. $\nabla\times F({\bf r})\neq 0$. Henceforth one can
observe that

$\nabla_{\bf r}{\bf v}+\nabla_{\bf v}\dot{\bf v}=\nabla_{\bf r}{\bf
v}+\nabla_{\bf v}F({\bf r})=0,\qquad{\bf v}=\dot{\bf r}.$

Although, in general, the dynamics associated with the curl forces are non-
dissipative. Therefore in presence of dissipation the phase space volume is
not preserved under the curl force, i.e. $\nabla\times F({\bf r})\neq 0$ and
$\nabla_{\bf r}{\bf v}+\nabla_{\bf v}F({\bf r})\neq 0$.

The curl forces are generated by the dynamical motion associated with saddle
potentials. A particle remains stable to its equilibrium if the surface
rotates around the vertical axis sufficiently fast. To visualize let us take a
saddle force field as $\big{(}F_{x}=x,F_{y}=-y\big{)}$. We now covert this to
a time-dependent one by implementing a rotation to each of the vector
counterclockwise with angular velocity $2\omega$. This describes the dynamical
motion of a particle under this force field with unit mass. From the
Earnshaw’s theorem we know that a static potential can not trap a charged
particle because any static potential, which satisfies the Laplace’s law,
lacks a potential minimal. Thus the trapping of a charged particle can be
achieved by rotating the saddle potential around a suitable axis with a
suitable angular frequency. This was the famous idea of Wolfgang Paul[14]. The
basic idea of Paul was to stabilize the saddle by means of a ‘vibrating’
electrostatic field by analogy with so called _Stephenson-Kapitsa_[15, 16]
pendulum. On the other hand, in the Penning trap, magnetic fields are used for
charged particle’s trapping[17, 18, 19]. In recent times, the study regarding
the stabilization and other geometric properties of these highly interesting
curl forces and the dynamics of nonlinear Hamiltonian curl forces have been
reported[13, 20, 21].

The dynamics of the curl forces are related to the theory of Kapitsa-Merkin
nonconservative positional forces. The linearized dynamics of a rotating shaft
formulated by Kapitsa[22] is given by

$\ddot{x}+ay+bx=0,\qquad\ddot{y}-ax+by=0.$ (1.1)

The corresponding characteristics equation reflects that, if we add a non-zero
nonconservative curl force ( i.e. $a\neq 0$) to a stable system associated
with a stable potential then it becomes unstable. This is associated to the
Merkin’s result[23, 24], which states that “the introduction of
nonconservative linear forces into a system with a stable potential and with
equal frequencies destroys the stability regardless of the form of nonlinear
terms”. It is worth mentioning that the positional force, i.e. the terms $ay$
and $-ax$ are proportional to $\omega^{2}$, where $\omega$ is the rotation
rate of the shaft. It is also possible to derive Eq.(1.1) associated with a
saddle potential via _Euler-Lagrange_ method. It is very straightforward to
check that the Lagrangian

$L=\frac{1}{2}(\dot{x}^{2}-\dot{y}^{2})-\frac{1}{2}a(x^{2}-y^{2})-bxy,$ (1.2)

yields Eq.(1.1). A symmetric saddle surface can be described by
${\tilde{U}}=a(x^{2}-y^{2})$, where $a$ is a geometrical parameter that
specifies the curvature of the saddle. It is note worthy to mention that an
ion trap potential is a rotating saddle surface on which a ball can be
trapped. The pondermotive potential of an ion trap is that of a saddle in $2D$
but the time evolution of the potential is one that flaps up and down. The
potential of the modified rotating system is

${\mathbb{U}}(x,y,t)=\frac{1}{2}(x^{2}-y^{2})\cos(2\omega t)-xy\sin(2\omega
t),$

where $\omega$ is the angular drive frequency of the spinning saddle.

The relativistic extension of the nonlinear dynamical equations are not a
straightforward job due to the intrinsic extra nonlinearity triggered by the
presence of the Lorentz factor[25, 26, 27]. The relativistic harmonic
oscillator is a topic of immense interest, although widely discussed, still
contains elements of interest. Here the Newtonian kinetic energy is replaced
by its special relativistic counterpart. This version of the relativistic
harmonic oscillator model has recently been probed experimentally also[28].
Recently, Haas[29] generalized Ermakov systems towards the special relativity
domain. In fact relativistic nonlinear dynamics is an open area of research
and deserves more attention and intense studies. In this present work we study
the relativistic analog of curl force and explore the relativistic Kapitza
equation (RKE) and trapping phenomena. One must note that the RKE is the
prototypical example of a relativistic curl force.

The paper is organized as follows: In Section $2$, we recapitulate, briefly,
the construction of the linear curl forces by following Berry and Shukla. We
also introduce the Kapitza equation and dynamics related to the higher saddles
there. Section $3$ is dedicated to the relativistic extension of the curl
forces and Kapitza equation. We also describe the trapping phenomena in that
section. Lastly, in Section $4$, we summarize our findings in the relativistic
domain with the comparison to the non-relativistic results along with the
possible application towards realistic scenarios.

##  2 Preliminaries: linear curl forces and Hamiltonization

The central fact about curl force theory is that only a small subset of all
curl forces are Hamiltonian cases and they find nice applications also. But it
is the non-Hamiltonian curl forces that are really distinctive and exhibit
various new features of dynamics.

Let us summarise the classical theory of linear curl forces[1, 2, 9]. We
recall from Ref.[1] that the form of the Hamiltonian in the case of curl
forces is given by

$H=\frac{1}{2}\alpha p_{x}^{2}+\beta p_{x}p_{y}+\frac{1}{2}\gamma
p_{y}^{2}+U(x,y),$ (2.1)

where the potential is defined by

$U(x,y)=\frac{1}{2}ax^{2}+bxy+\frac{1}{2}cy^{2}.$ (2.2)

The first set of Hamiltonian equations then gives

$\dot{x}=\alpha p_{x}+\beta p_{y},\qquad\dot{y}=\beta p_{x}+\gamma p_{y}.$
(2.3)

We can then get the forces from the second set of Hamiltonian equations as

$a_{x}=\ddot{x}=\alpha\dot{p_{x}}+\beta\dot{p_{y}}=-\alpha\frac{\partial
U(x,y)}{\partial x}-\beta\frac{\partial U(x,y)}{\partial y},$ (2.4)
$a_{y}=\ddot{y}=\beta\dot{p_{x}}+\gamma\dot{p_{y}}=-\beta\frac{\partial
U(x,y)}{\partial x}-\gamma\frac{\partial U(x,y)}{\partial y}.$ (2.5)

The curl is

$\nabla\times{\bf F}=(\alpha-\gamma)\frac{\partial^{2}U(x,y)}{\partial
x\partial y}{\bf i}+\beta\big{(}\frac{\partial^{2}U(x,y)}{\partial
x^{2}}-\frac{\partial^{2}U(x,y)}{\partial y^{2}}\big{)}{\bf j},$ (2.6)

which yields $\nabla\times{\bf F}=(\alpha-\gamma)b{\bf i}+\beta(c-a){\bf j}.$
To ensure that $\nabla\times F\neq 0$, it is necessary to take $\beta=0$ and
$\alpha=1=-\gamma$. Also the another choice we have is $\beta\neq 0$ and
$c=-a$. Henceforth the Hamiltonian turns out to be

$H=\frac{1}{2}(p_{x}^{2}-p_{y}^{2})+\frac{1}{2}(x^{2}-y^{2})+bxy,$ (2.7)

which is a Hamiltonian of the Kapitza equation.

###  2.1 Rotating saddle and Kapitza equation related to higher-saddle

Consider the motion of a particle in the rotating saddle potential in the
plane

$U({\bf x},t)=U_{0}(R^{-1}{\bf x}),\qquad
U_{0}=\frac{1}{2}(x^{2}-y^{2}),\qquad{\bf x}=(x,y);$ (2.8)

where

$R=R(\Omega t)=\left({\begin{array}[]{cc}\cos(\omega t)&\sin(\omega t)\\\
-\sin(\omega t)&\cos(\omega t)\\\ \end{array}}\right).$

The Lagrangian of the rotating saddle potential is given by

$L_{rotsaddle}=\frac{1}{2}(\dot{x}^{2}-\dot{y}^{2})-\frac{\Lambda}{2}\Big{(}\big{(}x^{2}-y^{2}\big{)}\cos(2\omega
t)+2xy\sin(2\omega t)\Big{)},\,\,\,\,\,\Lambda=2\frac{mgh_{0}}{r_{0}^{2}},$
(2.9)

and the corresponding equations of motion are

$\ddot{x}+\Lambda x\cos(2\omega t)+\Lambda y\sin(2\omega
t)=0,\qquad\ddot{y}+\Lambda y\cos(2\omega t)-\Lambda x\sin(2\omega t)=0.$
(2.10)

####  2.1.1 Higher saddle and generalized Kapitza equation

We can generalized rotating shaft equation formulated by Kapitza using higher-
order saddles. The Eq.(1.1) can be manufactured from the Euler-Lagrange
equation using simple saddle potential $g_{1}(x,y)=x^{2}-y^{2}$ and the
rotated version of the same surface, $g_{1}^{R}(x,y)=xy$. In other words,
latter one is obtained from the rotation of the saddle by $90$ degree. Let
$z=x+iy$, then $x^{2}-y^{2}=Re(z^{2})$ and $2xy=Im(z^{2})$. We follow this
guideline to construct equations of rotating shaft associated to monkey saddle

$g_{2}(x,y)=x^{3}-3xy^{2}=\hbox{ Re }(z^{3})$ (2.11)

and its rotated version

$g_{2}^{R}(x,y)=3x^{2}y-y^{3}=\hbox{ Im }(z^{3}).$ (2.12)

The generalized rotating shaft pair of equations associated to degree three
potential or monkey saddle potential is given by

$U_{3}=\frac{1}{3}\big{(}k_{1}g_{2}(x,y)+k_{2}g_{2}^{R}(x,y)\big{)}=\frac{1}{3}\big{(}k_{1}(x^{3}-3xy^{2})+k_{2}(3x^{2}y-y^{3}\big{)}.$

The corresponding equations of motion are given by

$\ddot{x}+k_{1}(x^{2}-y^{2})+2k_{2}xy=0,\quad\ddot{y}+2k_{1}xy-
k_{2}(x^{2}-y^{2})=0.$ (2.13)

##  3 Relativistic curl force

The relativistic Lagrangian in presence of scalar potential fields is given by

$L=-mc^{2}\sqrt{1-2\frac{\cal L}{mc^{2}}},\quad{\cal L}=\frac{m|{\bf
v}|^{2}}{2}-U(x).$ (3.1)

One way to describe the non-relativistic approximation is to apply the limits:
$U(x)<<mc^{2}$; $v<<c$,

$L=-mc^{2}\sqrt{\big{(}1-\frac{|{\bf
v}|^{2}}{c^{2}}\big{)}+\frac{2U}{mc^{2}}}=-mc^{2}\sqrt{1-\frac{|{\bf
v}|^{2}}{c^{2}}}\sqrt{1+\frac{2U}{mc^{2}}\big{(}1-\frac{|{\bf
v}|^{2}}{c^{2}}\big{)}^{-1}}$
$=-mc^{2}\gamma^{-1}\sqrt{1+\frac{2U}{mc^{2}}\gamma^{2}},\quad\gamma=\big{(}1-\frac{|{\bf
v}|^{2}}{c^{2}}\big{)}^{-\frac{1}{2}}.$

Expanding the expression within square-root binomially up to second order
gives us

$\tilde{L}=-mc^{2}\gamma^{-1}\big{(}1+\frac{U}{mc^{2}}\gamma^{2}\big{)}=-mc^{2}\gamma^{-1}-U\gamma.$
(3.2)

Although in _Goldstein_ ’s book the semi-relativistic Lagrangian is given by

$L_{sr}-mc^{2}\gamma^{-1}-U.$ (3.3)

If we assume

$\frac{|{\bf v}|}{c}=\alpha<<1,\Rightarrow U\gamma\sim
U+\frac{U}{2}\alpha^{2},$

for practical purpose; we can drop the second term for small value of $\alpha$
which immediately yields the semi-relativistic Lagrangian given in
_Goldstein_[30].

###  3.1 Relativistic Kapitza equation

The relativistic Lagrangian of the curl force is defined as

$L=-\frac{c^{2}}{\Gamma}-U(x,y),\quad\hbox{ with
}\quad\Gamma=\big{(}1-\frac{\dot{x}^{2}}{c^{2}}+\frac{\dot{y}^{2}}{c^{2}}\big{)}^{-\frac{1}{2}}.$
(3.4)

Here the saddle potential is assumed as

$U(x,y)=\frac{1}{2}k(x^{2}-y^{2})+bxy.$ (3.5)

###### Lemma 3.1

The Euler-Lagrange equation of the relativistic Lagrangian
$L=-\frac{c^{2}}{\Gamma}-\frac{1}{2}k(x^{2}-y^{2})-bxy$ yields

$\ddot{x}(1+\frac{\dot{y}^{2}}{c^{2}})-\frac{\dot{x}\dot{y}}{c^{2}}\ddot{y}=-\frac{kx}{\Gamma^{3}}-\frac{by}{\Gamma^{3}},$
(3.6)
$\ddot{y}(1-\frac{\dot{x}^{2}}{c^{2}})+\frac{\dot{x}\dot{y}}{c^{2}}\ddot{x}=-\frac{ky}{\Gamma^{3}}+\frac{bx}{\Gamma^{3}},$
(3.7)

where
$\Gamma=(1-\frac{\dot{x}^{2}}{c^{2}}+\frac{\dot{y}^{2}}{c^{2}})^{-\frac{1}{2}}$.

Proof It is straightforward to check that

$\frac{\partial
L}{\partial\dot{x}}=\dot{x}(1-\frac{\dot{x}^{2}}{c^{2}}+\frac{\dot{y}^{2}}{c^{2}})^{-\frac{1}{2}},\quad\frac{d}{dt}(\frac{\partial
L}{\partial\dot{x}})=\big{[}\ddot{x}(1+\frac{\dot{y}^{2}}{c^{2}})-\frac{\dot{x}\dot{y}}{c^{2}}\ddot{y}](1-\frac{\dot{x}^{2}}{c^{2}}+\frac{\dot{y}^{2}}{c^{2}})^{-\frac{3}{2}},$
$\frac{\partial
L}{\partial\dot{y}}=-\dot{y}(1-\frac{\dot{x}^{2}}{c^{2}}+\frac{\dot{y}^{2}}{c^{2}})^{-\frac{1}{2}},\quad\frac{d}{dt}(\frac{\partial
L}{\partial\dot{y}})=\big{[}-\ddot{y}(1-\frac{\dot{x}^{2}}{c^{2}})-\frac{\dot{x}\dot{y}}{c^{2}}\ddot{x}](1-\frac{\dot{x}^{2}}{c^{2}}+\frac{\dot{y}^{2}}{c^{2}})^{-\frac{3}{2}}.$

Then using the derivative of $\frac{\partial L}{\partial x}$ and
$\frac{\partial L}{\partial y}$ we obtain our result from the Euler-Lagrange
equation. We must note here that both the equations (3.6) and (3.7) contain
$\ddot{x}$ and $\ddot{y}$ terms. The equations of motion are therefore
obtained by separating these terms.

###### Proposition 3.1

The Euler-Lagrange equation for the relativistic Kapitza Lagrangian yields the
relativistic Kapitsa equation

$\ddot{x}=\frac{\dot{x}\dot{y}}{\Gamma
c^{2}}(-ky+bx)-\frac{1}{\Gamma\Gamma_{x}^{2}}(kx+by),$ (3.8)
$\ddot{y}=\frac{\dot{x}\dot{y}}{\Gamma
c^{2}}(kx+by)+\frac{1}{\Gamma\Gamma_{y}^{2}}(-ky+bx),$ (3.9)

where

$\Gamma_{x}=(1-\frac{\dot{x}^{2}}{c^{2}})^{-\frac{1}{2}},\quad\Gamma_{y}=(1+\frac{\dot{y}^{2}}{c^{2}})^{-\frac{1}{2}}.$
(3.10)

Proof Multiplying (3.6) by $(1-\frac{\dot{x}^{2}}{c^{2}})$ we obtain

$\ddot{x}\big{(}1+\frac{\dot{y}^{2}}{c^{2}}\big{)}\big{(}1-\frac{\dot{x}^{2}}{c^{2}}\big{)}-\frac{\dot{x}\dot{y}}{c^{2}}\ddot{y}\big{(}1-\frac{\dot{x}^{2}}{c^{2}}\big{)}=-\big{(}\frac{kx}{\Gamma^{3}}+\frac{by}{\Gamma^{3}}\big{)}\big{(}1-\frac{\dot{x}^{2}}{c^{2}}\big{)}$
$\frac{\ddot{x}}{\Gamma^{2}}=\frac{\dot{x}\dot{y}}{\Gamma}^{3}{c^{2}}\big{(}-ky+bx\big{)}-\frac{1}{\Gamma_{x}^{2}\Gamma^{3}}\big{(}kx+by).$

Similarly, we obtain the second equation via the multiplication of Eq.(3.7)
with $(1+\frac{\dot{y}^{2}}{c^{2}})$.

###### Corollary 3.1

The equations (3.16) and (3.16) for non-relativistic limit reduce to the
Kapitsa equation

$\ddot{x}+kx+by=0,\qquad\ddot{y}-ky+bx=0.$ (3.11)

The relativistic Kapitza equation can also be formulated using Hamiltonian
method.

###### Proposition 3.2

The (effective) Hamiltonian of the relativistic Kapitsa equation is

$H=\frac{p_{x}^{2}}{2\Gamma}-\frac{p_{y}^{2}}{2\Gamma}+\frac{1}{2}k(x^{2}-y^{2})+bxy,$
(3.12)

and corresponding Hamiltonian equations are given by

$\dot{x}=\frac{\partial H}{\partial
p_{x}}=\frac{p_{x}}{\Gamma},\quad\dot{p_{x}}=-\frac{\partial H}{\partial
x}=-kx-by,$ (3.13) $\dot{y}=\frac{\partial H}{\partial
p_{y}}=-\frac{p_{y}}{\Gamma},\quad\dot{p_{y}}=-\frac{\partial H}{\partial
y}=ky-bx.$ (3.14)

One can clearly see that $p_{x}=\dot{x}\Gamma$, $p_{y}=-\dot{y}\Gamma$ and the
rest follows from lemma 3.1. It can be easily shown that the relativistic
Kapitsa equation follows from Eqns.(3.13) and (3.14).

It is clear that for the non-relativistic limit the Hamiltonian equation
yields the Kapitza equation. One must note that non-relativistic Kapitza
equation has two integrals of motion, the Hamiltonian
$H=\frac{p_{x}^{2}}{2}-\frac{p_{y}^{2}}{2}+\frac{1}{2}k(x^{2}-y^{2})+bxy$ and
the Fradkin tensor $I=p_{x}p_{y}+\frac{1}{2}b(x^{2}-y^{2})-kxy$.
Unfortunately, for the relativistic case we have just one conserved quantity
due to the presence of the Lorentz factor.

###  3.2 Relativistic rotation of saddle

The Lagrangian of the rotating saddle potential is given by

$L_{\text{rel
saddle}}=-\frac{c^{2}}{\Gamma}-\frac{\Lambda}{2}\Big{(}\big{(}x^{2}-y^{2}\big{)}\cos(2\Gamma\omega
t)+2xy\sin(2\Gamma\omega t)\Big{)},$ (3.15)

where the constant coefficients $b$ and $k$ are replaced by the sinusoidal
terms, the corresponding equations of motion are

$\ddot{x}=\frac{\dot{x}\dot{y}}{\Gamma c^{2}}\Lambda\big{(}-\cos(2\Gamma\omega
t)y+\sin(2\Gamma\omega
t)x\big{)}-\frac{1}{\Gamma\Gamma_{x}^{2}}\Lambda\big{(}\cos(2\Gamma\omega
t)x+\sin(2\Gamma\omega t)y\big{)},$ (3.16)
$\ddot{y}=\frac{\dot{x}\dot{y}}{\Gamma c^{2}}\Lambda\big{(}\cos(2\Gamma\omega
t)x+\sin(2\Gamma\omega
t)y\big{)}+\frac{1}{\Gamma\Gamma_{y}^{2}}\Lambda\big{(}-\cos(2\Gamma\omega
t)y+\sin(2\Gamma\omega t)x\big{)},$ (3.17)

where

$\Gamma_{x}=(1-\frac{\dot{x}^{2}}{c^{2}})^{-\frac{1}{2}},\quad\Gamma_{y}=(1+\frac{\dot{y}^{2}}{c^{2}})^{-\frac{1}{2}}.$
(3.18)

(a) Plot of $\dot{x}$ vs $x$.

(b) Plot of $\dot{y}$ vs $y$.

(c) Plot of $\dot{x}$ vs $x$.

(d) Plot of $\dot{y}$ vs $y$.

Figure 1: Phase space diagrams (with normalized parameters) via Eq.(3.16) and
Eq.(3.17) with $\Lambda=0.1$ and $\omega=0.5$. The top two figures are for
non-relativistic cases with $v_{x}=0=v_{y}$ and the bottom two are for the
relativistic cases with $v_{x}=0.5c=v_{y}$. The initial conditions are
$x(0)=0$, $\dot{x}(0)=0.0$, $y(0)=0.0$, $\dot{y}(0)=0.001$. The normalized
parameters are as follows: $x\rightarrow x\omega_{c}/c$, $y\rightarrow
y\omega_{c}/c$, $\dot{x}\rightarrow\dot{x}/c$, $\dot{y}\rightarrow\dot{y}/c$,
$\ddot{x}\rightarrow\ddot{x}\omega_{c}c$,
$\ddot{y}\rightarrow\ddot{y}\omega_{c}c$,
$\Lambda\rightarrow\Lambda/\omega_{c}^{2}$,
$\omega\rightarrow\omega/\omega_{c}$ and $t\rightarrow t\omega_{c}$ where
$\omega_{c}$ can be taken as the cyclotron frequency in the typical microwave
range.

The phase space plots for the rotating saddle have been depicted in figure (1)
for Eqs.(3.16) & (3.17). In the non-relativistic scenario, it can be seen from
figure (1(a)) that the trajectory corresponds to limit cycle under one
specific saddle point (SP) and spanning over smaller to larger amplitudes. The
particle can be trapped inside those periodic boundaries for shorter times but
not completely. Figure (1(b)) corresponds to heteroclinic orbit where it moves
from one SP to another over time directs to the escaping of particles. In the
relativistic case, both figure (1(c)) and (1(d)) correspond heteroclinic orbit
and particle escaping.

In the non-relativistic limit Eqns.(3.16) and (3.18) reduce to

$\ddot{x}+\Lambda x\cos(2\omega t)-\Lambda y\sin(2\omega
t)=0,\qquad\ddot{y}+\Lambda y\cos(2\omega t)-\Lambda x\sin(2\omega t)=0.$

The relativistic generalized Kapitza equation corresponding monkey saddle is
the extension of the monkey saddle equation,

$\ddot{x}=\frac{\dot{x}\dot{y}}{\Gamma
c^{2}}\big{(}-2k_{1}xy+k_{2}(x^{2}-y^{2})\big{)}-\frac{1}{\Gamma\Gamma_{x}^{2}}\big{(}k_{1}(x^{2}-y^{2})+2k_{2}xy\big{)},$
(3.19) $\ddot{y}=\frac{\dot{x}\dot{y}}{\Gamma
c^{2}}\big{(}k_{1}(x^{2}-y^{2})+2k_{2}xy\big{)}+\frac{1}{\Gamma\Gamma_{y}^{2}}\big{(}-2k_{1}xy+k_{2}(x^{2}-y^{2})\big{)}.$
(3.20)

(a) Plot of $\dot{x}$ vs $x$.

(b) Plot of $\dot{y}$ vs $y$.

(c) Plot of $\dot{x}$ vs $x$.

(d) Plot of $\dot{y}$ vs $y$.

Figure 2: Phase space diagrams (with normalized parameters) via Eq.(3.19) and
Eq.(3.20) with $\Lambda=0.1$, $\omega=0.5$ and $\alpha_{1}=\alpha_{2}=1$. The
top two figures are for non-relativistic cases with $v_{x}=0=v_{y}$ and the
bottom two are for the relativistic cases with $v_{x}=0.5c=v_{y}$. The initial
conditions are $x(0)=0$, $\dot{x}(0)=0.0$, $y(0)=0.0$, $\dot{y}(0)=0.001$. The
normalization scheme is same as mentioned in figure (1) along with
$\alpha_{1}\rightarrow\alpha_{1}c/\omega_{c}$ and
$\alpha_{2}\rightarrow\alpha_{2}c/\omega_{c}$.

The phase space plots for the relativistic rotating monkey saddle have been
depicted in figure (2) for Eqs.(3.19) & (3.20). Figure (2(a)) and (2(c)) show
heteroclinic trajectories which remove the possibility of particle trapping
for both relativistic and non-relativistic cases. In contrary to this figure
(2(b)) and (2(d)) show limit cycle trajectories under one SP for both
relativistic and non-relativistic cases. As one can see that there are
periodic boundaries which are changing very slowly over time and closely
spaced, henceforth particle can be trapper over shorter time scale.

The relativistic rotating monkey saddle equation can be obtained by replacing
$k_{1}=\Lambda\alpha_{1}\cos(\Gamma 2\omega t)$ and
$k_{2}=\Lambda\alpha_{2}\sin(\Gamma 2\omega t)$.

###  3.3 Flapping and spinning of relativistic saddle potential and trap

Suppose we allow the potential $U=g(\frac{1}{2}(x^{2}-y^{2}))$ to vary
sinusoidally with time. this becomes

${\cal U}(x,y,t)=A_{RF}\cos(\omega t)g(\frac{1}{2}(x^{2}-y^{2}),$ (3.21)

where $A_{RF}$ is the amplitude of the AC component of the electric field
applied to create the trap, and $\omega$ is the frequency. This can be viewed
as a nonlinear generalization of ‘flapping saddle’. Latter occurs in a
quadrupole potential in which extreme oscillate between peaks and valleys.
Thus Newton’s law yields the following pair of equations

$\ddot{x}=-A_{RF}\cos(\omega
t)xg(\frac{1}{2}(x^{2}-y^{2}),\quad\ddot{y}=-A_{RF}y\cos(\omega
t)g(\frac{1}{2}(x^{2}-y^{2}).$ (3.22)

(a) Plot of $\dot{x}$ vs $x$.

(b) Plot of $\dot{y}$ vs $y$.

(c) Plot of $\dot{x}$ vs $x$.

(d) Plot of $\dot{y}$ vs $y$.

Figure 3: Phase space diagrams (with normalized parameters) via Eq.(3.3) and
Eq.(3.3) with $\Lambda=0.1$, $\omega=0.5$ and
$\beta_{1}=\beta_{2}=\beta_{3}=\beta_{4}=1$. The top two figures are for non-
relativistic cases with $v_{x}=0=v_{y}$ and the bottom two are for the
relativistic cases with $v_{x}=0.5c=v_{y}$. The initial conditions are
$x(0)=0$, $\dot{x}(0)=0.0$, $y(0)=0.0$, $\dot{y}(0)=0.001$. The normalization
scheme is same as mentioned in figure (1) along with
$\beta_{1}\rightarrow\beta_{1}\omega_{c}^{2}/c^{2}$,
$\beta_{2}\rightarrow\beta_{2}\omega_{c}^{4}/c^{4}$,
$\beta_{3}\rightarrow\beta_{3}\omega_{c}^{2}/c^{2}$ and
$\beta_{4}\rightarrow\beta_{4}\omega_{c}^{4}/c^{4}$. It must be noted here
that we have chosen
$g(x^{2}-y^{2})=1+\beta_{1}(x^{2}-y^{2})+\beta_{2}(x^{2}-y^{2})^{2}$ and
$f(xy)=1+\beta_{3}(xy)+\beta_{4}(xy)^{2}.$

The linear case coincides with the form of the Mathieu differential equations.
Our motivation is to study this new set of equations. In general it is
difficult to create flapping saddle on which the ball can move. So we bang on
the time-dependent spinning saddle at angular frequency $\omega$, which is
similar to the flapping potential

$\hat{{\cal U}}(x,y,t)=A\big{(}g(x^{2}-y^{2})\cos(2\omega
t)+2f(xy)\sin(2\omega t)\big{)},$ (3.23)

where $A$ is some constant. We now apply again the relativistic Lagrangian as

$L_{relflap}=-\frac{c^{2}}{\Gamma}-\hat{{\cal U}}(x,y,t),\quad\hbox{ where
}\hat{{\cal U}}(x,y,t)=A\big{(}g(x^{2}-y^{2})\cos(2\omega
t)+2f(xy)\sin(2\omega t)\big{)},$ (3.24)

to get the relativistic coupled differential equations as

$\displaystyle\ddot{x}=\frac{\dot{x}\dot{y}}{\Gamma
c^{2}}\Lambda\big{(}-yg^{\prime}(x^{2}-y^{2})\cos(2\Gamma\omega
t)+xf^{\prime}(xy)\sin(2\Gamma\omega t)\big{)}$
$\displaystyle-\frac{1}{\Gamma\Gamma_{x}^{2}}\big{(}xg^{\prime}(x^{2}-y^{2})\cos(2\Gamma\omega
t)+yf^{\prime}(xy)\sin(2\Gamma\omega t)\big{)},$ (3.25)
$\displaystyle\ddot{y}=\frac{\dot{x}\dot{y}}{\Gamma
c^{2}}\Lambda\big{(}xg^{\prime}(x^{2}-y^{2})\cos(2\Gamma\omega
t)+yf^{\prime}(xy)\sin(2\Gamma\omega t)\big{)}$
$\displaystyle+\frac{1}{\Gamma\Gamma_{y}^{2}}\big{(}-yg^{\prime}(x^{2}-y^{2})\cos(2\Gamma\omega
t)+xf^{\prime}(xy)\sin(2\Gamma\omega t)\big{)}.$ (3.26)

If we take $\tau=\omega t$ and $x^{\prime}=\frac{dx}{d\tau}$ we obtain

$\displaystyle x^{\prime\prime}=\frac{x^{\prime}y^{\prime}}{\Gamma
c^{2}}\Lambda\big{(}-yg^{\prime}(x^{2}-y^{2})\cos(2\Gamma\omega
t)+xf^{\prime}(xy)\sin(2\Gamma\omega t)\big{)}$
$\displaystyle-\frac{1}{\omega^{2}\Gamma\Gamma_{x}^{2}}\big{(}xg^{\prime}(x^{2}-y^{2})\cos(2\Gamma\omega
t)+yf^{\prime}(xy)\sin(2\Gamma\omega t)\big{)},$ (3.27) $\displaystyle
y^{\prime\prime}=\frac{x^{\prime}y^{\prime}}{\Gamma
c^{2}}\Lambda\big{(}xg^{\prime}(x^{2}-y^{2})\cos(2\Gamma\omega
t)+yf^{\prime}(xy)\sin(2\Gamma\omega t)\big{)}$
$\displaystyle+\frac{1}{\omega^{2}\Gamma\Gamma_{y}^{2}}\big{(}-yg^{\prime}(x^{2}-y^{2})\cos(2\Gamma\omega
t)+xf^{\prime}(xy)\sin(2\Gamma\omega t)\big{)}.$ (3.28)

The phase space plots for the relativistic flapping saddle have been depicted
in figure (3) for Eqs.(3.3) & (3.3). Like in the previous case of relativistic
rotating monkey saddle here also we see that both trapping in shorter time
scale and escaping is possible in both relativistic and non-relativistic
limits.

##  4 Summary

The relativistic analog of the Kapitza equation has been investigated and
associated trapping phenomena of the charged particles have been discussed. We
have presented an example of a relativistic curl force and the generalization
of the curl force associated with saddles potentials formulated by Berry and
Shukla. In case of relativistic, rotating saddle the phase space plots
correspond some heteroclinic orbits and particle escaping. With the inclusion
of the monkey saddle to the generalized Kapitza equation in relativistic
domain the phase plots also show heteroclinic trajectories with a possibility
of particle trapping in specific parametric domain in the shorter time scale.
In case of relativistic flapping and spinning saddle there is also a
possibility of trapping and escaping same as discussed in the previous one. At
last we have discussed about the relativistic generalization of the Kapitza
equation associated with the monkey saddle. The theoretical outcomes along
with the numerical results may direct and or give an idea about the possible
mechanism of particle trapping with finite curl forces in the relativistic
domain. In the earlier observations[11, 12], we see that there is a
possibility of trapping the charged particles under the curl force dynamics
with saddle potentials completely, whereas, in this work we have observed that
the trapping is not complete in the relativistic domain. As the results in the
relativistic domain direct us of incomplete trapping in a shorter time scale
therefore it becomes a field of relative interest to further investigate this
with the choice of different saddle potentials and to apply the outcomes
towards astrophysical scenarios.

## Acknowledgments

PG is immensely grateful to _Professor Sir Michael Berry_ for his critical and
meticulous reading of this draft and for making invaluable remarks and or
comments. PG and SG also convey their sincere thanks to _Professor Haret Rosu_
for his suggestions and comments and _Professors Praghya Shukla, Stefan Mancas
and Anindya Ghose-Choudhury_ for various discussions and correspondences. PG
thanks _Khalifa University of Science and Technology_ for its continued
support towards this research work under the grant number FSU-2021-014 and SG
thanks _Diamond Harbour Women’s University_ for providing the necessary
research environment with constant encouragement and support.

## Author contributions

All the authors contributed equally to this work

## Conflicts of interest

The authors declare that they have no conflict of interest.

## Data transparency

Not applicable.

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