Smooth trajectories on toroidal manifolds
In the recent paper trajectories of the natural motion on toroidal manifolds are considered. It is represented an information dealing with description of the rotation metric at the different methods of toroidal analytical fixing, description of geodesic integrals, links between global motion invaria...
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2002
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| citation_txt | Smooth trajectories on toroidal manifolds / S.S. Romanov // Вопросы атомной науки и техники. — 2002. — № 2. — С. 112-115. — Бібліогр.: 13 назв. — англ. |
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| description | In the recent paper trajectories of the natural motion on toroidal manifolds are considered. It is represented an information dealing with description of the rotation metric at the different methods of toroidal analytical fixing, description of geodesic integrals, links between global motion invariants of the trajectory with the toroidal manifolds parameters. Analytical representations of geodesics at the different values of the global invariants are given.
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SMOOTH TRAJECTORIES ON TOROIDAL MANIFOLDS
S.S. Romanov
National Scientific Center “Kharkov Institute of Physics and Technology”
1 Academicheskaya St., 61108 Kharkov, Ukraine
In the recent paper trajectories of the natural motion on toroidal manifolds are considered. It is represented an
information dealing with description of the rotation metric at the different methods of toroidal analytical fixing,
description of geodesic integrals, links between global motion invariants of the trajectory with the toroidal
manifolds parameters. Analytical representations of geodesics at the different values of the global invariants are
given.
PACS: 02. 40. – Hw, 52.20.Dq
1. INTRODUCTION
Progress in cyclic accelerators with high – current
density makes up interest to investigate geodesic
trajectories of charged particle beam in a stellatron [1,2].
One of example of manifolds where an infinite
particle motion exists are toroidal manifolds.
Electromagnetic potentials of these manifolds have to
satisfy some requirements [3] that the particle motion
will be in restrict volume. The same potentials are
generated by the current rods distributed in one or
another manner as a rule on the surface surrounding the
volume in which particle motion goes on. One of the
simplest methods of current distribution on the surface
of the manifolds is the linear dependence between
surface coordinates of the manifold [4-6]. Without
external forces the linear dependence between surfaces
coordinates leads to arising force acting in tangent to the
trajectory plane. The same force is absent at the motion
along geodesic trajectory. Motion in tangent plane goes
on with time independent velocity. Besides the
connection length between next periods of the geodesic
are minimal one.
Geodesic trajectories equation in quasicylinder
coordinates system has been obtained in [7,8].
In the mentioned coordinates system possessed for
certain “good” properties division of variables is
difficulty. The marked difficulty is absent in toroidal
coordinates [9].
2. METRIC OF TOROIDAL MANIFOLDS
Let x1, x2, x3 are the Cartesian coordinates of point P
of toroidal surface, and r, ф, θ are its spherical
coordinates. Let the plane P0x3 crosses the circle
22
2
2
1 cxx =+ , 03 =x in the points A and B. The
radius of the marked circle is a supreme verge upon
changing of distances from point P till axis x3. Adding
x3 values differed from zero we get distances less than c.
Let us denote the angle between directions under which
from point P you will see points A and B as
φ=∠ APB and ( ) 0ln η=PBPA (Fig. 1). It follows
from the ratio for the triangle APB area that
φsin2 3 PBAPcx ⋅= ,
Fig. 1. Definition for toroidal coordinates (η, ф, θ)
of point P on given Cartesian coordinates (x1, x2, x3) of
point P
and according in cosine theorem we have
φcos24 222 PBAPPBAPc ⋅−+= .
That is why ( )φηη cos2 0
22 −= − checPB , and
( )φηφ cossin 03 −= chcx .
For getting two rest connections linking Cartesian
coordinates x1, x2 of point P u it’s toroidal ones let us
consider triangles APB1 and B1PB. If ρ is the distance
from point P till axe x3 then 2
1
2
3
2 ABxAP =− ,
2
1
2
3
2 BBxPB =− , ρ+= cAB1 , ρ−= cBB1 .
Combining these connections we get
φη
ηρ
cos4 0
0
22
−
=−=
ch
csh
c
PBAP
.
Now representation φ of open set U on the 2 –
dimensional manifold R2, which is subset of set R3
32: RRU ⊂→ϕ , can be written in the form
( ) (
).sin,sin
,cos
cos
,,
φθη
θη
φη
θφηϕ
sh
sh
ch
c
−
=
112 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2002, № 2.
Series: Nuclear Physics Investigations (40), p. 112-115.
The coordinate surfaces in R3 are: tori const== 0ηη
( ) 0
222
3
2
0
2
2
2
1 ηη shcxccthxx =+−+ ,
spheres const== 0φφ
( ) 0
222
03
2
2
2
1 sin φφ ccctgxxx =−++ ,
planes const== 0θθ 120 xxtg =θ . To clear the
geometrical meaning of scale factor c we need other
than η, ф, θ parameterization for toroidal manifold
( ) ( ( )
( ) )υθυ
θυθυϕ
sin,sincos
,coscos,,
0
0
rR
rRr
−
−=
image of which is Descartes’ product of two circles, that
are: the circle of radius R0 in (x1,x2) – plane (this circle
is named azimuthally one, axis x3 is the toroidal axis)
and in (ρ,x3) – plane, radius of which is equal r. The last
circles lie in θ=const plane and are toroidal meridians
(Fig. 2). In this parameterization the coordinate surfaces
in R3 are: tori r=r0=const,
( ) 2
0
2
3
2
2
2
2
10 rxxxR =++− ,
const== 0υυ are cones
( ) 02
030
2
2
2
1 =−−+ υcthxRxx ,
vertexes of which have coordinates (0, 0,R/ctgυ0).
Therefore it is valid correlations
00 ηccthR = , 00 ηshcr = ,
and so the scale number 2
0
2
0 rRc −= equals
digitally length of tangent in the plane θ=const to
meridian led from the torus center. From the other side,
chη0=R0/r0, ( )
−+= 1ln
22
0
2
0000 rRrRη .
Just as hyperbolic cosine is the quantity greater the
unity then the values 0<η<η0 correspond the area
outside of a torus and values η>η0 – the inside of a
torus. The value η→∞ corresponds to the azimuthally
axe of a torus on which z=0, ρ=c. Besides at r0>R0
hyperbolic cosine will be less than the unity that makes
impossible single – valued attribution for every point
coordinates η, ф, θ at ρ<c. Some more property
differing coordinates η, ф, θ from r, υ, θ consists in that
that the parabolic lines of a torus correspond the
quantities υ=±π/2 and in toroidal coordinates
1
12
0
0
+
−±=
η
ηφ
ch
charctg .
In the table numerical relations for some ф and υ so and
typical values x3 and ρ are given for a torus. We have
for the metric
2
3
222 dxddg ++= θρρ
following expression in coordinates η, ф, θ:
( ) ( )2222
2
2
cos
θηφη
φη
dshdd
ch
cg ++
−
= .
Fig. 2. Closed geodesic with p=1, q=3
ф υ z ρ
0 π 0 1
1
0
0
−
+
η
η
ch
chc
1
12
0
0
+
−
η
η
ch
charctg
2
π
0ηshc 0ηccth
2
π
00
002
rR
rRarctg
+
−
0ηchc 0ηcth
π 0 0
1
1
0
0
+
−
η
η
ch
chc
3. LOCAL EGUATION FOR GEODESIC
According known metric we can write local equation
for geodesic on toroidal manifold η=η0=const [10] as a
function arc length s
αφηφ coscos0
c
ch
ds
d −= ,
α
η
φηθ sincos
0
0
csh
ch
ds
d −= ,
αφα sinsin
cds
d = .
The set of equations for geodesic has two integrals.
The first one, which composes Clairaut’s theorem
content ( )φηα cossin 0 −= chh describes variation
113
of angle α of geodesic inclination to a meridian
η=const, θ=const. Later we miss zero indexes at η.
Angle α between geodesic and meridian is maximum
( )1sin max += ηα chh at the crossing by geodesic of
torus inner equator. Minimal angle value equals
( )1sin min −= ηα chh and attained at passing by
geodesic through outer torus equator. Integration
constant h along geodesic has finite magnitude
according to Clairaut’s theorem. The first integral
permits to lead geodesic through two neighboring
points.
Representation about geodesic behavior in a whole
can be received by using the second integral of set of
equations for geodesics [11]
( ) ( )
( )∫
−−
−=
22 cos1
cos
φηη
φφηφθ
chhsh
dchh
.
Depending on value of integration constant h geodesic
will be tight or periodic on a toroidal manifold. Since it
is examined the smooth geodesic then the periodicity
means and closeness of geodesic.
To receive presentation about geodesic behavior on
a toroidal manifold let us consider some partial
quantities of integrate constant values.
At ( ) 11 −−= ηchh Clairaut’s theorem is valid only
at 0=φ and the second integral does not belong to the
set R of the real – valued functions. Hence in this case
there are not smooth geodesics.
Let Clairaut’s constant is a small value. Then the
second integral of geodesics has a form
( ) ).sin(
η
φηφφθ
sh
cthh −≈
In this case geodesic gets cover tightly a toroidal
surface. The smallness of h means that inclination
angle of geodesic to meridian of torus is small.
Another possibility for geodesic to be sometimes in
toroidal surface region where Gauss curvature is
positive so and in a region where it is negative consists
in that that the angle α at the crossing by geodesic the
inners equator of torus differs slightly from the right
one. Integration constant in this case is equal
( ) δη −+≅ − 11chh .
For the estimation of integral for geodesics it is handy to
introduce a new variable ( )2φξ tg= and rewrite it in
the form
( ) ( ) ( )2112 TTch
sh
h ++= η
η
ξθ .
The first integral
( )
( ) ( )( )∫ ⋅
+−+−−
=
21
1111 ξηη
ξξ
chhchh
dT
( ) ( )( ) 21111
1
ξηη +++−+ chhchh
we expand in a series on degrees of small δ and
abandoning only the first term has
( ) ( )kF
ch
T ,
2
1
1 ϕ
η
ξ
+
≅ .
F(φ,k) is an elliptic function [12], its argument
−
+= ξ
η
ηϕ
1
1
ch
charctg ,
and square of additional module is equal
( ) ηηδ chchkk 1
2
1 22 +≅−=′ .
The second integral
( )
( )
( ) ( )( )
( ) ( )( ) 2
2
12
2
1111
1
1111
1
1
2
ξηη
ξηη
ξξ
η
ξ
+++−+
⋅
+−+−−
+
⋅
+
−=
∫
−
chhchh
chhchh
d
ch
T
we wrote down in a series on δ abandoning only the
first term
( ) ( )
( )
( )
.
1
1
ln
2
1
1
1
1
1
2
2
2
12
2
ξξηη
ξξηη
ξ
η
η
ξξ
η
ξ
−++
+++
−
=
+
+
+
+
−= ∫
−
chch
chch
ch
ch
d
ch
T
After substitution T1 and T2 the second integral of
geodesics takes the form
( )
( )
( )
.
1
1ln
1
1
4ln121
2
2
−++
+++−
−
−
+⋅
⋅
′
+≅
ξξηη
ξξηη
ξ
η
η
η
πη
ξθ
chch
chch
ch
charctg
k
ch
sh
4. CLOSED GEODESICS
Abovementioned Clairaut’s constant values allows
to describe geodesics which gets cover tightly a torus
surface. Here we link Clairaut’s constant with the
global invariants for the closed geodesics.
Let any geodesic which is closed performs p rounds
around torus axis (Fig. 2) and round a toroidal surface q
114
times around the azimuthally circle axis. Then
according Darboux’s theorem [13] equality
( )
( )∫ =
−−
−π
π
φηη
φφη
0
22 cos1
cos
q
p
chhsh
dchh
is fulfilled at possible h (p and q are integral numbers).
At ( ) 1cos < <− φηchh equation for the closed
geodesics can be written in the form
−≅
η
φφθ
chq
p sin
.
Clairaut’s constant as it is followed from closeness
conditions is equal
ηsh
q
ph ≅ .
Hence the value
( ) 11 < <+ηη chth
q
p
should be small.
The marked condition is satisfied slightly when
p<<q. It is not valid at p=q=1.
The closed geodesic at the small inclination to the
meridian angle makes up to closeness the revolution
number around azimuthally axis of a torus, which is
significantly greater than the geodesic round number of
toroidal axis.
Some weaker looks limitation on the ratio p and q
when the closed geodesic trajectory goes on the inners
equator of a torus under the angle of near to zero
degrees. In this case the small magnitude is
( )
.1
11
11
ln
1
1
12exp321
< <
−+
++
+
+
+
−−=+
η
η
η
ηπ
η
ηδ
ch
ch
ch
ch
q
p
ch
ch
The closed geodesic trajectory itself
( )
2
sin
2
sin
2
sin
2
sin
ln
1
2
1
11
11
ln12
2
2
φφη
φφη
η
φ
η
η
η
η
ηπ
φθ
−+
++
⋅
⋅−
+
−+
++
+≅
ch
ch
sh
tg
ch
charctg
ch
ch
shq
p
is a smooth curve without any self-crossings.
5. RESULTS
1. Parameterization for a toroidal manifold in
form of Cartesian product of two circles with radii
R0 and r0 makes possible the link between scale
number of toroidal coordinate system and radii of
formed torus circles.
2. To the parabolic lines of toroidal manifold
correspond different values of parameters υ and ф
(see Table).
3. Toroidal coordinate system with rotation
metric permits to write in evident forms the equation
of smooth geodesic trajectory.
4. The angle between torus meridian and geodesic
trajectory is maximum when geodesic goes on the
inner equator of torus and minimum at crossing the
outer one.
5. At the small angle between closed geodesic
trajectory and meridian of torus loxodrome makes
more rounds around torus axis till closing than
geodesics.
6. When closed geodesic trajectory crosses the
inner torus equator at the small angle geodesic is a
smooth curve without any self - crossings.
REFERENCES
1. D. Chernin. Beam dynamics in high – current
cyclic electron accelerations // Phys. Fluids. 1990,
v. 2, №6, p. 1478-1485.
2. C.A. Kapetanakos, L.K. Lin, T. Smith at all.
Compact high – current accelerators and their
prospective applications // Phys. Fluids. 1991, v. 3,
№8, p. 2396-2402.
3. W. Paul. Electromagnetic traps for charged
and neutral particles // Usp. Fiz. Nauk. 1990,
v. 160, p. 109-127 (in Russian).
4. V.F. Aleksin. Magnetic field of helical
currents flowing on toroidal surface. In the book
“Plasma physics and controlled nuclear fusion
problems”, issue 3. Kiev: Publishing house Akad.
of Sci. of USSR, 1963, p. 216-224.
5. V.K. Melnikov. On magnetic field lines of
helical current flowing upon toroidal surface //
Dokl. Akad. Nauk. SSSR. 1963, v. 149, p. 1056-
1059 (in Russian).
6. L.M. Kovrizschnych. Magnetic surfaces of
toroidal helical field // Zh. Tekh. Fiz. 1963, v. 33,
p. 377-381 (in Russian).
7. S.S. Romanov. Current lines on a torus of
circular cross – section // Dokl. Akad. Nauk. Ukr.
2001, v. 9, p. 84-88 (in Russian).
8. S.S. Romanov. Smooth trajectories on
toroidal manifolds. Int. Workshop “Innovative
Concepts and Theory of Stellarators.” Scientific
Center “Institute for Nuclear Research” (Kyiv,
Ukraine, May 28-31, 2001), Programme, Abstracts,
List of participations. P. 34
9. E.W. Hobson. The theory of spherical and
ellipsoidal harmonics. Cambridge at the University
press, 1931, 476 p.
10. S.P. Finikov. Course of differential
geometry. Moskva: GITTL, 1952, 343 p. (in
Russian).
11. S.S. Romanov. Smooth trajectories on
toroidal manifolds // Scientific paper of the
Institute for Nuclear Research. 2001, v. 4(6),
p. 189-193.
115
12. H.B. Dwight. Tables of integrals and other
mathematical data. N.Y., the Macmillan Company,
1961, 180 p.
13. A.L. Besse. Manifolds all of whose
Geodesics are closed. Berlin, Heidelberg, N.Y.,
1978, 325 p.
116
National Scientific Center “Kharkov Institute of Physics and Technology”
|
| id | nasplib_isofts_kiev_ua-123456789-80132 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T15:25:51Z |
| publishDate | 2002 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Romanov, S.S. 2015-04-12T06:58:36Z 2015-04-12T06:58:36Z 2002 Smooth trajectories on toroidal manifolds / S.S. Romanov // Вопросы атомной науки и техники. — 2002. — № 2. — С. 112-115. — Бібліогр.: 13 назв. — англ. 1562-6016 PACS: 02. 40. – Hw, 52.20.Dq https://nasplib.isofts.kiev.ua/handle/123456789/80132 In the recent paper trajectories of the natural motion on toroidal manifolds are considered. It is represented an information dealing with description of the rotation metric at the different methods of toroidal analytical fixing, description of geodesic integrals, links between global motion invariants of the trajectory with the toroidal manifolds parameters. Analytical representations of geodesics at the different values of the global invariants are given. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Theory and technics of particle acceleration Smooth trajectories on toroidal manifolds Гладкие траектории на тороидальных многообразиях Article published earlier |
| spellingShingle | Smooth trajectories on toroidal manifolds Romanov, S.S. Theory and technics of particle acceleration |
| title | Smooth trajectories on toroidal manifolds |
| title_alt | Гладкие траектории на тороидальных многообразиях |
| title_full | Smooth trajectories on toroidal manifolds |
| title_fullStr | Smooth trajectories on toroidal manifolds |
| title_full_unstemmed | Smooth trajectories on toroidal manifolds |
| title_short | Smooth trajectories on toroidal manifolds |
| title_sort | smooth trajectories on toroidal manifolds |
| topic | Theory and technics of particle acceleration |
| topic_facet | Theory and technics of particle acceleration |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80132 |
| work_keys_str_mv | AT romanovss smoothtrajectoriesontoroidalmanifolds AT romanovss gladkietraektoriinatoroidalʹnyhmnogoobraziâh |