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Motion of a charge in a magnetic dipole field. I. Painlevé analysis and a conservative numerical scheme
Motion of a Charge in a Magnetic Dipole Field. I. Painlevk Analysis and a Conservative Numerical Scheme Luis VAzquez Research Center Bielefeld-Bochum-Stochustics University of Bielefeld D-4800 Bielefeld 1, F. R. G. and Salvador Jimenez Departamento de Fisica Tebrica Facultad de Ciencias Fiskas Universidud Complutense 28040 Madrid, Spain
Transmittedby John Casti
ABSTRACT
We present a new numerical scheme to integrate the equations of motion of a charged particle in a magnetic dipole field. The scheme is time-inversion symmetry preserving, and it shows a discrete version of the two conserved quantities of the underlying continuous equation. We also p:esent the Painlevit analysis of the system by using an approach which to our best knowledge is new.
I.
INTRODUCTION
The study of the motion of a charged particle in the magnetic dipole field of the earth is a basic tool to understand the magnetospheric and radiation belt phenomena. Since Starmer [l], the problem has been studied by several authors [2, 31. It must be remarked that the Van Allen radiation of the earth cannot be explained in the framework of a pure magnetic dipole field, even considering either higher multipole moments or short-term dynamic fluctuations in the magnetic field of the earth. In this context, a stochastic model [4] has recently been developed to explain the formation of the Van Allen 207
APPLIEDMATHEMATICSANDCOMPUTATION 25207-217 (1988) 0 Elsevier Science Publishing Co., Inc., 1988 52 Vanderbilt Ave., New York, NY 10017
oo9&m3/88/$03.50
208
LUIS ViiZQUEZ AND SALVADOR JIMkNEZ
radiation around the planets, and this modei accounts for important observational facts. Now, our purpose is to study a new stochastic model for the Van Allen radiation belt related to the above one. As a preparatory step to studying the new model, we analyze a new numerical scheme for the equations of motion of a charge in a magnetic dipole field. This scheme shows the same dynamics as the underlying continuous equation, in the sense that it is energy and timeinversion symmetry preserving. The features of the equations as well as the numerical schemes are described in Sections II and III together with some numerical results. Finally, the Painlevi! analysis is applied to the motion equations in order to get some insight into the integrability of the system (Section IV), but we get a negative result.
II.
EQUATIONS
The equation field is
OF MOTION
IN THE MAGNETIC
of motion of a nonrelativistic
charged
DIPOLE particle
FIELD
in a magnetic
dV e m-=-VxB, dt c where e and m are the charge and the mass of the particle, velocity of the light. The dipole magnetic field is given by B=v
and c is the
XA,
A = A( - Y, x,0), A=
-P
(x2 + y2 +
z2y2.
In this case, there are two conserved quantities: the energy, and the azimuthal angular momentum, due to the invariance of the Hamiltonian under rotations about the z-axis [2]. From this the three-dimensional problem is reduced to the two-dimensional motion of a particle in the pz plane [p = (x2 + ~~)l/~]. On introducing dimensionless variables, the motion equations become
d2p _=-dt2
aU ap ’
(3)
Motionof a Charge in a Dipole Field. 1
209
where 2
1
P qp,+;
;-
(p2+
i
z2)3’2 -
This is known as the Stermer problem, and it has been widely studied [2, 31. The dimensionless Hamiltonian is
H=S2+b2)+U(p,z), and the system shows trapped orbits for
III.
(5)
H < $ and untrapped ones for H > &.
A CONSERVATIVE NUMERICAL NUMERICAL RESULTS
SCHEME:
Up to now the finite-difference schemes [3] used to integrate the equations of motion (3) are useless for studying the long-time behavior of the trajectory, due to the cumulative error for long integration runs. A way to avoid this problem is to use numerical schemes showing a conserved discrete energy and which are time-inversion symmetry preserving like the underlying continuous equation. In this framework we have constructed the following two energy preserving difference schemes for the system (3): A.
Scheme z/p:
u( p”, 2”+2) - u( pn, 2”)
z”+2-222”+1+z” At2
=-
P “+2 - 2P”+l
z
=
-
P
@a)
-.z”
v( p”+2, z “f2)
-P”
At2
,
n+2
n+2
- u( p”, zn+2) - P”
(ebb)
And the associated discrete conserved energy is
E”=;(
““+;;““)2+f( p”‘bl”)
+;[U(p”+l,z”+l)+U(p”,z”)].
(7)
LUIS VAZQUEZ
210
AND SALVADOR
JIMkNEZ
At each step the finite-difference scheme (6) requires solving the functional equation (6a) for the unknown z n + 2, and next solving the equation (6b) for the unknown pn+2. This can be accomplished by Newton’s method. B. Scheme p/z: Pn+2 - 2P”+’
q P”+2, z”) - u( p”, 2”)
+ P”
At2
=-
n+2
z n+2_2z”+1_z”
u(pn+2,
At2
=
-pn
P
-
2”+2) .z
n+2
- U(p”+2, -
@a)
’
z”) @b)
zn
The associated discrete conserved energy is the same as given by (7). The difference from scheme z/p is that now we have first to find P”+~ by solving (8a) and next z n + 2 from the equation (8b). The usefulness of each scheme is related to the characteristics trajectories in the p-z plane.
P:lO? Z=O E = 0.0024992
+“”
-04
A0
2 -0.0355
, N’6780
.~.I..~~~~~.~I~~~,‘~~~~I~~~~‘~~~.I”’ -0 3
of the
-02
-01
0
‘,,,,I..“““‘I,““,“‘I”“““‘~ 1
FIG. 1. Trapped orbit in the z-p plane.
I
3
d
Z
Motion of a Charge in a Dipole Field. 1
07 44
“’
211
~..I~““.“~I~~~.‘~~~~I~~.~I~.~~I~~~,1~~~~I~~..”~.~I~~~“~~~.I~...~““l 43
-01
-01
FIG. 2.
0
Trapped
. ,
2
3
Z 4
orbit in the .zp plane.
These schemes are a generalization of the conservative scheme for systems with one degree of freedom considered by Vaquez [5]. In the computations we report in this work the mesh size At = 0.01 was chosen. In Figures 1 and 2 we represent two trapped orbits in the pz plane, while in Figures 3, 4, 5, and 6 we show four untrapped trajectories. In each figure, the initial conditions, the energy and the number of points N (iteration steps) are indicated. The numerical results have been obtained in double precision, and for the whole computed trajectories the energy is conserved up to the tenth decimal digit.
III.
THE PAINLEVI? ANALYSIS
The Painleve analysis has been used as a test of integrability for Hamiltonian systems [6-81. In this section, we apply it to the Hamiltonian (5). Such a system has a very rich structure [3]: it is not integrable globally, but only
212
LUIS VAZQUEZ AND SALVADOR JIMkNEZ
FIG. 3.
Untrapped orbit in the z-p plane.
locally. In the untrapped region the system is completely integrable, and also it shows regional integrabihty for the case of trajectories sufficiently near to certain periodic trajectories. It means that the integrability is energy dependent, in this way it seems very natural to apply the Painleve analysis in order to get some information about the relation between the integrability and the energy. Moreover, since the motion equations (3) are not rational functions, we must transform them in order to apply the Painlevi: analysis. To accomplish that we will consider two special changes, one of which incorporates explicitly the dependence on the energy of the system. First of all, let us summarize the Painleve analysis. We consider a system of two ordinary differential equations of the form
g(t, x, y, i, 0, c>=0,
6-J)
with f and g analytical in t and rational in the other variables. Then the
213
Motion of a Charge in a Dipole Field. I P
5T
Pzl.5
,
P:o 1
Z:O ,
izo.1
E-0034709
N=4000
FIG. 4.
Untrapped
orbit in the a-p plane.
Painleve analysis is performed basically as follows: (a) First we consider the variables extended to the complex plane and we insert in (9) the expansions
i=l
y=/?(t-toy+
f
aj(t-toy+j.
j=l
(b) We find all the possible values of the leading-term exponents p and q. (c) If all those values are integer, we must check the coefficients which remain arbitrary and we choose them and t,, as the constants of the motion. (d) If there are enought constants [four in the case of the system (9)], the system shows the Painleve property, i.e., all its singularities located at the arbitrary to (movable singularities) are at most poles.
214
LUIS V;lZQUEZ AND SALVADOR JIMliNEZ
FIG. 5.
Untrapped
orbit in the Z-P plane.
The PainlevC conjecture implies that the system is integrable if the property in (d) is satisfied [9, lo]. A weaker formulation of this conjecture has been considered recently [9]. Let us come back to our system. According to (4) the motion equations are the following:
f+
1 6-- i -P
-(p2+ z2)5'2 ( P
1
P
(p2+z2)3'2
P
1
3PZ
7+ I(
P
(p2+ z2)3'2)
z2-
2p2
(p2+ z2)5'2
= 0,
i = 0.
These equations are not rational in p and z. We have considered two ways to
Motion of a Charge ina Dipole Field. I
E~0040001.
215
N:2330
transform them into a rational system: (A) Let the variable change be
(z, P)-+(c P),
r2= (p2+ z2y2.
Then the motion equations become
++p-jj-/jL
ti-p ----+3-r2-p2
By considering the expansions p - a?
r2;”
li
5
=(), i
(11)
and r - prq, where 7 = t - to we
216
LUIS VAZQUEZ AND SALVADOR JIMkNEZ
have found four cases: (a) p = q = 0, a and p arbitrary; 1 (b) p = i, q = 0, a = - 4, /? arbitrary; (c) p = $9, q > $, a2 = * p3; (d) p = q = ;, a2 = [(Z 5 a)/31
1’3 [l - 2(2 k fi)],
p = 31’3 (2 + J?;)“‘.
Case (c) allows, for instance, irrational values of p and q; thus we get no conclusion about the integrability. (B) Instead of changing variables, we change the nonrational terms in (10) by using the constancy of the energy. We obtain
1 p-p
P-
:(z”+p2)+; i
i
I
-H
>: i
=O. II
Among other cases we have found p >/ i, 9 > p, a6 = - 9, and p arbitrary, so irrational values for p and 9 are allowed; thus no information about the integrabihty of the system is obtained even if the energy appears explicitly in the system. As a conclusion, the Painlevi: analysis does not give us any information about the regional integrability of the Stermer problem. One of the authors (L. V.) is very grateful to Professors S. Albeverio, Ph. Blunchard, and L. Streit for their invitation to join the Research Center Bielefeld-Bochum-Stochastics, where this work was finished.
Motion of a Charge in a Dipole Field. Z
217
REFERENCES
5 6 7 8 9 10
C. Sknmer, The Polar Aurora, Oxford, Clarendon, London, 1955. A. J. Dragt, Reu. Geophys. 3:255 (1965). A. J. Dragt and J. M. Finn, I. Ceophys. Res. 81:2327 (1976). Ph. Blanchard, Acta Phys. Austriaca Suppl. XXVI:185 (1984) and references therein. L. Vkzquez, Long time behavior in numerical solutions of certain dynamical systems, An. Fb. to appear. Y. F. Chang, M. Tabor, and J. Weiss, I. Math. Phys. 23531 (1982). T. Bountis, H. Segur, and F. Vivaldi, Phys. Reu. A. 251257 (1982). S. Jim$nez and L. Vkzquez, An. F/s. 82A:lQ (1986). B. Grammaticos, B. Dorizzi, and A. Ramani, J. Math. Phys. 242282 (1983). A. F. RaEada, A. Ramani, B. Dorizzi, and B. Grammaticos, J. Math. Phys. 26:708 (1985).
Transmittedby John Casti
ABSTRACT
We present a new numerical scheme to integrate the equations of motion of a charged particle in a magnetic dipole field. The scheme is time-inversion symmetry preserving, and it shows a discrete version of the two conserved quantities of the underlying continuous equation. We also p:esent the Painlevit analysis of the system by using an approach which to our best knowledge is new.
I.
INTRODUCTION
The study of the motion of a charged particle in the magnetic dipole field of the earth is a basic tool to understand the magnetospheric and radiation belt phenomena. Since Starmer [l], the problem has been studied by several authors [2, 31. It must be remarked that the Van Allen radiation of the earth cannot be explained in the framework of a pure magnetic dipole field, even considering either higher multipole moments or short-term dynamic fluctuations in the magnetic field of the earth. In this context, a stochastic model [4] has recently been developed to explain the formation of the Van Allen 207
APPLIEDMATHEMATICSANDCOMPUTATION 25207-217 (1988) 0 Elsevier Science Publishing Co., Inc., 1988 52 Vanderbilt Ave., New York, NY 10017
oo9&m3/88/$03.50
208
LUIS ViiZQUEZ AND SALVADOR JIMkNEZ
radiation around the planets, and this modei accounts for important observational facts. Now, our purpose is to study a new stochastic model for the Van Allen radiation belt related to the above one. As a preparatory step to studying the new model, we analyze a new numerical scheme for the equations of motion of a charge in a magnetic dipole field. This scheme shows the same dynamics as the underlying continuous equation, in the sense that it is energy and timeinversion symmetry preserving. The features of the equations as well as the numerical schemes are described in Sections II and III together with some numerical results. Finally, the Painlevi! analysis is applied to the motion equations in order to get some insight into the integrability of the system (Section IV), but we get a negative result.
II.
EQUATIONS
The equation field is
OF MOTION
IN THE MAGNETIC
of motion of a nonrelativistic
charged
DIPOLE particle
FIELD
in a magnetic
dV e m-=-VxB, dt c where e and m are the charge and the mass of the particle, velocity of the light. The dipole magnetic field is given by B=v
and c is the
XA,
A = A( - Y, x,0), A=
-P
(x2 + y2 +
z2y2.
In this case, there are two conserved quantities: the energy, and the azimuthal angular momentum, due to the invariance of the Hamiltonian under rotations about the z-axis [2]. From this the three-dimensional problem is reduced to the two-dimensional motion of a particle in the pz plane [p = (x2 + ~~)l/~]. On introducing dimensionless variables, the motion equations become
d2p _=-dt2
aU ap ’
(3)
Motionof a Charge in a Dipole Field. 1
209
where 2
1
P qp,+;
;-
(p2+
i
z2)3’2 -
This is known as the Stermer problem, and it has been widely studied [2, 31. The dimensionless Hamiltonian is
H=S2+b2)+U(p,z), and the system shows trapped orbits for
III.
(5)
H < $ and untrapped ones for H > &.
A CONSERVATIVE NUMERICAL NUMERICAL RESULTS
SCHEME:
Up to now the finite-difference schemes [3] used to integrate the equations of motion (3) are useless for studying the long-time behavior of the trajectory, due to the cumulative error for long integration runs. A way to avoid this problem is to use numerical schemes showing a conserved discrete energy and which are time-inversion symmetry preserving like the underlying continuous equation. In this framework we have constructed the following two energy preserving difference schemes for the system (3): A.
Scheme z/p:
u( p”, 2”+2) - u( pn, 2”)
z”+2-222”+1+z” At2
=-
P “+2 - 2P”+l
z
=
-
P
@a)
-.z”
v( p”+2, z “f2)
-P”
At2
,
n+2
n+2
- u( p”, zn+2) - P”
(ebb)
And the associated discrete conserved energy is
E”=;(
““+;;““)2+f( p”‘bl”)
+;[U(p”+l,z”+l)+U(p”,z”)].
(7)
LUIS VAZQUEZ
210
AND SALVADOR
JIMkNEZ
At each step the finite-difference scheme (6) requires solving the functional equation (6a) for the unknown z n + 2, and next solving the equation (6b) for the unknown pn+2. This can be accomplished by Newton’s method. B. Scheme p/z: Pn+2 - 2P”+’
q P”+2, z”) - u( p”, 2”)
+ P”
At2
=-
n+2
z n+2_2z”+1_z”
u(pn+2,
At2
=
-pn
P
-
2”+2) .z
n+2
- U(p”+2, -
@a)
’
z”) @b)
zn
The associated discrete conserved energy is the same as given by (7). The difference from scheme z/p is that now we have first to find P”+~ by solving (8a) and next z n + 2 from the equation (8b). The usefulness of each scheme is related to the characteristics trajectories in the p-z plane.
P:lO? Z=O E = 0.0024992
+“”
-04
A0
2 -0.0355
, N’6780
.~.I..~~~~~.~I~~~,‘~~~~I~~~~‘~~~.I”’ -0 3
of the
-02
-01
0
‘,,,,I..“““‘I,““,“‘I”“““‘~ 1
FIG. 1. Trapped orbit in the z-p plane.
I
3
d
Z
Motion of a Charge in a Dipole Field. 1
07 44
“’
211
~..I~““.“~I~~~.‘~~~~I~~.~I~.~~I~~~,1~~~~I~~..”~.~I~~~“~~~.I~...~““l 43
-01
-01
FIG. 2.
0
Trapped
. ,
2
3
Z 4
orbit in the .zp plane.
These schemes are a generalization of the conservative scheme for systems with one degree of freedom considered by Vaquez [5]. In the computations we report in this work the mesh size At = 0.01 was chosen. In Figures 1 and 2 we represent two trapped orbits in the pz plane, while in Figures 3, 4, 5, and 6 we show four untrapped trajectories. In each figure, the initial conditions, the energy and the number of points N (iteration steps) are indicated. The numerical results have been obtained in double precision, and for the whole computed trajectories the energy is conserved up to the tenth decimal digit.
III.
THE PAINLEVI? ANALYSIS
The Painleve analysis has been used as a test of integrability for Hamiltonian systems [6-81. In this section, we apply it to the Hamiltonian (5). Such a system has a very rich structure [3]: it is not integrable globally, but only
212
LUIS VAZQUEZ AND SALVADOR JIMkNEZ
FIG. 3.
Untrapped orbit in the z-p plane.
locally. In the untrapped region the system is completely integrable, and also it shows regional integrabihty for the case of trajectories sufficiently near to certain periodic trajectories. It means that the integrability is energy dependent, in this way it seems very natural to apply the Painleve analysis in order to get some information about the relation between the integrability and the energy. Moreover, since the motion equations (3) are not rational functions, we must transform them in order to apply the Painlevi: analysis. To accomplish that we will consider two special changes, one of which incorporates explicitly the dependence on the energy of the system. First of all, let us summarize the Painleve analysis. We consider a system of two ordinary differential equations of the form
g(t, x, y, i, 0, c>=0,
6-J)
with f and g analytical in t and rational in the other variables. Then the
213
Motion of a Charge in a Dipole Field. I P
5T
Pzl.5
,
P:o 1
Z:O ,
izo.1
E-0034709
N=4000
FIG. 4.
Untrapped
orbit in the a-p plane.
Painleve analysis is performed basically as follows: (a) First we consider the variables extended to the complex plane and we insert in (9) the expansions
i=l
y=/?(t-toy+
f
aj(t-toy+j.
j=l
(b) We find all the possible values of the leading-term exponents p and q. (c) If all those values are integer, we must check the coefficients which remain arbitrary and we choose them and t,, as the constants of the motion. (d) If there are enought constants [four in the case of the system (9)], the system shows the Painleve property, i.e., all its singularities located at the arbitrary to (movable singularities) are at most poles.
214
LUIS V;lZQUEZ AND SALVADOR JIMliNEZ
FIG. 5.
Untrapped
orbit in the Z-P plane.
The PainlevC conjecture implies that the system is integrable if the property in (d) is satisfied [9, lo]. A weaker formulation of this conjecture has been considered recently [9]. Let us come back to our system. According to (4) the motion equations are the following:
f+
1 6-- i -P
-(p2+ z2)5'2 ( P
1
P
(p2+z2)3'2
P
1
3PZ
7+ I(
P
(p2+ z2)3'2)
z2-
2p2
(p2+ z2)5'2
= 0,
i = 0.
These equations are not rational in p and z. We have considered two ways to
Motion of a Charge ina Dipole Field. I
E~0040001.
215
N:2330
transform them into a rational system: (A) Let the variable change be
(z, P)-+(c P),
r2= (p2+ z2y2.
Then the motion equations become
++p-jj-/jL
ti-p ----+3-r2-p2
By considering the expansions p - a?
r2;”
li
5
=(), i
(11)
and r - prq, where 7 = t - to we
216
LUIS VAZQUEZ AND SALVADOR JIMkNEZ
have found four cases: (a) p = q = 0, a and p arbitrary; 1 (b) p = i, q = 0, a = - 4, /? arbitrary; (c) p = $9, q > $, a2 = * p3; (d) p = q = ;, a2 = [(Z 5 a)/31
1’3 [l - 2(2 k fi)],
p = 31’3 (2 + J?;)“‘.
Case (c) allows, for instance, irrational values of p and q; thus we get no conclusion about the integrability. (B) Instead of changing variables, we change the nonrational terms in (10) by using the constancy of the energy. We obtain
1 p-p
P-
:(z”+p2)+; i
i
I
-H
>: i
=O. II
Among other cases we have found p >/ i, 9 > p, a6 = - 9, and p arbitrary, so irrational values for p and 9 are allowed; thus no information about the integrabihty of the system is obtained even if the energy appears explicitly in the system. As a conclusion, the Painlevi: analysis does not give us any information about the regional integrability of the Stermer problem. One of the authors (L. V.) is very grateful to Professors S. Albeverio, Ph. Blunchard, and L. Streit for their invitation to join the Research Center Bielefeld-Bochum-Stochastics, where this work was finished.
Motion of a Charge in a Dipole Field. Z
217
REFERENCES
5 6 7 8 9 10
C. Sknmer, The Polar Aurora, Oxford, Clarendon, London, 1955. A. J. Dragt, Reu. Geophys. 3:255 (1965). A. J. Dragt and J. M. Finn, I. Ceophys. Res. 81:2327 (1976). Ph. Blanchard, Acta Phys. Austriaca Suppl. XXVI:185 (1984) and references therein. L. Vkzquez, Long time behavior in numerical solutions of certain dynamical systems, An. Fb. to appear. Y. F. Chang, M. Tabor, and J. Weiss, I. Math. Phys. 23531 (1982). T. Bountis, H. Segur, and F. Vivaldi, Phys. Reu. A. 251257 (1982). S. Jim$nez and L. Vkzquez, An. F/s. 82A:lQ (1986). B. Grammaticos, B. Dorizzi, and A. Ramani, J. Math. Phys. 242282 (1983). A. F. RaEada, A. Ramani, B. Dorizzi, and B. Grammaticos, J. Math. Phys. 26:708 (1985).