Regularization study with harmonic polynomial functions
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Transcript of Regularization study with harmonic polynomial functions
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Regularization study withharmonic polynomial functions
by I. Szucs-Csillik, R. RomanRomanian Academy Cluj-Napoca
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Abstract
The regularization of the celestial bodiesmotion is significant and well-studied inspace dynamics.
Levi-Civita (1920) used firstly theharmonic polynomial function (of order 2)for regularization.
Generalizing the coordinatetransformations, we found newregularization methods of n-th order.
Applying and then comparing these newregularization methods the study ofcollision and escape orbits become moredetailed.
These numerical methods are fast,because we have no singulariti
es in the
equations of motion.
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Introduction What is the regularization? (singularity,
collision, manifold)
The collision must be slowed down by a timetransformation. So that the approach of the actualvelocity can be handled by infinity, and blow up by thecoordinate transformation.
What is harmonic conjugated polynomialfunctions?
The harmonic conjugate to a given polynomial functionf(Q1,Q2) is a polynomial functiong(Q1,Q2) such that theholomorphic function u(z)=f(Q1,Q2)+ig(Q1,Q2) is
complex differentiable and satisfies the Cauchy-Riemann equations
wheref,g, Q1, Q2are real,zcomplex variable,
z=Q1+iQ2 .
1221 Q
g
Q
fand
Q
g
Q
f
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Harmonic and conjugatepolynomial functions
We can find harmonic and conjugate functions, by using the theory of
complex functions. We denotez = Q1+iQ2a single complex variable andh:C, h(z) = h(Q1+iQ2) =f(Q1,Q2) + ig(Q1,Q2) a complex-valuedfunction. From the theory of complex numbers we know that: if h(z) is acomplex function, than its real and imaginary parts are harmonic functions.That means:
Table 1. Polynomial functions of nth order
To generalize the harmonic polynomial function consider thefunction h(z) = z. Thenth order polynomial functions are h(zn) = zn,n is positive integers,andzn= (Q1+ iQ2)n. The harmonic
polynomials are given in the table above.
002
2
2
2
1
2
2
2
2
2
1
2
Q
g
Q
gand
Q
f
Q
f
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R3BP restricted 3 body problemFor simplicity, we consider in the following that the third body moves
into the orbital plane. Denoting S1 and S2 the components of the binary
system (masses m1 and m2), the equations of motion of the test particle inthe coordinate system xS1y are
These equations have singularities in terms 1/r1 and 1/r2.
This situation corresponds to collision of the test particle
with S1 and S2. If the test particle approaches very closely
to one of the primaries, such an event produces large
gravitational force and sharp bends of orbit. The removing
of these singularities can be done by regularization.
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Regularization The first step of regularizationrepresents a
conformal mapping. It contains the geometricinformation and it controls the accuracy of theshape of the orbit. In our case the nth order transformations are
the harmonic polynomial functions given in thetable 1.
The second step of regularizationis theessential one, since it controls the kinematics'aspects and it performs the regularization. The new time, namely, the fictitious time, is
introduced in the following way:
The third step of regularization is the Jacobiintegral, it controls the energy preservation:
nnrr
d
dt21
Crq
q
rqq
q
qq
dt
qd
dt
qd
21
2
2
2
1
1
1
21
1
2
12
2
2
2
1
2
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nth order equations of motionIn order to obtain canonical equations, whenf andg are
harmonic and conjugate polynomial functions, we have to
write first the corresponding Hamiltonian equation. Let us
consider the complex variablez = Q1+i Q2, which can be
written in the trigonometric form:
The new nth order Hamiltonian
the new nth order
canonical equations
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Application in Earth-Moon system
In order to obtain 'similar' trajectories, the canonical
equations of motion of the test particle must be integrated,using initial conditions. We denote
the initial conditions for the canonical equations in thephysical plane, and
the initial conditions for the canonicalequations in the regularized plane. Theconnection between these initial
conditions is given by the equations:
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The motion in physical planeInitial conditions:
q=0.0123 (Earth-Moon system)
q10=0.6, q20=0.4, p10=0.1, p20=0.6 (in physical plane)
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S1 and S2 position S1, S2 positions and the initial conditions for
different methods of regularization (the case n = 1,polynomial function, corresponds to the physicalplane). Applying the transformation of coordinatesgiven by f and g, not only the trajectory will bechanged, but the positions of the components of the
binary system S1 and S2 will be changed too. Thebolded coordinates are those used to obtain thetrajectories of the test particle in this article.
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After regularization
Compare the next figures, which presented the nth
order trajectories for each type of functionsf andg
presented in the table.
All trajectories are represented for a time interval
equal with an orbital period.
As we can see, changing the functionsf andg, not
only the initial positionP0 is changed, but the
shape of the trajectory changes too.
Iff andg are polynomial functions of n degree,
the greater is n, the greater is the distance ofP0 to
the origin of the coordinate system.
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Conclusions The unique condition imposed to the generating functions
f andg was that those functions are harmonic and
conjugated. Using the theory of complex analysis, wegenerated nth order harmonic and conjugated functions.
We obtained in each case the canonical equations ofmotion of the test particle.
We integrated these equations, using initial conditions,obtained from the initial conditions used in the physical
plane. Using the regularization we realize:
that the new canonical equations of motion arewithout singularity (regular), so the numericalintegrator is faster,
the trajectories conserve the shapes of the orbit (near
the collision the manifold blow up), the motion is slowed down.
In many situations, the distance from the trajectory'spoints to the more massive star of the binary system (S1)increases. In the case of the polynomial functions, thegreater is the degree of the polynomials, the greater is the
distance. This remark can be useful in some applications(for example if some "objects" are located near S1).
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Thank you for your attention!
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A Selective Bibliography Birkhoff, G.D.: The restricted problem of three bodies. Rend. Circ. Mat.
Palermo 39, 1-70 (1915) Boccaletti, D., Pucacco, G.: Theory of orbits, vol. 1. Springer-Verlag,
Berlin Heidelberg New York (1996) Burrau, C.: ber einige in Aussicht genommene Berechnung, betreffend
einen Specialfall des Dreikrperproblems, Vierteljahrschrift Astron. Ges.41, 261-266 (1915)
Carathodory, C.: Theory of functions of a complex variable, vol. 1. AMSChelsea Publishing, Providence, Rhode Island (2001)
Csillik, I.: Regularization methods in celestial mechanics. House of the
Book of Science, Cluj (2003) Lematre, G.: Regularization of the three-body problem, Vistas in
Astronomy 1,207-213 (1955) Levi-Civita, T.: Sur la rvolution qualitative du probleme restreint de trois
corps, Acta Mathematica, 30, 305-327 (1906) Roman, R., Szcs-Csillik, I.: Regularization of the circular restricted
three-body problem using similar coordinate systems. Astrophysics andSpace Science 338(2), 233-243 (2012)
Szcs-Csillik, I., Roman, R.: New regularization of the restricted three-body problem. RoAJ 22(2), 135-145 (2012)
Stiefel, L., Scheifele, G.: Linear and regular celestial mechanics.Springer, Berlin (1971)
Szebehely, V.: Theory of orbits. Academic Press, New York 1967 Thiele, T.N.: Recherches numriques concernant des solutions
priodiques d'un cas spcial du probleme des trois corps. Astron. Nachr.138, 1-10 (1896)