regularization study with harmonic polynomial functions

8/10/2019 Regularization study with harmonic polynomial functions

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Regularization study withharmonic polynomial functions

by I. Szucs-Csillik, R. RomanRomanian Academy Cluj-Napoca

[email protected], [email protected]


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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)

regularization study with harmonic polynomial functions

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