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AAS 12-213

A CLOSED FORM SOLUTION OF THE TWO BODY PROBLEM

IN NON-INERTIAL REFERENCE FRAMES

Daniel Condurache* and Vladimir Martinusi

A comprehensive analysis, together with the derivation of a closed form solu-tion to the two-body problem in arbitrary non-inertial reference frames aremade within the present work. By using an efficient mathematical instrument,which is closely related to the attitude kinematics methods, the motion in thenon-inertial reference frame is completely solved. The closed form solutionsfor the motion in the non-inertial frame, the motion of the mass center, and therelative motion are presented in the paper. Dynamical characteristics analogueto the linear momentum, angular momentum and total energy are introduced.In the general situation, these quantities may be determined as functions oftime, and their derivation is presented within the paper. In the situation wherethe non-inertial frame has only a rotation motion, these quantities become firstintegrals in a larger sense, with respect to an adequately defined differentiationrule.

INTRODUCTION

The paper approaches the two-body problem in an arbitrary non-inertial reference frame, offeringits solution in the most general case, where the non-inertial frame has an arbitrary rotation andtranslation. To the knowledge of the authors, such a solution is not present in textbooks [1, 2, 3, 4],perhaps due to the unusual expression of the rst integrals.

The study of the problem in a non-inertial frame, in place of the classical study in an inertial one,has two simple motivations here: rst, it is a problem which models real situations, for examplethe way a telescope orbiting Earth sees the relative motion between two attracting bodies, or it isthe way the same motion is seen by an Earth-xed observer, from an Earth-xed reference frame,which rotates around the polar axis, and also translates in an orbit around the sun. Second, it isa very interesting mathematical problem, which has not been approached comprehensively in theliterature.

Assuming that the motion of the non-inertial frame is known, the solution is determined by usinga tensorial instrument introduced in 1995 by one of the authors [5]. This method is similar to theapproach used in rigid body kinematics, based on proper orthogonal and skew-symmetric tensorvalued functions, related by the Darboux equation, also known as the attitude kinematics equation.The paper is organized as follows. After the presentation of the non-linear initial value problemswhich model the motion of two interacting bodies with respect to a general non-inertial frame,the tensor method is presented, together with a vector differentiation operator, which makes the

Professor, Department of Theoretical Mechanics, Technical University Gheorghe Asachi, 700050, Iasi, Romania.Member AIAA, AAS. E-mail: daniel.condurache@gmail.com.

Post-doctoral Fellow, Faculty of Aerospace Engineering, Technion - Israel Institute of Technology, 32000, Haifa, Israel.E-mail: vladmartinus@gmail.com

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connection between the derivative in a rotating frame and the derivative in the inertial frame. Onehas to remark that all the computations are made in the non-inertial frame, and the inertial one isused only as a catalyst in solving the problem. The study of the two-body problem in non-inertialreference frames is made rst by studying the motion of the center of mass of the system, then bystudying the relative motion of one body with respect to a reference frame attached to the other one,and the relative motion of both bodies with respect to a reference frame attached to the center ofmass. The closed form solution for the motion of each body with respect to the original frame isalso presented. The motion with respect to the frame attached to the mass center is comprehensivelystudied, and new rst integrals of the motion are obtained. These rst integrals allow to offer ageometrical visualization of the motion.

In the end, dynamical characteristics of the motion, analogue to the classical linear momentum,angular momentum and total energy are introduced, and a method to determine them is also offered.In the situation where the non-inertial frame is only a rotational one, they become rst integralsof the motion, in a larger sense, and a generalized potential energy function is also introduced.Particularizing to the case where the reference frame has a constant instantaneous angular velocity,the aforementioned function becomes the classical potential energy. The present work constitutesthe basis for future research, where the full-body problem will be studied, where the internal torquesof the system are going to be incorporated into the mathematical model.

PROBLEM FORMULATION

The mathematical model for the two body problem in non-inertial reference frames is representedby the initial value problems (IVPs):

m1r1 + 2m1 r1+m1 ( r1) +m1 r1 +m1aF = F12,{r1 (t0) = r

01

r1 (t0) = v01

(1)

m2r2 + 2m2 r2+m2 ( r2) +m2 r2 +m2aF = F21,{

r2 (t0) = r02

r2 (t0) = v02

(2)

where rk denotes the position vector, rk the velocity vector of the particle Pk, k = 1, 2, related tothe reference frame where the motion takes place, mk are the masses of the two particles, aF isthe acceleration and is the angular velocity of the non-inertial reference frame where the motiontakes place. The vector map of real variable is supposed to be differentiable, and aF is supposedto be continuous. Fij denotes the force that acts upon the particle Pi due to the interaction withparticle Pj , i = j, i, j = 1, 2. The approach is based on the assumption:

Fij = Fij (|ri rj |) , i = j, i, j = 1, 2. (3)From the principle of reciprocal interactions, it follows that:{

F12 + F21 = 0;F12 (r1 r2) = 0. (4)

Within this paper, it is going to be proved that the two-body problem in non-inertial referenceframes may be solved exactly like in the inertial case, by solving two single-particle problems:

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1. The motion of the center of the center mass, denoted with C; its position vector with respectto the non-inertial frame F is denoted with rC (also see Figure 1);

2. The relative motion of one particle related to a reference frame associated to the other one,for example the motion of particle P2 related to P1. This motion is described by the vectormap r = r2 r1, where r2 is the solution to IVP (2) and r1 is the solution to IVP (1).

Figure 1. The two-body problem in a non-inertial reference frame.

The closed form solution will be presented with the help of a tensor instrument, rst introduced inRef. [5]. It is based on proper orthogonal and skew-symmetric tensor maps of real variable. A keyrole is played by a vector differentiation operator, which relates the motion in a rotating referenceframe to the motion in an inertial frame.

By using this tensor instrument, the following results are presented herein:

An explicit solution for the general motion of the center of mass; A representation theorem of the relative motion; the problem is reduced to the study of the

motion of one mass point in an arbitrary central force eld;

novel rst integrals, replicas to the linear momentum, angular momentum and energy conser-vation laws.

TENSORIAL CONSIDERATIONS

This section introduces the main mathematical instruments used in this paper. A tensorial mapand a vectorial differential operator will be dened. The following denotations are introduced:

V3 the three-dimensional space of free vectors; VR3 the set of functions of real variable, with values in V3; SO3 the special orthogonal group of second order tensors:

SO3 ={Q | QTQ = I3, det (Q) = 1

}; (5)

SOR3 the set of functions of real variable, with values in SO3;

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so3 the Lie-algebra of skew-symmetric second order tensors:

SO3 ={ | T =

}; (6)

soR3 the set of functions of real variable, with values in so3.

A Tensor Operator

The precession with arbitrary angular velocity is related to proper orthogonal tensor maps ofreal variable by a tensor IVP, similar to the one from attitude kinematics, which is also referred toas the Darboux equation.

Lemma 1 Consider the IVP:

Q = Q; Q (t0) = I3, t0 0. (7)

Then there exist a unique solution Q SOR3 , for any continuous map soR3 .

Proof. From the existence and uniqueness theorem, it follows that IVP (7) has a unique solutionQ = Q (t). One has to prove that Q is in SOR3 , meaning that Q

TQ = I3 and det (Q) = 1.We have that ddt

(QQT

)= QQ

T+ QQ

T= QQT QQT = 03, so QQT is a constant

differentiable function that satises(QQT

)(t0) = I3. Then, QQT = I3. Using that det (Q) is

also a continuous function which satises det (Q) {1, 1} and det (Q (t0)) = det I3 = 1, itfollows that det (Q) = 1. So Q SOR3 .

Remark 2 Lemma 1 is the well-known Darboux problem (also named the attitude kinematicsequation) (also see Ref [6]): determining the rotation tensor when the instantaneous angular ve-locity is known. The link between the rotation tensor map and the skew-symmetric tensor associatedto the angular velocity vector is given by IVP (7).

The solution to IVP (7) will be denoted F. The next result presents the properties of thistensorial orthogonal map.

Lemma 3 The map F satises:

1. F is invertible;

2. Fu Fv = u v, ()u,v VR3 ;3. |Fu| = |u| , ()u VR3 ;4. F (u v) = Fu Fv, ()u,v VR3 ;

5.d

dtFu = F

( u+ u

), ()u VR3 , differentiable;

6.d2

dt2Fu = F

( u+ 2 u+ ( u) + u

), ()u VR3 , differentiable.

The proof of Lemma 3 may be performed by elementary manipulations, therefore it is omitted.

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Comments and Remarks

1. The following denotation is introduced:

(F)1 = R (8)

Since F is the solution to IVP (1), it follows that R is the proper orthogonal tensor mapassociated to the instantaneous angular velocity , therefore it obeys the IVP:

R + R = 0; R (t0) = I3. (9)

2. There exists a numerical solution of the initial value problem (9) when the instantaneousangular velocity is an arbitrary continuous vector function. This is known as the Peano-Baker solution, it is obtained by iteration [7] and it is presented as a limit of innitesimalintegrals.

R (t) = I3 +n=1

1

n!

tt0

dt1...

tt0

dtnT [ (t1) , ..., (tn)] , (10)

with

T [ (t1) , ..., (tn)]=

(1)n2n

P(n)

n1k=1

(t(k) t(k+1)

) np=1

(t(p)

) , (11) (t) =

{0, t t01, t > t0

, (12)

and P (n) denotes the group of permutations of the set {1, ..., n} , n 2.

In case has xed direction, = , with constant unit vector and : R R,since (t1) (t2) = (t2) (t1), () t1,2 R, (also see Refs. [5, 8, 9]), then Rhas the explicit expression:

R (t) = exp

tt0

(s) ds

= I3 sin (t)

+ [1 cos (t)](

)2, (13)

where

(t) =

tt0

(s) ds. (14)

If is constant, then R has the explicit expression:

R(t) = exp [ (t t0) ] = I3sin [ (t t0)]

+{1 cos [ (t t0)]}

(

)2.

(15)

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If vector has a regular precession with angular velocity 1 around a xed axis, ex-pressed mathematically like:

= R10; 0 = (t0) ; R1 = exp [(t t0) 1] , (16)then the IVP (9) still has a timeexplicit solution [10, 11], expressed like:

R (t) = exp [(t t0) 1] exp [ (t t0) (1 + 0)] , (17)and written explicitly:

R =

{I3 + sin [1 (t t0)] 1

1+ {1 cos [1 (t t0)]}

(11

)2} (18){

I3 sin [ (t t0)]

+ {1 cos [ (t t0)]}(

)2},

where: = 1 + 0. (19)

A comprehensive study, together with a closed-form solution to the IVP (9) in the gen-eral case, may be found in Ref. [10]

Remark 4 Equations (13), (15) and(18) provide the closed form solution to the Darboux equationin the situation where vector has xed direction, it is constant, and it has a regular precession,respectively.

A Vector Differentiation Operator

A vector differential operator related to the angular velocity is introduced. It relates the deriva-tive of a vector valued function in an inertial reference frame to the derivative of the same vectorfunction expressed in a rotating reference frame. Like the regular derivative, it admits an inverseoperator, dened within this section. This derivation rule will prove to be useful in the study of themotion with respect to a non-inertial reference frame.

Dene the vector valued function differentiation rule ( ) : VR3 VR3 by:

( ) =( ) + ( ) (20)

For any arbitrary vectorial map u : R VR3 , it stands:

u = u + u (21)

The next result presents the properties of this operator, together with the link to the previouslydened tensor valued function F.

Lemma 5 The following afrmations hold:

1. =;

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2. (u + v) = u + v, ()u,v C2 (V R3 ) .3. (u) = u+u, ()u C2 (V R3 ) , () : R R, differentiable.4. (u v) = u v + u v, ()u,v C2 (V R3 ) .5. u v + u v = u v + u v = ddt (u v) , ()u,v C2

(V R3).

6. u = u+ 2 u+ ( u) + u, ()u C2 (V R3 ) .7.

d

dt(Fu)= F (u

) , ()u C2 (V R3 ) .8. Fu|t=t0 = u (t0) ;

d

dt(Fu)

t=t0

= u (t0) + (t0) u (t0) ;

The proof to Lemma 5 may be made by elementary mathematical manipulations, therefore it willnot be presented here.

The vector differentiation dened in Equation (20) makes the connection between the derivativeof a vector referred to a reference frame which rotates with angular velocity (denoted with a dotabove) and the derivative of the same vector referred to an inertial reference frame (denoted withprime), as it is depicted in Figure 2.

Figure 2. The vector differentiation operator transports the derivative from the ro-tating reference frame to the inertial reference frame.

The anti-derivation rule associated to the differentiaition rule ( ) is presented below.

Lemma 6 Consider b = b (t) a continuous vector valued function. Then the solution to the IVP:

u = b, u (t0) = u0 (22)

is expressed like:

u = R

u0 + tt0

RT (s)b (s) ds

(23)where R is dened in Equation (8).

Proof. Apply the tensor operator F to IVP (22) and take into account point (8.) from Lemma 5. Itfollows that:

d

dt(Fu) = Fb; Fu|t=t0 = u0 (24)

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By using point (7) from Lemma 5, together with the initial conditions from Equation (22), it follows:

Fu = u0 +

tt0

RT (s)b (s) ds. (25)

Equation (23) is obtained by applying R to the equality (25). The proof is nalized.

Remark 7 From Lemma 6 it follows that if a vector map u : R+ VR3 obeys the IVP:u = 0,u (t0) = u0, (26)

then vector u is the rotation with angular velocity of a constant vector u0 = u (t0):u = Ru0. (27)

This remark will help to offer a geometrical interpretation of the rst integrals of this particulardynamical system.

THE STUDYOFTHETWOBODYPROBLEM INNON-INERTIALREFERENCEFRAMES

This is the main section of the paper, presenting the theoretical study of the two-body problemwith respect to a non-inertial reference frame. The motion of the center of mass, the relative motionof a particle with respect to an adequately chosen reference frame attached to the other one, theprime integrals of the dynamical system formed by the two mass points, the relative motion ofthe two particles with respect to a reference frame attached to the center of mass are presented,with closed form equations of motion and novel features. The essential result is presented withinTheorem 9, which relates the inertial and the non-inertial approaches of the studied problem via anproper orthogonal tensor map of real variable.

The Motion of The Center of Mass

This Section presents the closed form solution for the motion of the center of mass. Introduce thedenotation:

rC =m1r1 +m2r2m1 +m2

. (28)

Vector rC models the position of the center of mass of the dynamical system comprising the twomass points with respect to the non-inertial reference frame to which the motion is referred. The thefollowing afrmation hold:

rC =m1r1 +m2r2m1 +m2

(29)

By summarizing eqs (1) and (2) and taking into account Equations (1) and (2), it follows that thevector function rC obeys the IVP:

rC + 2 rC + ( rC) + rC + aF = 0,rC (t0) =

m1r01 +m2r

02

m1 +m2

= r0C

rC (t0) =m1v

01 +m2v

02

m1 +m2

= v0C

(30)

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Theorem 8 The solution to IVP (30) is:

rC (t) = R

r0C + (v0C + 0 r0C) (t t0)t

t0

st0

RT ()aF () d

ds , t t0.

(31)

where 0 = (t0).

Proof. Remark that IVP (30) may be written with the help of the differential operator ( ) like:rC = aF ,{

rC (t0) = r0C ,

rC (t0) = v0C +

0 r0C .(32)

Apply the tensor operator F to IVP (32) and make the change of variable:

C= FrC ; (33)

then IVP (32) transforms into: C = FaF ,{

C (t0) = r0C ,

C (t0) = v0C +

0 r0C .(34)

By solving IVP (34) through direct integration, the solution to IVP (32) is obtained by taking intoaccount substitution (33), and it is expressed like:

rC (t) = R

r0C + (v0C + 0 r0C) (t t0)t

t0

st0

RT ()aF () d

ds . (35)

The proof is nalized.

REMARKS

In the situation when the acceleration of the non-inertial frame {F} is null, aF = 0, since|Ru| = |u| , ()u

(V R3), it follows that at any moment of time the mass center is

situated on a variable sphere that has an increasing radius:

R (t) =r0C + (v0C + 0 r0C) (t t0) , t t0. (36)

If v0C + 0 r0C = 0, then the motion of the center of mass takes place on a sphere with

constant radius r0C .

In case vector has a xed direction, or it has a regular precession, then the law of motionof the mass center rC = rC (t) may be written explicitly, also by taking into account theexpressions (13) and (18) of the tensor map R.

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Figure 3. If aF = 0, the motion of the center of mass takes place on a ruled (conical)surface, generated by a straight line which rotates with angular velocity .

If aF = 0, then the hodograph of the vector map that models the motion of the mass center isa curve that is situated on a ruled surface generated by the rotation with angular velocity of the straight line:

r = r0C +(v0C +

0 r0C)(t t0) , t t0. (37)

In this situation, the motion of the mass center may be decomposed into two:

a rectilinear uniform motion with velocity v0C + 0 r0C ;

a rotation with angular velocity of the straight line where the rectilinear motiontakes place.

The trajectory generated by these two independent motions is situated on a conical surface(surface generated by a straight line with a xed point, see Figure 3).

The Mathematical Model of The Relative Motion

The motion of particle P1 (m1) related to particle P2 (m2) , referred to the non-inertial frame{F} , is modeled by the vector valued function:

r = r2 r1, (38)where r1, r2 are the solutions to IVPs (1) and (2). After simple manipulations, it follows that vectorr obeys the IVP:

r+ 2 r + ( r) + r =F21m{

r (t0) = r02 r01 = r0

r (t0) = v02 v01 = v0

, (39)

where:m

=

m1m2m1 +m2

(40)

denotes the reduced mass of the system. The present approach will refer to the situation where F21is a central force, namely it may be written like:

F21 = f (r) r, (41)

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where f : R+ R is a scalar continuous function of real positive variable and r denotes the unitvector associated to r, dened by: r=r/r. It follows that the relative motion of P1 (m1) related toP2 (m2) , referred to frame {F} , is modeled by the IVP:

r+ 2 r + ( r) + r = f (r)m

r,

{r (t0) = r

0

r (t0) = v0

(42)

The Closed Form Solution of The Relative Motion

The IVP (42) models the motion of a particle having the mass dened in Equation (40); themotion is related to the non-inertial reference frame {F}. The solution to IVP (42) is determinedwithin the next result (also see Ref. [12,13]).

Theorem 9 The solution to IVP (42) is obtained by applying the tensor operator R to the solu-tion to the IVP:

=f ()

m

{ (t0) = r

0

(t0) = v0 + 0 r0

, (43)

where 0 = (t0) .

Proof. Eq (42) may be written using the previous considerations:

r =f (r)

mr. (44)

Apply operator F to Equation (44) and perform the same change of variable dened in Equation(33), =Fr; then IVP (43) is obtained by simple manipulations, also by taking into accountLemma 3. The proof is nalized.

Theorem 9 offers a simple way to solve the two body problem in non-inertial reference frames,by following the steps: (i) solve IVP (43); (ii) apply R to the solution to IVP (43)to determinethe solution to IVP (42).

The First Integrals of The Relative Motion

The rst integrals associated to IVP (42) are determined. They are deduced with the help of thetensor instrument previously introduced within the present work. The following denotation is made:

h0= r0 (v0 + 0 r0) . (45)

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Theorem 10 The rst integrals of IVP (42) are:

r (r+ r) = Rh0 = h (46)(replica to the specic angular momentum conservation law);

mr2

2+m (, r, r) +

m

2( r)2

f (r) dr = constant = E (47)

(replica to the specic energy conservation law).

Proof. By using operator ( ) previously introduced, it follows that:(r r) = r r = 0 (48)

and by using Lemma 6 one may write:

r r = R[(r r)

t=t0

]= Rh0. (49)

The existence of the rst integral dened in Equation (47) is proved by direct computations, bydifferentiating with respect to the time variable t the quantity from the left hand side.

Remarks

1. The rst integral (46) shows that the hodograph of the vector map h dened in Equation(46) is a spherical curve, which is drawn by the extremity of vector h0 = h (t0) whileprecessing with the instantaneous angular velocity (see Figure 4, left). If vector has a

Figure 4. Left: The hodograph of h is a spherical curve; Right: If has a xeddirection, the hodograph of h is a circular section, and it sweeps the surface of acircular cone.

xed direction, dened by the constant unit vector , then the hodograph of h (t) is a circularsection, drawn by the extremity of vector h0 = h (t0) while precessing around a xed axisdened by the unit vector . Vector h sweeps the lateral surface of a right circular conewith the instantaneous angular velocity (see Figure 4, right). As it follows from Equation(13), its explicit expression is:

h = (h0 ) sin (t) ( 0) cos (t) [ ( h0)] , (50)

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where

(t) =

tt0

() d. (51)

2. The rst integral (47) has an energetic meaning. It emphasizes the existence of a generalizedpotential energy function [14]:

V (t, r, r) = m (, r, r) +m

2( r)2

f (r) dr. (52)

The rst integral (47) may be rewritten:

m

2

r2+ V

(t, r,

r)= constant. (53)

If has a constant direction, then (t) = (t) ,with constant. Since r (r+ r) =Rh0, by dot-multiplying this relation with , it follows:(

, r,r)+ ( r)2 = Rh0 = h0 (R)T = h0 (54)

(If has constant direction, then R = RT = ). Consequently,(, r,

r)+

( r)2 = h0 , and in correlation with (52) it follows that:

V = m

[h0 1

2( r)2

]

f (r) dr. (55)

In this situation, V = V (t, r).

If is a constant vector, from (55) it follows that V = V (r) (i.e. V represents the classicalspecic potential energy).

The Laws of Motion in the Non-Inertial Reference Frame

From Equations (28) and (28) it follows that:r1 (t) = rC (t) m2

m1 +m2r (t) ;

r2 (t) = rC (t) +m1

m1 +m2r (t) .

(56)

The vector function rC (t) is the solution to IVP (30) and it has the explicit expression given inEquation (31). The vector function r (t) is the solution to IVP (42).

The First Integrals of the Two-Body System in The Non-Inertial Frame

The global dynamical characteristics of the system of two particles is studied herein. By introduc-ing the analogues of the classic dynamical characteristics of a system of particles (linear momentum,angular momentum, kinetic energy), time-varying quantities are described in place of the classic in-ertial conservation laws. By solving the differential equations derived within this Section, one may

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determine the expressions for the analogue quantities dened herein. However, a complete set ofrst integrals may be determined if the non-inertial frame {F} where the motion is referred to hasonly a rotation motion, namely when aF = 0.

It is known that the classic (inertial) dynamical characteristics of a two-particle system are:

P =

2k=1

mkrk (57)

(linear momentum);

K =2

k=1

mk (rk rk) (58)

(angular momentum);

Ekin =1

2

2k=1

mkr2k (59)

(kinetic energy).

In a non-inertial reference frame, the conservation laws of the quantities dened above are notvalid anymore.

Dene:

H =

2k=1

mk (rk + rk) (60)

(generalized linear momentum);

L =

2k=1

mk [rk (rk + rk)] (61)

(generalized angular momentum);

T =1

2

2k=1

mk (rk + rk)2 (62)

(generalized kinetic energy).

It is natural to search for a link between generalized linear momentum, angular momentum andenergy dened in Equations (60) (62) and their classic counterparts expressed in Equations (57) (59). One may write:

H = P+ (m1 +m2) rCL = K+ I0T = Ekin + VC

(63)

where the following notations were used:

I0 =2

k=1

mkrkrTk ; (64)

VC =

2k=1

[mk (, rk, rk) +mk

( rk)22

](65)

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(rC is dened in Equation (28), I0 is analogous with the the inertia tensor related to the non-inertialframe).

The following result may be stated:

Theorem 11 The following afrmations hold:

H+ H = (m1 +m2)aF ; (66)

L+ L = (m1 +m2) rC aF ; (67)d

dt

[T

f (r) dr

]= H aF . (68)

Proof. Remark that the differential equations from the two IVPs (1,2) which model the motion ofthe two particles with respect to {F} may be written compactly:

mkrk +mkaF = (1)k F21, k = 1, 2. (69)

Remark that the vector function H dened in Equation (60) may be rewritten like:

H =2

k=1

mkrk. (70)

By adding the two relations expressed in Equation (69), Equation (66) is obtained, also keeping inmind the denition of the vector differential operator ( ) from Equation (20).

By cross multiplying with rk in Equation (69) and then adding the two relations, it follows:

L+ L = L = 2

k=1

mkrk aF + (r2 r1) F21 = (m1 +m2) rC aF , (71)

where Equations (4) and (28) were taken into account.

Notice that:

T =2

k=1

mkrk (

d

dtrk

)=

2k=1

mkrk (

d

dtrk + rk

)=

2k=1

mkrk rk. (72)

By dot-multiplying with rk in Equation (69) and adding the obtained equalities, it follows that:

T = (

2k=1

mkrk

) aF +

(r2 r1

) F21 = H aF + f (r) r r, (73)where Equations (38), (40) and (41) were taken into account. Now take into account that:

f (r) r r =f (r) r rr

=ddr

(f (r) dr

)drdt

=ddt

(f (r) dr

). (74)

By replacing Equation (74) into (73), Equation (68) is obtained. The proof is nalized.

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Remark 12 Assuming that the acceleration aF of the non-inertial frame is known, from Equations(66) (68), one may determine H,L, T by direct integration, by using Lemma 6.

Corollary 13 In the situation when the non-inertial reference frame has only a rotation motion, i.e.aF = 0, then the following afrmations hold:

H+ H = 0 (75)(replica to the linear momentum conservation law);

L+ L = 0 (76)(replica to the angular momentum conservation law);

T

f (r) dr = constant (77)

(replica to the energy conservation law).

Corrolary 13 shows that in the situation when aF = 0, the vector functions H and L precesswith the angular velocity :

H = RH0 = R

[2

k=1

mk(v0k +

0 r0k)]

; (78)

L = RL0 = R

{2

k=1

mk[r0k

(v0k +

0 r0k)]}

. (79)

The following result offers a new insight to the variation of the classical dynamical quantitiesP,K,Ekin in the situation where aF = 0.

Corollary 14 The rst integrals of the two-particle system in a rotating reference frame {F} maybe expressed like:

|P+ (m1 +m2) rC | = constant; (80)K+2

k=1

mk [rk ( rk)] = constant; (81)

Ekin +

2k=1

[mk (, rk, rk) +mk

( rk)22

]

f (r) dr = constant. (82)

If vector has a xed direction, then the vectorial rst integrals (75) and (76) have explicitformulations:

H = (H0 ) sin (t) H0 cos (t) ( H0) ; (83)L = (L0 ) sin (t) L0 cos (t) ( L0) , (84)

where H0 = H (t0) , L0 = L (t0) . The hodographs of vectors H and L are circular sections. Hand L sweep the lateral surface of circular cones, with angular velocity (see Figure 5).

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Figure 5. If vector has a xed direction, vectors H,L sweep the lateral surface of circular cones.

The Motion Referred to the Mass Center Reference Frame

Consider the general situation where the translation acceleration of the non-inertial referenceframe {F} is non-null, aF = 0. When referred to a non-inertial reference frame {FC} originatedin the center of mass of the the system, having the same axes orientation like the original frame{F} , the motion of P1 (m1), P2 (m2) may be described completely by the solutions to the IVPs(30) and (42). Denote:

rk = rk rC , k = 1, 2. (85)Then from Equations (56) it follows that:

rk = (1)kmk

m1 +m2r, k = 1, 2. (86)

Equations (86) describe the motion of particles P1 (m1), P2 (m2) related to the non-inertial refer-ence frame of the mass center. The vector valued functions rk, k = 1, 2 obey the IVPs:

rk + 2 rk + ( rk) + rk =f (r)

mkrk,

{rk (t0) = r

0k r0C ;

rk (t0) = v0k v0C .

(87)

The dynamical global characteristics of the system in this reference frame are:

HC=2

k=1

mk (rk + rk) (88)

(generalized linear momentum);

LC=2

k=1

mk [rk (rk + rk)] (89)

(generalized angular momentum);

TC =1

2

2k=1

mk (rk + rk)2 (90)

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(generalized kinetic energy).

From Equations (88)(90), also by taking into account Equations (86), it follows that:

HC= 0; (91)

LC=m [r (r+ r)] ; (92)

TC =1

2m (r+ r)2 , (93)

where m is the reduced mass of the system, dened in Equation (40), and r is the solution to theIVP (42).

It follows that the vectorial rst integral LC obeys:

LC= RLC0 , (94)

Equation (94) shows that the hodograph of vector L is a spherical curve. If has constant direction,this hodograph is a circular section; vector L sweeps the surface of a circular cone, with angularvelocity . From (92) it follows:

r LC = 0, (95)therefore a geometrical visualization of the motion may be given: at any moment of time, the twoparticles are situated in a variable plane that is normal on vectorL. It also follows that the trajectoriesof the two particles are curves which are homotetical to the center of mass C, proven by:

r1 = m2m1

r2. (96)

The homotety ratio is m2/m1.The trajectories are plane curves if and only if vector LC has xed direction (also see Figure 6),

as follows from (95). The following result may be stated:

Lemma 15 The trajectories in the two-body problem in the mass center non-inertial referenceframe {FC} are planar if and only if the following conditions below are simultaneously satised:

(i) Vector has a xed direction, = (t) , with (t) continuous and constant unit vector;

(ii) LC0 = 0, where LC0 = LC (t0) .

Proof. If the trajectory is a plane curve contained in plane , it follows that vector LC isconstant, as it is orthogonal on plane and has a constant magnitude. It follows that RLC0 =constant, therefore d

dt

(RLC0

)= 0. It follows that RLC0 = 0, so RLC0 = 0. Further:

RLC0 = 0. It implies: LC= 0. (97)

Since LC is constant, then has xed direction, that of LC . Since = (t) , Equation (97)implies u LC0 = 0.

If conditions (i) and (ii) are satised, then:From (i), it follows: R = and (R)1 = R.

1667

From (ii), it follows: 0 = u LC0 = (Ru) LC0 = (Ru) [(R)1 LC

]= (Ru) (

RLC). It follows that: 0 = (Ru)

(RL

C)= R

(u LC) , and consequently u LC =

0. Then vector LC has a constant direction. Since it also has a constant magnitude, it is a constantvector, therefore the trajectory is a planar curve. The proof is nalized.

Figure 6. The trajectories are planar homothetical curves, situated in a variableplane, orthogonal on vector LC .

From the assumption (3), it follows that a scalar rst integral with energetic meaning may bededuced:

TC

f (r) dr = constant. (98)

Equation (98) may be rewritten like:

m

2r2 +m (, r, r) +

m

2( r)2

f (r) dr = constant. (99)

Dene:Ekin =

m

2r2, (100)

which represents the kinetic energy of the system related to the non-inertial frame {FC} , and de-note:

V (t, r, r) = m (, r, r) +m

2( r)2

f (r) dr (101)

the generalized potential energy. Then the conservation law (99) may be written like:

Ekin + V (t, r, r) = constant. (102)

Equation (102) shows that in the two-body problem referred to the rotating non-inertial referenceframe of the mass center {FC}, there exists the rst integral (102), with the generalized potentialenergy introduced in Equation (101).

If the angular velocity has constant direction, Equation (102) becomes:

Ekin + V (t, r) = constant, (103)

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V (t, r) = m

[LC0

1

2( r)2

]

f (r) dr, (104)

and if is constant:

Ekin + V (r) = constant, V (r) = m2( r)2

f (r) dr. (105)

CONCLUSIONS

The study of the two-body problem in an arbitrary non-inertial reference frames was approachedcomprehensively, and its closed form solution was determined. The motion of the center of mass,the relative motion of one body with respect to a frame attached to the other one, the relative mo-tion with respect to the center of mass and the motion of the two bodies in the non-inertial framewere approached by using the same tensor instrument, together with a vector differentiation ruleadequately dened. The results were presented in vectorial coordinate-free expressions. First inte-grals, equivalent to the classical dynamical characteristics of the motion (linear momentum, angularmomentum, total energy), for each body, as well as for the two-body system, were determined. Thepresent approach generalizes the classic two body-problem. The interaction force between the twobodies was considered to be a general central positional force, and the reference frame to which allthe quantities were referred to was considered to have both rotational and translational motion, bothassumed to be known a priori. Future works will study the same context for particular forms of theinteraction force (including the classic gravitational case), as well as several particular situations ofother central forces. Also, the system of full bodies will be studied, by incorporating the internaltorques which might act upon the considered celestial objects.

REFERENCES[1] T. Levi Civita and U. Amaldi, Lezioni di Mecanica Razionale. Nicola Zanichelli (Ed.), 19221926.[2] H. Goldstein and C. P. Poole, Classical Mechanics. Addison Wesley, 2001.[3] V. I. Arnold, V. Kozlov, and A. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics.

Springer, 3rd ed., 2006.[4] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry. Springer, 2nd ed., 1999.[5] D. Condurache, New Symbolic Procedures in The Study of the Dynamical Systems. Ph.D. Thesis, Tech-

nical University Gheorghe Asachi, Iasi, Romania, 1995.[6] G. Darboux, Lecons sur la theorie generale des surfaces et les applications geometriques du calcul

innitesimal, Vol. 1. Gauthier-Villars, Paris, 1887.[7] A. Fasano and S. Marmi, Analytical Mechanics: An Introduction. Oxford University Press, 2006.[8] D. Condurache and V. Martinusi, A Quaternionic Exact Solution to The Relative Orbital Motion Prob-

lem, Journal of Guidance, Control and Dynamics, Vol. 30, No. 1, 2010, pp. 201213.[9] J. Angeles, Fundamentals of Robotic Mechanical Systems. Springer, 2nd ed., 2002.

[10] V. Martinusi, Lagrangian and Hamiltonian Formulations in Relative Orbital Dynamics. Applicationsto Spacecraft Formation Flying and Satellite Constellations. Iasi, Romania: Ph.D. Thesis, TechnicalUniversity Gheorghe Asachi, January 2010.

[11] D. Condurache and V. Martinusi, A Tensorial Explicit Solution to Darboux Equation, The 2nd Interna-tional Conference Advanced Concepts in Mechanical Engineering, Technical University GheorgheAsachi, Iasi, Romania, June 2006.

[12] D. Condurache and V. Martinusi, Relative Spacecraft Motion in A Central Force Field, Journal ofGuidance, Control, and Dynamics, Vol. 30, No. 3, 2007, pp. 873876.

[13] D. Condurache and V. Martinusi, Exact solution to the relative orbital motion in a central force eld,2nd IEEE/AIAA International Symposium on Systems and Control in Aerospace and Astronautics, Shen-zen, China, December 2008.

[14] D. Condurache and V. Martinusi, Foucault Pendulum-like problems: A tensorial approach, Interna-tional Journal of Non Linear Mechanics, Vol. 43, 2008, pp. 743760.

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