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arXiv:physics/0511086v1 [physics.class-ph] 9 Nov 2005

Constant of motion, Lagrangian and Hamiltonian of the gravitational

attraction of two bodies with variable mass

G. L´ opez

Departamento de F´ ısica de la Universidad de Guadalajara

Apartado Postal 4-137

44410 Guadalajara, Jalisco, M´ exico

PACS: 03.20.+i

Octuber, 2005

ABSTRACT

The Lagrangian, the Hamiltonian and the constant of motion of the gravita-

tional attraction of two bodies when one of them has variable mass is con-

sidered. The relative and center of mass coordinates are not separated, and

choosing the reference system in the body with much higher mass, it is possible

to reduce the system of equations to 1-D problem. Then, a constant of motion,

the Lagrangian, and the Hamiltonian are obtained. The trajectories found in

the space position-velocity,(x,v), are qualitatively different from those on the

space position-momentum,(x,p).

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1. Introduction

Mass variable systems has been important since the foundation of the classical

mechanics and have been relevant in modern physics [1]. Among these type

of systems one could mentioned: the motion of rockets [2], the kinetic theory

of dusty plasmas [3], propagation of electromagnetic waves in dispersive and

nonlinear media [4], neutrinos mass oscillations [5], black holes formation [6],

and comets interacting with solar wind [7]. The interest in this last system

comes from the concern about to determinate correctly the trajectory of the

comet as its mass is changing. This system belong to the so called two-bodies

problem. The gravitational two-bodies system is one of the must well known

systems in classical mechanics [8] and is the system which made a revolution

in our planetary and cosmological concepts. Normally, one assumes that the

masses of these two bodies are fixed and unchanged during the dynamical

interaction [9]. However, this can not be true any more when one consider

comets as one of the bodies. Comets loose part of their mass as traveling

around the sun (or other star) due to their interaction with the solar wind

which blows off particles from their surfaces. In fact, it is possible that the

comet could disappear as it approaches to the sun [10]. So, one should consider

the problem of having one body with variable mass during its gravitational

interaction with other body.

In this paper, one considers the problem of finding the constant of motion,

Lagrangian, and Hamiltonian, for the gravitational interaction of two bodies

when one of them is loosing its mass during the gravitational interaction. The

mass of one of the bodies is assumed much larger than the mass of the other

body. Choosing the reference system on big-mass body, the three-dimensional

two-bodies problem is reduced to a one-dimensional problem. Then, one uses

the constant of motion approach [11] to find the Lagrangian and the Hamil-

tonian of the system. A model for the mass variation is given for an explicit

illustration of form of these quantities. With this model, one shows that the

trajectories in the space position-velocity (defined by the constant of motion)

are different than the trajectories on the space position-momentum (defined

by the Hamiltonian).

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2. Reference system and constant of motion

Newton’s equations of motion for two bodies interacting gravitationally, seen

from arbitrary inertial reference system, are given by

?

dt

d

dt

m1dr1

?

= −Gm1m2

|r1− r2|3(r1− r2) (1a)

and

d

dt

?

m2dr2

dt

?

= −Gm1m2

|r2− r1|3(r2− r1) ,(1a)

where m1and m2are the masses of the bodies, r1= (x1,y1,z1) and

r2= (x2,y2,z2) are the vectors position of the two bodies from our reference

system, G is the gravitational constant, and

|r1− r2| = |r2− r1| =

is the Euclidean distance between the two bodies. It will be assumed that m1

is constant and that m2varies with respect the time. Taking into consideration

this mass variation, Eqs. (1a) and (1b) are written as

?

(x2− x1)2+ (y2− y1)2+ (z2− z1)2

m1d2r1

dt2= −Gm1m2

|r1− r2|3(r1− r2) (2)

and

m2d2r2

dt2= −Gm1m2

|r2− r1|3(r2− r1) − ˙ m2dr2

dt

, (3)

where it has been defined ˙ m2as ˙ m2= dm2/dt. Now, let us consider the usual

relative, r, and center of mass, R, coordinates defined as

r = r2− r1, and

R =m1r1+ m2r2

m1+ m2

. (4)

Let us then differentiate twice these coordinates with respect the time, taking

into consideration the equations (2) and (3). Thus, the following equations

are obtained

¨ r = −(m1+ m2)G

r3

and

r −

˙ m2

m2

˙ r2

(5)

¨R =

− ˙ m2

m1+ m2

˙ r2+

2m2˙ m2

(m1+ m2)2˙ r +(m1+ m2)m1¨ m2− 2m1˙ m2

2

(m1+ m2)3

r .(6)

One sees that the relative motion does not decouple from the center of mass

motion. So, these new coordinates are not really useful to deal with mass

variation systems. In fact, using (4) , one has

r2= R +

m1

m1+ m2

r , and

˙ r2=˙R +

m1

m1+ m2˙ r −

m1˙ m2

(m1+ m2)2r .(7)

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Substituting these expressions in (5) and (6), one can see more clearly this

coupling,

¨ r =

?m1+ m2)G

r3

+

m1˙ m2

2

m2(m1+ m2)2

?

r −

˙ m2

m2

?

˙R +

m1

m1+ m2˙ r

?

(8)

and

¨R =

− ˙ m2

m1+ m2R +

m1˙ m2

(m1+ m2)2˙ r +(m1+ m2)m1¨ m2− m1˙ m2

2

(m1+ m2)3

r .(9)

However, one can consider the case for m1≫ m2(which is the case star-comet),

and consider to put our reference system just on the first body (r1=?0). In

this case, Eq. (3) becomes

m2d2r

dt2= −Gm1m2

r3

r − ˙ m2˙ r ,(10)

where r = r2= (x,y,z). Using spherical coordinates (r,θ,ϕ),

x = rsinθcosϕ , y = r sinθsinϕ , z = r cosθ ,(11)

Eq. (10) can be written as

m2d2r

dt2= −

?Gm1m2

r2

+ ˙ m2˙ r

?

? r + ˙ m2

?

r˙θ?θ + r ˙ ϕsinθ ? ϕ

?

, (12)

where ? r,?θ and ? ϕ are unitary directional vectors,

? r = (sinθcosϕ,sinθsinϕ,cosθ) ,

?θ = (cosθcosϕ,cosθsinϕ,−sinθ) ,

and

with

˙? r =˙θ?θ + ˙ ϕsinθ ? ϕ ,

˙?θ = −˙θ ? r + ˙ ϕcosθ ? ϕ

˙? ϕ = sinθ ? r + cosθ?θ .

(13a)

with

(13b)

? ϕ = (−sinθ,cosθ,0) ,

Since one has that r = r? r, it follows that

+(2˙ r ˙ ϕsinθ + r¨ ϕsinθ + 2 ˙ ϕ˙θcosθ)? ϕ ,

and Eq. (12) is discomposed in the following three equations

m2(¨ r − r˙θ2+ r ˙ ϕsin2ϕ) = −Gm1m2

m2(2˙ r˙θ + r¨θ + r ˙ ϕsinθcosθ) = ˙ m2r˙θ ,

with(13c)

¨ r = (¨ r − r˙θ2+ r ˙ ϕsin2ϕ)? r + (2˙ r˙θ + r¨θ + r ˙ ϕsinθcosθ)?θ

(14)

r2

− ˙ m2˙ r ,(15a)

(15b)

and

m2(2˙ r ˙ ϕsinθ + r¨ ϕsinθ + 2 ˙ ϕ˙θcosθ) = ˙ m2r ˙ ϕsinθ .(15c)

Thus, one has obtained coupling among these coordinates due to the term ˙ m2.

Nevertheless, one can restrict oneself to consider the case ˙ m2r ≈ 0. For this

case, it follows that ˙ ϕ = 0, and the resulting equations are

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m2(¨ r − r˙θ2) = −Gm1m2

r2

− ˙ m2˙ r ,(16a)

and

m2(2˙ r˙θ + r¨θ) = 0 .(16b)

Let mobe the mass of the second body when this one is very far away from

the first body (when a comet is very far away from the sun, the mass of the

comet remains constant). Since m2?= 0 on (16b), the expression inside the

parenthesis must be zero. In addition, one can multiply this expression by

mor to get the following constant of motion

Pθ= mor2˙θ . (17)

Using this constant of motion in (16a), one obtains the equation

d2r

dt2= −Gm1

r2

+

P2

m2

θ

or3−

˙ m2

m2

?dr

dt

?

.(18)

This equation represents a dissipative system for ˙ m2> 0 and anti-dissipative

system for ˙ m2< 0. Suppose now that m2is a function of the distance between

the first and second body, m2= m2(r). Therefore, it follows that

dm2

dt

=dm2

dr

dr

dt,

(19)

and Eq. (18) can be written as

d2r

dt2= −Gm1

r2

+

P2

m2

θ

or3−m′

2

m2

?dr

dt

?2

, (20)

where m′

dynamical system [12]

2= dm2/dr. This equation can be seen as the following autonomous

dr

dt= v ,

dv

dt= −Gm1

r2

+

P2

m2

θ

or3−m′

2

m2v2.(21)

A constant of motion for this system is a function K = K(r,v) such that the

following partial differential equation is satisfied [13]

v∂K

∂r+

?−Gm1

r2

+

P2

m2

θ

or3−m′

2

m2v2

?∂K

∂v

= 0 .(22)

This equation can be solved by the characteristic method [14] from which the

following characteristic curve results

C(r,v) = m2

2(r)v2+ 2Gm1

?m2

2(r) dr

r2

−2P2

m2

θ

o

?m2

2(r) dr

r3

, (23)

and the general solution of (22) is given by

K(r,v) = F(C(r,v)) , (24)

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where F is an arbitrary function of the characteristic curve. One can have a

constant of motion with units of energy by selecting F as F = C/2mo. That

is, the constant of motion is given by

K(r,v) =m2

2(r)

2mo

v2+Gm1

mo

?m2

2(r) dr

r2

−P2

m3

θ

o

?m2

2(r) dr

r3

. (25)

2. Lagrangian and Hamiltonian

Given the time independent constant of motion (25), the Lagrangian of the

system (20) can be obtained using the following known expression [11]

L(r,v) = v

?K(r,v) dv

v2

. (26)

Thus, the Lagrangian is given by

L(r,v) =m2

2(r)

2mo

v2−Gm1

mo

?m2

2(r) dr

r2

+P2

m3

θ

o

?m2

2(r) dr

r3

. (27)

The generalized linear momentum (p = ∂L/∂v) is

p =m2

2(r)

mo

v ,(28)

and the Hamiltonian is

H(r,p) =

mop2

2m2

2(r)+Gm1

mo

?m2

2(r) dr

r2

−P2

m3

θ

o

?m2

2(r) dr

r3

. (29)

Note from (25) and (29) that the constant of motion and Hamiltonian can be

written as

K(r,v) =m2

2mo

and

mop2

2m2

where Veffis the effective potential energy defined as

2(r)

v2+ Veff(r) (30)

H(r,p) =

2(r)+ Veff(r) ,(31)

Veff(r) =Gm1

mo

?m2

2(r) dr

r2

−P2

m3

θ

o

?m2

2(r) dr

r3

.(32a)

This potential energy has an extreme value at the point

r∗=

P2

θ

Gm1m2

o

(32b)

which depends on mobut it does not depend on the model for m2(r). One can

see that this extreme value is a minimum for m2(r∗) ?= 0, since one has that

?d2Veff

dr2

r=r∗

?

=(Gm1mo)4mom2

2(r∗)

P6

θ

> 0 .

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On the other hand, because of the expression (28), one could expect different

behavior of a trajectory in the phase space (r,v) and the phase space (r,p).

The trajectory r(θ) is found using the relation dr/dt = (dr/dθ)˙θ, and the Eq.

(17) in (30) to get

?θ

θodθ =

Pθ

2m3

?

o

?r

ro

m2(r) dr

r2?

K − Veff(r)

, (34a)

where K and Pθare determinate by the initial conditions, K = K(ro,vo) and

Pθ= mor2

from Eq. (30) as

1

√2mo

r1

o˙θo. The time of half of cycle of oscillation, T1/2, is obtained directly

T1/2=

?r2

m2(r) dr

?

K − Veff(r)

, (34b)

where r1 and r2 are the two return points deduced as the solution of the

following equation

Veff(ri) = K ,i = 1,2 .(34c)

3. Model of Variable Mass

As a possible application of (25) and (29), consider that a comet looses material

as a result of the interaction with star wind in the following way (for one cycle

of oscillation)

√1 − e−αr

m2(r) =

moo

incoming (v < 0)

mieα(r1−r)+ mf(1 − e−αr) outgoing (v > 0)

(35)

where mooor mf (where mf = 2mi− mooby symmetry) is the mass of the

comet very far away from the star (in each case), miis the mass of the comet

at the closets approach to the star (distance r1), mi= moo

is a factor that can be adjusted from experimental data. Thus, the effective

potential (32a) has the following form for the incoming case (mo= moo)

√1 − e−αr1, and α

V(in)

eff(r) = −Gm1moo

r

(1 − e−αr) +

P2

θ

2moor2(1 − e−αr)

Ei(−αr) +αP2

+

?

GM1mooα +α2P2

θ

2moo

?

θe−αr

2moor

, (36a)

where Ei(x) is the exponential-integral function [15].

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For the outgoing case, one has mo= mfand

V(out)

eff(r) = −Gm1mf

r

+

˜P2

θ

2mfr2

+Gm1(mieαr1− mf)2

mf

˜P2

m3

?

−e−2αr

−e−2αr

?

?

r

− 2αEi(−2αr)

2r2+αe−2αr

r

−e−αr

r

−e−αr

2r

?

−

θ(mieαr1− mf)2

f

?

+ 2Ei(−2αr)

?

+α2

2Ei(−αr)

?

+2Gm1(mieαr1− mf)

−2˜P2

m2

− αEi(−αr)

2r2+αe−αr

θ(mieαr1− mf)

f

?

, (36b)

where˜Pθis defined now as˜Pθ= mfr2˙θ. The extreme point of the effective

potential (32b) for the incoming and outgoing cases is given by

r∗

in=

P2

θ

Gm1m2

oo

,r∗

out=

P2

θ

Gm1m2

f

. (37)

Given the definition (35), the constant of motion, Lagrangian, generalized

linear momentum, and Hamiltonian are given by

K(i)(r,v) =m2

2(r)

2mo

v2+ V(i)

eff(r) ,(38)

L(i)(r,v) =m2

2(r)

2mo

v2− V(i)

eff(r) ,(39)

p(i)(r,v) =m2

2(r)

mo

v , (40)

and

H(i)(r,p) =

mop2

2m2

2(r)+ V(i)

eff(r) , (41)

where i = in for the incoming case, and i = out for the outgoing case. As

an example of illustration of this model, let us use the following parameters

to estimate the dependence of several physical quantities with respect the

parameter α,

G = 6.67 × 10−11m3/Kg sec ;

Pθ

= 1017Kg m2/sec ;

moo= 106Kg ;

K = −8 × 1023Joules .

mf= 0.1 moo;

and (42)

Fig. 1 shows the curves of Veff(r) for several values of α (incoming case). As

one can see from this figure, the location of the minimum does not change,

but the minimum value of Vefftends to disappear as α goes to zero. Also for

the incoming case, Fig. 2 shows how the minimum distance of approximation

of the two bodies, r1, and maximum distance, r2, behave as a function of the

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parameter α. As one can guess, the following limit is satisfied limα→0r1 =

limα→0r2 = r∗which will become a inflexion point for Veff. Fig. 3 shows

the velocity (v) and normalized linear momentum (p/mo) as a function of r

for several values of α and for the incoming case. All the trajectories start at

r2= 200 and finish at r1(α). One can see the difference of the trajectories in

(a) with respect to (b) due to position dependence of the momentum, relation

(40).

Conclusions

The Lagrangian, Hamiltonian and a constant of motion of the gravitational

attraction of two bodies when one of them has variable mass were given. One

found that the relative and center of mass coordinates are coupled due to

this mass variation. However, chosen the reference system in the much more

massive body, it was possible to reduce the system to 1-D problem. Then, the

constant of motion, Lagrangian and Hamiltonian were obtained. One main

feature of these quantities was the appearance of an effective potential, which

is reduced (when ˙ m2= 0) to the usual gravitational effective potential of two

bodies with fixed masses. Other feature was the distance dependence of the

generalized linear momentum. A model for comet-mass-variation was given

which depends on the parameter α. A study was made of the dependence with

respect to α of Veff, minimum and maximum distance between the two bodies,

and the trajectories in the spaces (r,v) and (r,p). Of course, the problem of

the interaction comet-star with the variation of mass deserves more complete

analysis. The intention here with this example was to show explicitly the form

of the constant of motion, Lagrangian, and Hamiltonian and to point out the

different trajectories behavior in the spaces (r,v) and (r,p) arising from the

constant of motion and Hamiltonian.

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Figure Captions

Fig. 1 V(in)

α = 0.01 (2); and α = 0.005 (3).

eff(r) with the values of the parameters given on (42), for α = 1 (1);

Fig. 2 Maximum (r2) and minimum (r1) distances between the two bodies as

a function of the parameter α.

Fig. 3 (a): Trajectories in the plane (r,v); (b): Trajectories in the plane (r,p).

α = 1 (1), α = 0.01 (2), and α = 0.005 (3).

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0 100200 300400

500

r

-8×1023

-6×1023

-4×1023

-2×1023

0

2×1023

4×1023

(1)

(2)

(3)

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0

50

100

α

150

200

0

20

40

60

80

r1

r2

Page 14

25 50 75

100

125 150 175

200

r

0

-V

255075

100

125150175

200

r

0

-P/m0

(1)

(1)

(2)

(2)

(3)

(3)

(a)

(b)

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