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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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References
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G. L´ opez, Ann. of Phys., 251,2 (1996),372.
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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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0100 200300400
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
255075
100
125150 175
200
r
0
-V
255075
100
125 150 175
200
r
0
-P/m0
(1)
(1)
(2)
(2)
(3)
(3)
(a)
(b)