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
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
Mass variable systems has been important since the foundation of the classical
mechanics and have been relevant in modern physics . Among these type
of systems one could mentioned: the motion of rockets , the kinetic theory
of dusty plasmas , propagation of electromagnetic waves in dispersive and
nonlinear media , neutrinos mass oscillations , black holes formation ,
and comets interacting with solar wind . 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  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 . 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 . 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  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).
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
|r1− r2|3(r1− r2) (1a)
|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
|r1− r2|3(r1− r2) (2)
|r2− r1|3(r2− r1) − ˙ m2dr2
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
Let us then differentiate twice these coordinates with respect the time, taking
into consideration the equations (2) and (3). Thus, the following equations
¨ r = −(m1+ m2)G
− ˙ m2
(m1+ m2)2˙ r +(m1+ m2)m1¨ m2− 2m1˙ m2
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 +
r , and
˙ r2=˙R +
m1+ m2˙ r −
(m1+ m2)2r .(7)
Substituting these expressions in (5) and (6), one can see more clearly this
¨ r =
m1+ m2˙ r
− ˙ m2
m1+ m2R +
(m1+ m2)2˙ r +(m1+ m2)m1¨ m2− m1˙ m2
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
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
+ ˙ m2˙ r
? r + ˙ m2
r˙θ?θ + r ˙ ϕsinθ ? ϕ
where ? r,?θ and ? ϕ are unitary directional vectors,
? r = (sinθcosϕ,sinθsinϕ,cosθ) ,
?θ = (cosθcosϕ,cosθsinϕ,−sinθ) ,
˙? r =˙θ?θ + ˙ ϕsinθ ? ϕ ,
˙?θ = −˙θ ? r + ˙ ϕcosθ ? ϕ
˙? ϕ = sinθ ? r + cosθ?θ .
? 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 ,
¨ r = (¨ r − r˙θ2+ r ˙ ϕsin2ϕ)? r + (2˙ r˙θ + r¨θ + r ˙ ϕsinθcosθ)?θ
− ˙ m2˙ r ,(15a)
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
m2(¨ r − r˙θ2) = −Gm1m2
− ˙ m2˙ r ,(16a)
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
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
and Eq. (18) can be written as
dynamical system 
2= dm2/dr. This equation can be seen as the following autonomous
dt= v ,
A constant of motion for this system is a function K = K(r,v) such that the
following partial differential equation is satisfied 
= 0 .(22)
This equation can be solved by the characteristic method  from which the
following characteristic curve results
C(r,v) = m2
and the general solution of (22) is given by
K(r,v) = F(C(r,v)) , (24)
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
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 
L(r,v) = v
Thus, the Lagrangian is given by
The generalized linear momentum (p = ∂L/∂v) is
and the Hamiltonian is
Note from (25) and (29) that the constant of motion and Hamiltonian can be
where Veffis the effective potential energy defined as
v2+ Veff(r) (30)
2(r)+ Veff(r) ,(31)
This potential energy has an extreme value at the point
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
> 0 .
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
K − Veff(r)
where K and Pθare determinate by the initial conditions, K = K(ro,vo) and
from Eq. (30) as
o˙θo. The time of half of cycle of oscillation, T1/2, is obtained directly
K − Veff(r)
where r1 and r2 are the two return points deduced as the solution of the
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
√1 − e−αr
incoming (v < 0)
mieα(r1−r)+ mf(1 − e−αr) outgoing (v > 0)
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
eff(r) = −Gm1moo
(1 − e−αr) +
2moor2(1 − e−αr)
where Ei(x) is the exponential-integral function .
For the outgoing case, one has mo= mfand
eff(r) = −Gm1mf
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
Given the definition (35), the constant of motion, Lagrangian, generalized
linear momentum, and Hamiltonian are given by
v , (40)
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
G = 6.67 × 10−11m3/Kg sec ;
= 1017Kg m2/sec ;
moo= 106Kg ;
K = −8 × 1023Joules .
mf= 0.1 moo;
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
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
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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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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