Received 12 August 1997; accepted 14 April 1998. We consider the derivation of Jacobi's integral of motion for a particle moving in a uniformly rotating potential ...
Noether’s theorem, rotating potentials, and Jacobi’s integral of motion C. M. Giordano and A. R. Plastino Citation: American Journal of Physics 66, 989 (1998); doi: 10.1119/1.19011 View online: http://dx.doi.org/10.1119/1.19011 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/66/11?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in A Linear/Rotational Motion Exercise with a Tape Deck Counter Phys. Teach. 40, 394 (2002); 10.1119/1.1517876 Solving boundary-value electrostatics problems using Green’s reciprocity theorem Am. J. Phys. 69, 1280 (2001); 10.1119/1.1407256 A note on integration of scalar products Phys. Teach. 39, 270 (2001); 10.1119/1.1375463 Using great circles to understand motion on a rotating sphere Am. J. Phys. 68, 1097 (2000); 10.1119/1.1286858 Continued fractions and the harmonic oscillator using Feynman’s path integrals Am. J. Phys. 65, 390 (1997); 10.1119/1.18545
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Noether’s theorem, rotating potentials, and Jacobi’s integral of motion C. M. Giordano and A. R. Plastino Facultad de Ciencias Astrono´micas y Geofı´sicas de la Universidad Nacional de La Plata, La Plata, Argentina
~Received 12 August 1997; accepted 14 April 1998! We consider the derivation of Jacobi’s integral of motion for a particle moving in a uniformly rotating potential, as a simple illustrative application of Noether’s theorem. We also provide some examples of Jacobi’s integral. In particular, we show that some elementary problems usually discussed in texts of classical mechanics can be interpreted in terms of Jacobi’s integral. © 1998 American Association of Physics Teachers.
I. INTRODUCTION Noether’s theorem constitutes one of the most profound and beautiful results of theoretical physics.1–6 For systems whose equations of motion are derivable from Hamilton’s principle, it provides a general and systematic method for obtaining conservation laws from the symmetries of the corresponding action. A nontechnical exposition of the meaning of the theorem, together with some interesting information about the life and times of Emmy Noether can be found in Ref. 6. Frequently, Noether’s theorem is presented in the context of field theory.4,7,8 However, it can also illuminate the relation between symmetries and integrals of motion in the case of discrete particle dynamics. Bobillo-Ares5 ~see also Ref. 3! provided a clear and elegant derivation of Noether’s theorem for particle dynamics together with some interesting examples. More mathematically oriented formulations, presented in the language of differential geometry, can be found in Refs. 9 and 10. Neuenschwander and Starkey11 studied the relation between Noether’s theorem and adiabatic invariants. These last results were later applied to the adiabatic invariants of plasma physics.12 989
Am. J. Phys. 66 ~11!, November 1998
The conservation laws most commonly discussed within Noether’s approach are the conservation of energy, associated with the invariance under time translations; the conservation of linear momentum, associated with the invariance under space translation; and the conservation of angular momentum, related to the invariance under rotations. BobilloAres provided some further examples of symmetries associated with conservation laws. The aim of the present article is to consider Jacobi’s integral of motion ~also referred to as Jacobi’s constant of motion!13–17 in the framework of Noether’s theorem. Jacobi’s constant of motion appears whenever a particle moves within a uniformly rotating ~with respect to an inertial reference system! potential.15 Jacobi’s constant is given by C J5
m ˙2 ˙2 ˙2 m ~ X 1Y 1Z ! 1m f 2 v 2 ~ X 2 1Y 2 ! , 2 2
~1!
where (X,Y ,Z) are the Cartesian coordinates, with respect to a rotating reference frame, of a particle of mass m moving in the rotating gravitational potential f and v denotes the con© 1998 American Association of Physics Teachers
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stant angular velocity. Jacobi’s integral has many astronomical applications, particularly in the fields of celestial mechanics17–28 and stellar dynamics.15 This integral was first introduced by Jacobi in his study of the circular restricted three-body problem.13,14,16,26,29 This important problem of celestial mechanics involves a system of three gravitationally interacting particles, the mass of one of them being much smaller than the masses of any of the other two ~principal! particles. The principal masses are assumed to describe circular orbits around their common center of mass, under their mutual gravitational attraction, without being perturbed by the small particle. This last particle, in turn, moves in the gravitational field generated by the two principal bodies. Clearly, the small particle moves in a gravitational potential field that rotates uniformly with respect to an inertial reference frame. The restricted three-body problem constitutes a very useful model for the study of the motion of asteroids and comets. In these cases, the two principal bodies are the Sun and planet Jupiter. Jacobi’s constant also finds interesting applications in galactic dynamics.15 Many galaxies have a nonaxisymmetric mass distribution that appears to rotate rigidly. These ones are called barred galaxies, since they usually show a barshaped concentration of mass. Jacobi’s constant plays a fundamental role in the analysis of the orbits described by individual stars within barred galaxies.15 Jacobi’s integral is usually derived in dynamical astronomy texts14–18,20–26,29,30 by recourse to the explicit equations of motion of the system. Although it is seldom discussed in physics texts ~a notable exception is the text by Symon13!, it constitutes an important example of an integral of motion in particle dynamics. In the present work we show that Jacobi’s integral can be presented as a simple illustration of Noether’s procedure. This fact, as far as we know, has not been considered in any textbook of classical mechanics or dynamical astronomy or in any of the tutorial articles devoted to Noether’s theorem. Furthermore, we show that some elementary problems of mechanics that appear in many texts can be reinterpreted in terms of Jacobi’s integral. The paper is organized as follows. In Sec. II we give a brief review of Noether’s theorem. In Sec. III we consider the case of uniform rotating potentials and obtain, by recourse to Noether’s approach, the Jacobi integral of motion. In Sec. IV we discuss some elementary mechanical problems that can be interpreted in terms of Jacobi’s integral. There we also consider the case of potentials with helicoidal symmetry. Some conclusions are drawn in Sec. V.
E
I5
t1
Here we provide an outline of Noether’s theorem in particle dynamics, as required for our present purposes. A detailed account, within a more general framework, can be found in Refs. 3 and 5. Let us consider a classical dynamical system characterized by the Lagrangian ~2!
where xPR 3 is the vector position, with respect to an inertial reference system, of a particle of mass m, and f denotes the potential energy per unit mass. The concomitant action integral Am. J. Phys., Vol. 66, No. 11, November 1998
~3!
x8 5x1 e K~ x,x8 ,t !
~4!
t 8 5t1 e q ~ x,x8 ,t ! .
~5!
and
The invariance assumption for I means that, to first order in the small parameter e,
d I5I 8 2I50,
~6!
where I 8 denotes the action integral associated with the transformed trajectory x8 (t 8 ), I 85
E
t 28
t 81
L ~ x8 ,x˙8 ,t 8 ! dt 8 .
~7!
By recourse to the relation dt 8 511 e q˙ , dt
~8!
we can change the integration variable in ~7!, and recast the invariance of the action as
E
t2
t1
˙ ~ 12 e q˙ ! ,t1 e q !~ 11 e q˙ ! @ L ~ x1 e K,x˙1 e K 2L ~ x,x˙,t ! ]dt50.
~9!
Keeping in Eq. ~9! only linear terms in e, and assuming that the original trajectory x~t! is a real physical one that verifies Lagrange’s equations
S D
d ]L ]L , 5 dt ] x˙ ]x
~10!
we easily arrive at
e
E
t2
t1
d $ K–p2 q E % dt50. dt
~11!
In this last equation p stands for the vector whose components are the conjugate momenta of the components of x,
]L , ] x˙
~12!
and E is given by
II. NOETHER’S THEOREM
990
L ~ x,x˙,t ! dt,
evaluated along a trajectory x(t), is assumed to remain invariant, up to first order in the small parameter e, under the infinitesimal transformation defined by
p5
L ~ x,x˙,t ! 5 21 mx˙2 2m f ~ x,t ! ,
t2
E5
]L –x˙2L. ] x˙
~13!
The quantity E, when expressed as a function of the coordinates x i and the corresponding conjugated momenta p i , is the Hamiltonian function of the dynamical system. However, E is not necessarily equal to the total energy of the system, in the sense that E may not be equal to the kinetic energy T plus the potential energy V. It is worth realizing that the conditions for E being the total energy are independent of the requisites for E being conserved ~see Goldstein,8 pp. 61 and 62!. In the case of Lagrangians of the form ~2!, that we focus on in the present work, it is easy to verify that E is C. M. Giordano and A. R. Plastino
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indeed the total energy, even if f depends explicitly on t and the energy is not conserved.8 Since Eq. ~11! holds for any time interval @ t 1 ,t 2 # , we can conclude that d $ K–p2 q E % 50, dt
~14!
so that the Noether invariant is K–p2 q E.
~15!
In this paper we will only discuss applications involving uniform time translations, t 8 5t1 e ,
~16!
and consequently
On the other hand, the time derivatives of the inertial and rotating Cartesian coordinates are connected by x˙ 5X˙ cos v t2Y˙ sin v t2 v X sin v t2 v Y cos v t, y˙ 5X˙ sin v t1Y˙ cos v t1 v X cos v t2 v Y sin v t, z˙ 5Z˙ . The velocity vector of the particle in the inertial reference system has components (x˙ ,y˙ ,z˙ ) with respect to that system. However, the mentioned vector also can be resolved along the axis of the instantaneous rotating frame. If we call (w x ,w y ,w z ) the corresponding components ~i.e., with respect to the rotating frame!, it is easy to verify that ~25! w 5X˙ 2 v Y , w 5Y˙ 1 v X, w 5Z˙ . x
dt 8 51. dt
~17!
We notice that in such cases, the invariance of the Lagrangian L under the infinitesimal transformation ~4!–~5!, L ~ x,x˙,t ! 5L ~ x8 ,x˙8 ,t 8 ! ,
~24!
y
z
It is important to stress that (x˙ ,y˙ ,z˙ ) and (w x ,w y ,w z ) stand, respectively, for the coordinates of the very same vector in two different Cartesian systems. These two sets of coordinates are related, of course, by a standard rotation transformation,
~18!
w x 5x˙ cos v t1y˙ sin v t,
implies that the action integral I remains invariant as well.
w y 52x˙ sin v t1y˙ cos v t,
~26!
w z 5z˙ .
The kinetic energy of the particle is given by III. ROTATING POTENTIALS We now consider a uniformly rotating potential f (X,Y ,Z). Here f denotes the potential energy per unit mass and (X,Y ,Z) stands for a Cartesian coordinate frame that rotates with constant angular velocity v around the Z axis. The rotating reference frame is related to an inertial one (x,y,z), with both the same origin and z axis as (X,Y ,Z), by X5x cos v t1y sin v t, Y 52x sin v t1y cos v t,
~19!
T5 5
f 5 f ~ x cos v t1y sin v t,2x sin v t1y cos v t,z ! . ~20! An example of this kind of time-dependent potential is given by the gravitational potential generated by a rigidly and uniformly rotating mass distribution. The behavior of a particle of mass m moving within this kind of potential is governed by the Lagrangian L5 21 m ~ x˙ 2 1y˙ 2 1z˙ 2 ! 2m f ~ x cos v t1y sin v t, 2x sin v t1y cos v t,z ! ,
~21!
]f , ]x
y¨ 52
]f , ]y
z¨ 52
]f . ]z
~22!
Let us briefly describe the usual derivation of the Jacobi integral. First, one obtains the ~Cartesian! equations of motion in the rotating reference frame (X,Y ,Z). The inertial coordinates (x,y,z) are given in terms of the rotating ones by x5X cos v t2Y sin v t,
It is convenient to recall now an important property of Lagrange’s equations of motion. Given a dynamical system characterized by a set q 1 ,...,q n of independent generalized coordinates with a Lagrangian L(q,q˙ ,t), the form of the concomitant Lagrangian’s equations are invariant under an arbitrary ~even time-dependent! point transformation ~Ref. 8 p. 33, exercise 15!, q i 5q i ~ s 1 ,...,s n ,t ! ,
y5X sin v t1Y cos v t, 991
z5Z.
Am. J. Phys., Vol. 66, No. 11, November 1998
~29!
i51,...,n,
where s i stand for a new set of generalized coordinates. This property means that the equations of motion for s i have the usual Lagrangian form
S D
d ]L ]L 5 , ˙ dt ] s i ]si
~30!
where L is the Lagrangian function expressed in terms of the new variables by recourse to the transformation equation ~29!. It is clear that the rotation ~23! is a particular example of a time-dependent point transformation, hence Lagrange’s equations still hold true in terms of the rotating coordinates, d
~23!
~27!
L5 21 m @~ X˙ 2 v Y ! 2 1 ~ Y˙ 1 v X ! 2 1Z˙ 2 # 2m f ~ X,Y ,Z ! . ~28!
which leads to the equations of motion x¨ 52
m ~ w 2x 1w 2y 1w 2z ! . 2
By recourse to Eq. ~25! and the last line of Eq. ~27!, we can write the Lagrangian of our dynamical system ~i.e., the particle moving in the potential f! in terms of the rotating Cartesian coordinates (X,Y ,Z),
Z5z. Then, in terms of the inertial coordinates, the rotating potential function has the form
m 2 ~ x˙ 1y˙ 2 1z˙ 2 ! 2
dt
S D
]L ]L 5 , ˙ ]X ]X
d dt
S D
]L ]L 5 , ˙ ]Y ]Y
d dt
S D
]L ]L 5 . ˙ ]Z ]Z
C. M. Giordano and A. R. Plastino
~31! 991
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We can easily obtain the pertinent equations of motion in the rotating reference frame,
]f , X¨ 22 v Y˙ 5 v 2 X2 ]X ]f , Y¨ 12 v X˙ 5 v 2 Y 2 ]Y
~32!
]f Z¨ 52 . ]Z
Notice that the equations of motion expressed in terms of the rotating coordinates incorporate a centrifugal force of components ( v 2 X, v 2 Y ,0), as well as a Coriolis force of components (2 v Y˙ ,22 v X˙ ,0). Multiplying ~i! the first equation of Eqs. ~32! by X˙ , ~ii! the second one by Y˙ , and ~iii! the third one by Z˙ , and finally summing things up term by term, one easily arrives at
F
G
d 1 ˙2 ˙2 ˙2 1 ~ X 1Y 1Z ! 1 f 2 v 2 ~ X 2 1Y 2 ! 50, dt 2 2
~33!
so that one verifies that the system admits the following integral of motion: C J5
m ˙2 ˙2 ˙2 m ~ X 1Y 1Z ! 1m f 2 v 2 ~ X 2 1Y 2 ! . 2 2
~34!
By expressing C J in terms of an inertial reference frame (x,y,z), one is led to the remarkable result15,28 C J5
m 2 ~ x˙ 1y˙ 2 1z˙ 2 ! 1m f ~ x,y,z ! 2 v m ~ xy˙ 2yx˙ ! 2
5E2 v L z ,
~35!
which implies that Jacobi’s integral turns out to be a linear combination of the energy and the z component of angular momentum. It should be clear that neither E nor L z is individually conserved. For the sake of clarity we should emphasize that the introduction of the rotating frame is not necessary for the physical analysis of this kind of system. We have introduced the rotating coordinates (X,Y ,Z) just to review the usual derivation of Jacobi’s integral. Moreover, in the discussion that follows, dealing with Jacobi’s constant in connection with Noether’s theorem, we are going to work entirely within an inertial reference frame. We now examine Noether’s approach to Jacobi’s constant. It is easy to realize that, because of the uniform rotation of the potential, the Lagrangian L(x,x˙,t) @and consequently the action integral I corresponding to a trajectory x(t)# remains invariant under a time translation t 8 5t1 e , performed together with a rotation in an angle ve around the z axis. In the infinitesimal case, this combined transformation is given by x 8 5x2 v e y,
y 8 5y1 v e x,
z 8 5z,
t 8 5t1 e , ~36!
which, recalling the general expression ~14! for Noether’s conservation law, implies that d @ 2 v p x y1 v p y x2E # 50, dt
~37!
and provides us with Jacobi’s integral C J 5E2 v L z .
~38!
It is instructive to consider the limit situation of a nonrotating potential. In this case the angular velocity v vanishes 992
Am. J. Phys., Vol. 66, No. 11, November 1998
and the Jacobi integral C J coincides with the total energy E. In other words, the conservation law for the Jacobi constant reduces to the standard energy conservation law for a timeindependent potential. Of course, if the ~now! nonrotating potential has axial symmetry around the z axis, the component L z of the angular momentum is also conserved. It is worth remarking that Noether’s theorem allows us to obtain Jacobi’s integral without the explicit manipulation of the particular equations of motion of the system,4 even if we need to assume that Lagrange equations hold. We only have to identify the relevant symmetries of the Lagrangian. IV. EXAMPLES We now consider some examples of systems admitting a Jacobi integral of motion. By recourse to the general discussion given in Sec. III we obtain the expression for Jacobi’s constant for each particular problem studied. Thus one can obtain useful information about the behavior of the system without considering the explicit form of the concomitant equations of motion. Additionally, in some particular cases, this approach provides a straightforward derivation of the equations of motion. A. Systems with uniformly rotating constraints A particle that is compelled to remain on a uniformly ~and rigidly! rotating surface or curve can be treated by recourse to Jacobi’s integral. The concomitant potential energy is taken to be 1` in those regions of space not allowed by the constraints. In order to make this clear, we now consider two simple examples involving rotating constraints: a particle constrained to move in a rotating horizontal straight line, and a particle whose motion is constrained to a rotating vertical circumference. Both examples are usually discussed in elementary texts of mechanics, though not in connection with Jacobi’s integral. 1. Particle moving in a rotating horizontal straight line Let us assume that a particle of mass m is constrained to move on a horizontal straight line that rotates with constant angular velocity v around a fixed point O. In this case the total energy is given just by the kinetic energy E5E K 5
m 2 @ r˙ 1 v 2 r 2 # , 2
~39!
where r stands for the coordinate of the particle within the rotating line ~i.e., the distance along the radial line from the axis of rotation!. The z component of the angular momentum is L z 5m v r 2 .
~40!
From the general equation ~38!, we obtain the Jacobi integral for this problem, C J5
m 2 @ r˙ 2 v 2 r 2 # . 2
~41!
The conservation of C J , d 2 @ r˙ 2 v 2 r 2 # 50, dt
~42!
immediately yields the equation of motion for r(t), r¨ 5 v 2 r.
~43! C. M. Giordano and A. R. Plastino
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2. Particle moving in a vertical rotating circle Let us now have a particle of mass m constrained to move on a vertical circle of radius R and ~fixed! center O that rotates with constant angular velocity v around its vertical diameter. The total energy of the particle is E5
m @ R 2 v 2 sin2 a 1R 2 a˙ 2 12gR cos a # , 2
~45!
Therefore, Jacobi’s constant is given by C J5
m @ R 2 a˙ 2 2 v 2 R 2 sin2 a 12gR cos a # , 2
~46!
and the associated conservation law is d @ a˙ 2 2 v 2 sin2 a 12 ~ g/R ! cos a # 50, dt
~47!
which furnishes the equation of motion for a (t),
a¨ 2 v 2 sin a cos a 2 ~ g/R ! sin a 50.
~48!
B. Rotating billards Uniformly rotating billards constitute another interesting application of the present approach. Let us consider the motion of a particle inside a twodimensional box that rotates uniformly around a fixed point O, with angular velocity v. The corresponding potential energy vanishes inside the box and is equal to 1` on its boundary. The point O is assumed to be the origin of the inertial reference frame (x,y,z), the box being contained within the plane (x,y). If (X,Y ,Z) stands for the rotating coordinate system ~as in Sec. III!, the boundary of the billard is given by a fixed curve in the plane (X,Y ). Here we have used the rotating frame (X,Y ,Z) just to clarify what we mean by a ‘‘rotating billard.’’ Within the physical discussion that follows, and in order to apply the results of Sec. III, we work in the inertial frame (x,y,z). The only relevant contribution to the total energy is given by the kinetic energy m E5E K 5 ~ x˙ 2 1y˙ 2 ! , 2
~49!
and the z component of the angular momentum is L z 5m ~ xy˙ 2yx˙ ! .
~50!
Hence, in this case, Jacobi’s constant is C J5
m 2 @ x˙ 1y˙ 2 22 v ~ xy˙ 2yx˙ !# . 2
~51!
Let us notice that this last constant of motion for the rotating billard was obtained taking into account neither the specific shape of the billard nor the detailed features of the collisions of the particle with the box wall. Jacobi’s conservation law can be recasted as 993
Am. J. Phys., Vol. 66, No. 11, November 1998
~52!
where v is the velocity of the particle in the inertial reference frame and (r, a ) are the polar coordinates associated with the Cartesian variables (x,y).
~44!
where g is the acceleration of gravity and a stands for the angular coordinate characterizing the position of the particle, measured along the circle and from its top point. The z component of the angular momentum of the particle is L z 5m v R 2 sin2 a .
d ~ v2 22 v r 2 a˙ ! 50, dt
C. Rotating systems with cylindrical helicoidal symmetry We now discuss the problem of a particle of mass m moving in a uniformly rotating potential f ~as described in Sec. III! with cylindrical helicoidal symmetry. In this case f depends on the rotating reference frame coordinates (X,Y ,Z) only through the quantities r and (Z2 ka ),
f 5 f ~ r ,Z2 ka ! ,
~53!
where ( r , a ,Z) constitute a cylindrical coordinate system within the rotating frame, so that X5 r cos a ,
Y 5 r sin a .
~54!
As in the previous example, the rotating frame has been introduced just to provide a clear description of the dynamical system considered. From now on we work within the inertial frame (x,y,z). Clearly the Lagrangian associated with this problem is invariant under a rotation in an angle e around the z axis ~which coincides with the Z axis!, combined with a translation in ke along the same axis. In the infinitesimal case this transformation is given, in terms of the inertial coordinates (x,y,z), by x 8 5x2 e y,
y 8 5y1 e x,
z 8 5z1 ke .
~55!
This symmetry yields the Noether invariant C h 5 k p z 1L z .
~56!
Additionally, since we are dealing with a uniformly rotating potential ~as discussed in Sec. III!, we still have the Jacobi integral C J 5E2 v L z .
~57!
We notice that a linear combination of these two invariants furnishes the integral C p 5E1 v k p z .
~58!
It is interesting to realize that the constant of motion C p can also be obtained by recourse to Noether’s procedure. The cylindrical helicoidal symmetry of the potential @see Eq. ~45!#, together with its uniform rotation around the z axis, imply that f remains unchanged under a time increment in e together with a z displacement in 2 e v k . Hence, we can easily see that the Lagrangian ~and the action integral! is invariant under the infinitesimal transformation defined by t 8 5t1 e ,
~59!
z 8 5z2 e v k ,
~60!
and
that yields the Noether integral C p given in Eq. ~58!. C. M. Giordano and A. R. Plastino
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D. N-body system in a rotating background potential We are now concerned with the motion of N gravitationally interacting masses m i (i51,...,N), in a uniformly rotating ~external! potential ~potential energy per unit mass!
f 5 f ~ x cos v t1y sin v t,2x sin v t1y cos v t,z ! . ~61! In this case we do not have only one particle moving in a rotating potential, as in Sec. III, but a system of N interacting particles. However, the ideas of that section are easily extended to apply to the present problem. The Lagrangian for this system reads L5T2V N
1 5 2
N
(
m i x˙~i 2 ! 1G
i51
N
N
m im j 2 m f ~ xi,t ! , i xi2xji i51 i
( ( i, j j51
(
~62! where xi stands for the vector position ~with respect to an inertial reference frame! of the mass m i , and G is Newton’s gravitational constant. The potential energy N
V52G
N
( ( i, j j51
N
m im j 1 m f ~ xi,t ! , i xi2xji i51 i
(
~63!
has two terms, the first one corresponds to the gravitational interaction between the N particles, and the second one takes into account the external potential f. Due to the mutual attraction of the masses the separate Jacobian integrals associated with each particle are lost. However, the system still allows for an integral which can easily be derived by means of Noether’s theorem. In fact, the infinitesimal transformation x 8i 5x i 2 v e y i ,
i51,...,N,
y i8 5y i 1 v e x i ,
i51,...,N,
z i8 5z i ,
~64!
i51,...,N,
t 8 5t1 e , leaves invariant the Lagrangian ~and the associated action integral!, and the corresponding conservation law, according to ~14!, is d dt
F(
G
N
~ 2 v p x i y i 1 v p y i x i ! 2E 50,
i51
~65!
where E is the total energy, 1 E5 2
N
(
i51
N
m i x˙~i 2 ! 2G
N
( ( i, j j51
N
m im j 1 m f ~ xi,t ! . i xi2xji i51 i ~66!
(
Hence, in this case Noether’s formalism leads again to Jacobi’s integral C J 5E2 v L z ,
~67!
L z being the z component of the total angular momentum of the N particles N
L z5 994
( ~ x i p y 2y i p x ! .
i51
i
i
Am. J. Phys., Vol. 66, No. 11, November 1998
~68!
The integral ~67! has been obtained elsewhere ~without using Noether’s Theorem! within the framework of the restricted problem of (21N) bodies.31 Here the classical restricted three-body problem is generalized so that several gravitationally interacting bodies with small masses, instead of a single negligible mass, move in the gravitational field of two finite masses on circular orbits. V. CONCLUSIONS We have shown that Jacobi’s integral of motion for rotating potentials constitutes a simple but important illustrative example of application of Noether’s theorem. Jacobi’s constant is a linear combination of the total energy and the z component of the angular momentum, that arises from the invariance of the action under a time translation performed together with a rotation around the z axis. We have illustrated these ideas by some examples, and shown that some elementary problems often discussed in texts of mechanics can be interpreted in terms of Jacobi’s constant. ACKNOWLEDGMENTS The support of PROFOEG and the Comisio´n Nacional de Investigaciones Cientı´ficas y Te´cnicas de Argentina ~CONICET! is gratefully acknowledged. Also the authors are indebted to an anonymous referee for his or her valuable comments and suggestions. 1
E. Noether, ‘‘Invarinate Variationsprobleme,’’ Nachr. Akad. Wiss. Goett. II, Math.-Phys. K1 II, 235–257 ~1918!. 2 M. A. Tavel, ‘‘Invariant variation problems,’’ Transp. Theory Stat. Phys. 1, 186–207 ~1971!. 3 J. D. Logan, Invariant Variational Principles ~Academic, New York, 1977!. 4 E. L. Hill, ‘‘Hamilton’s Principle and the Conservation Theorems of Mathematical Physics,’’ Rev. Mod. Phys. 23, 253–260 ~1951!. 5 N. Bobillo-Ares, ‘‘Noether’s theorem in discrete classical mechanics,’’ Am. J. Phys. 56 ~2!, 174–177 ~1988!. 6 A. Zee, Fearful Symmetry, Collier Books ~MacMillian, 1986!, Chap. 8. 7 T. H. Boyer, ‘‘Derivation of Conserved Quantities from Symmetries of the Lagrangian in Field Theory,’’ Am. J. Phys. 34, 475–478 ~1966!. 8 H. Goldstein, Classical Mechanics ~Addison–Wesley, Reading, MA, 1980!, 2nd ed., pp. 588–596. 9 Neil S. Rasband, Dynamics ~Wiley-Interscience, New York, 1983!, pp. 136–141. 10 V. I. Arnold, Dynamical Systems III, Encyclopaedia of Mathematical Sciences Vol. 3 ~Springer-Verlag, Berlin, 1988!, p. 78. 11 D. Neuenschwander and S. Starkey, ‘‘Adiabatic invariance derived from the Rund–Trautman identity and Noether’s theorem,’’ Am. J. Phys. 61, 1008–1013 ~1993!. 12 D. Neuenschwander and G. Taylor, ‘‘The adiabatic invariants of plasma physics derived from the Rund–Trautman identity and Noether’s theorem,’’ Am. J. Phys. 64 ~11!, 1428–1430 ~1996!. 13 K. R. Symon, Mechanics ~Addison–Wesley, Reading, 1971!, 3rd ed., pp. 286–291. 14 J. M. A. Danby, Fundamentals of Celestial Mechanics ~Bell, London, 1992!, p. 254. 15 J. Binney and S. Tremaine, Galactic Dynamics ~Princeton U.P., Princeton, NJ, 1987!, p. 135. 16 L. G. Taff, Celestial Mechanics. A Computational Guide for the Practitioner ~Wiley-Interscience, New York, 1985!, pp. 195–196. 17 F. R. Moulton, Periodic Orbits ~Carnegie Institution of Washington, Washington, DC, 1963!, p. 156. 18 Y. Hagihara, Celestial Mechanics, Dynamical Principles and Transformation Theory Vol. I ~MIT, Cambridge, MA, 1970!, p. 245. 19 Y. Hagihara, Celestial Mechanics, Periodic and Quasi-Periodic Solutions Vol. IV, Part 1 ~Japan Society for the Promotion of Science, 1975!, p. 2. 20 C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics ~SpringerVerlag, Berlin, Heidelberg, 1971!, p. 149. C. M. Giordano and A. R. Plastino
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H. Pollard, Mathematical Introduction to Celestial Mechanics ~PrenticeHall, Englewoods Cliffs, NJ, 1966!, pp. 53–54. 22 S. W. McCuskey, Introduction to Celestial Mechanics ~Addison–Wesley, Reading, MA, 1963!, pp. 110–111. 23 D. Brouwer and G. M. Clemence, Methods of Celestial Mechanics ~Academic, New York, 1961!, p. 255. 24 H. C. Plummer, An Introductory Treatise on Dynamical Astronomy ~Dover, New York, 1960!, p. 236. 25 E. Finlay-Freundlich, Celestial Mechanics ~Pergamon, New York, 1958!, pp. 45–46. 26 F. R. Moulton, An Introduction to Celestial Mechanics ~Macmillan, New York, 1923!, p. 280. 27 A. Brunini, C. M. Giordano, and R. B. Orellana, ‘‘Capture Conditions in 21
995
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the Restricted Three-Body Problem,’’ Astron. Astrophys. 314, 977–982 ~1996!. 28 C. M. Giordano, A. R. Plastino, and A. Plastino, ‘‘Comment on ‘A Simple, Conservative, Understanding of many Time-Driven Systems,’ by Harvey Kaplan @Am. J. Phys. 62, 1097–1099 ~1994!#,’’ Am. J. Phys. ~accepted for publication!. 29 V. Szebehely, Theory of Orbits. The Restricted Problem of Three Bodies ~Academic, New York, 1967!, pp. 16–18. 30 R. H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics ~AIAA Education Series, American Institute of Aeronautics and Astronautics, Inc., New York, 1987!, p. 372. 31 V. Szebehely and A. L. Whipple, ‘‘Generalizations of the Jacobian Integral,’’ Celest. Mech. 34, 125–133 ~1984!.
C. M. Giordano and A. R. Plastino
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This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.215.9.20 On: Mon, 23 Nov 2015 19:35:33
