5 Dynamics of Systems of Particles
5.6 Closing Comments 151
Making use of the potential energy function U, we can express this equation as E˙ = µ1(er1· v1+ eR2· v2)+ µ2E3· v1− µd||µ2E3|| vrel
||vrel|| · v1,
where the total energy E= T + U. Thus, as v1 = vreland is normal to E3, and µ2 = m1g, we surmise that
E˙= µ1(er1· v1+ eR2· v2)− µdm2g||v1||.
However, er1· v1+ eR2· v2 = ˙r1+ ˙R2, and this sum is zero because r1+ R2= L0. In conclusion,
E˙ = −µdm2g||v1||.
Notice that ˙E≤ 0 as expected because of the friction force.
Conservations of Angular Momenta
If friction is absent, then we find from (5.14)2,4 that HO1· E3= m1r21˙θ1 and HO2· E3= m2(L0− r1)2sin2(φ2) ˙θ2 are conserved. That is, the angular momentum of each particle relative to O in the E3direction is conserved.
Configuration Manifold and Its Geometry
The configuration manifold for this system is a four-dimensional subspace ofE6 pa-rameterized by r∈ (0, L0), θ1∈ [0, 2π), θ2∈ [0, 2π), and φ2∈ (0, π). The kinematical line-element ds for this manifold is given by
ds=
2 ˜T2
m1+ m2
dt,
where we find ˜T2from T by imposing the constraints and collecting all those terms that are quadratic in the generalized velocities:
T˜2 = m1 2
˙r21+ r21˙θ21
+m2
2
˙r21+ (L0− r1)2sin2(φ2) ˙θ22+ (L0− r1)2φ˙22
.
To visualize the configuration manifold, one would give a two-dimensional picture of a plane with the coordinates r1cos (θ1)–r1sin (θ1). This would be supplemented by a three-dimensional image featuring a sphere of radius 1 parameterized by φ2and θ2.
5.6 Closing Comments
Problems involving systems of particles have played a key role in the development of dynamics. Specifically, mention is made here of a model for the celestial system of the Sun, Earth, and Moon, known as the three-body problem. In this problem, the three bodies are modeled as particles subject to the mutual interaction that is due to
m1
m1
m1
m2 m2
m2
m3
m3
m3
(a) (b)
(c)
Figure 5.6. Representative orbits from the three-body problem: (a), (b) examples of La-grange’s equilateral triangle solutions, and (c) the figure-eight solution. For the solutions shown in this figure m1= m2= m3.
a Newtonian gravitational force field. That is, the potential energy for the system is [cf. (4.9)]
Un= −Gm1m2
||r2− r1||− Gm3m1
||r1− r3||− Gm2m3
||r3− r2||, (5.15) where r1, r2, and r3 are the position vectors of the particles of masses m1, m2, and m3, respectively.
Famous exact solutions to special cases of the three-body problem range from the equilateral triangle solution by Lagrange [117] in 1772∗ to the figure-eight so-lution that was only recently found numerically by Moore [147] and Moore and Nauenberg [148] and proven to exist by Chenciner and Montgomery [37, 145] (see Figure 5.6).† Apart from its paucity of exact solutions, the three-body problem is also well known because of the profound analysis of this system by Henri Poincar ´e (1854–1912) in the late 1880s (see [4, 13, 45]). His analysis is considered to be the first description of chaos in mathematical models for physical systems and formed one of the cornerstones for the field of chaos in dynamical systems that achieved popular attention some 100 years later in the late 1980s.
The three-body and two-body problems are special cases of the more general n-body problem. In celestial mechanics, the n-body problem is synonymous with models for our Solar System and has attracted some of the most celebrated scien-tists in history. It was also the problem that led Hamilton to discover his famous
∗ A discussion of this famous solution can be found in many texts on celestial mechanics, for example, [93, 150, 220].
† The interested reader is referred to the on-line article by Casselman [31], where simulations of several three-body problems can be found.
Exercises 5.1–5.3 153
equations of motion in [88] and his variational principle in [89]. Unfortunately, we do not have the opportunity to explore the three-body and n-body problems in any detail here; the interested reader is referred to the previously cited texts.
The problems just mentioned do not feature constraints on the motions of the particles, and they are often formulated without using Lagrange’s equations of motion. However, problems featuring particles connected by rigid links often feature in simple models for artificial satellites orbiting a celestial body and in various pendulum systems. For these models, Lagrange’s equations of motion are ideally suited to the task of establishing a set of governing ordinary differential equations that are free from constraint forces. In the following exercises, problems featuring systems of particles of this type are emphasized.
EXERCISES
5.1. Consider the systems of particles discussed in Section 5.2. Suppose a time-dependent force P(t)E1acted on the particle m2.∗What are the equations of motion for each of these systems?
5.2. Again, consider the systems of particles discussed in Section 5.2. Suppose, in addition to the springs, there are viscous dashpots in these systems.†Then, what are the equations of motion?
5.3. Here, we are interested in establishing a particular representation for the equa-tions governing the motion of two unconstrained particles. In a subsequent exercise, one can impose constraints to yield the equations of motion of a pendulum system.
Consider the system of particles shown in Figure 5.7. The particles are free to move inE3under the influences of resultant external forces F1and F2, respectively.
m1 m2
O g
E1
E2
E3
Figure 5.7. A system of two particles.
(a) To establish the equations of motion for the single particle, we use a cylindrical polar coordinate system{r1, θ1,z1} for the particle of mass m1. For the second particle, it is convenient to describe its motion with the assistance of the relative position vector r21 = r2− r1. We describe this
∗ This force is not conservative.
† The forces from these dashpots are not conservative.
vector by using a spherical polar coordinate system{R2, φ2, θ2}. Show that the position vector of the single particle is
r= r1cos(θ1)e1+ r1sin(θ1)e2+ z1e3 + (r1cos(θ1)+ R2sin(φ2) cos(θ2))e4
+ (r1sin(θ1)+ R2sin(φ2) sin(θ2))e5+ (z1+ R2cos(φ2))e6.
(b) Using r and the curvilinear coordinate system it induces onE6,
q1 = r1, q2 = θ1, q3= z1, q4= R2, q5= φ2, q6 = θ2,
show that the six covariant basis vectors aJ =∂q∂rJ are
a1= ∂r
∂r1
= cos(θ1)e1+ sin(θ1)e2+ cos(θ1)e4+ sin(θ1)e5,
a2= ∂r
∂θ1 = −r1sin(θ1)e1+ r1cos(θ1)e2− r1sin(θ1)e4+ r1cos(θ1)e5, a3= ∂r
∂z1 = e3+ e6, a4= ∂r
∂R2 = sin(φ2) cos(θ2)e4+ sin(φ2) sin(θ2)e5+ cos(φ2)e6, a5= ∂r
∂φ2 = R2cos(φ2) cos(θ2)e4+ R2cos(φ2) sin(θ2)e5− R2sin(φ2)e6, a6= ∂r
∂θ2 = −R2sin(φ2) sin(θ2)e4+ R2sin(φ2) cos(θ2)e5.
(c) Show that the six contravariant basis vectors have the following represen-tations:
a1 = cos(θ1)e1+ sin(θ1)e2, a2 = −sin(θ1)
r1
e1+cos(θ1) r1
e2, a3 = e3,
a4 = sin(φ2)(cos(θ2)(e4− e1)+ sin(θ2)(e5− e2))+ cos(φ2)(e6− e3), a5 =cos(φ2)
R2
cos(θ2)(e4− e1)+ sin(θ2)(e5− e2)
−sin(φ2) R2
(e6− e3),
a6 = − sin(θ2)
R2sin(φ2)(e4− e1)+ cos(θ2)
R2sin(φ2)(e5− e2).
Exercises 5.3–5.4 155
(d) Show that the kinetic energy T of the particle of mass m = m1+ m2is T= m1+ m2
2
˙r21+ r21˙θ21+ ˙z21
+m2
2
R˙22+ R22φ˙22+ R22sin2(φ2) ˙θ22
+ m2cos(φ2)R˙2˙z1+ ˙φ2˙r1R2cos(θ21)+ ˙φ2˙θ1r1R2sin(θ21)
− m2sin(φ2)
R2φ˙2˙z1− r1R2˙θ1˙θ2cos(θ21)− ˙r1R˙2cos(θ21)
− m2sin(φ2)
−r1R˙2˙θ1sin(θ21)+ ˙r1θ˙2R2sin(θ21) ,
where we have used the abbreviation θ21= θ2− θ1. This expression for the kinetic energy follows from the definition
T=m
2v· v = m1
2 v1· v1+m2 2 v2· v2.
(e) If the forces acting on the particles are F1= −m1gE3 and F2= −m2gE3, then what are the force and potential energy U associated with this force?
(f) What are the six Lagrange’s equations governing the motion of the particle of mass m?∗
5.4. As shown in Figure 5.8, two particles of mass m1 and m2 are connected by a rigid massless rod of length L2. The rod is connected to m1 by a ball-and-socket joint. In addition, the particle of mass m1 is connected by a rigid massless rod of length L1 to a fixed point O. The connection between the rod and the point O is through a pin joint and is such that the motion of m1is in the E1− E2plane.
m1
m2
O g
E1
E2 E3
Figure 5.8. A planar double pendulum.
(a) What are the three constraints on the motion of the particle of mass m?
(b) Using Lagrange’s prescription, what is the constraint forcecacting on the particle of mass m? You should also, if possible, verify that the components of this force are physically realistic.
∗ You should refrain if possible from expanding the time derivative here – it will entail a considerable amount of algebra.
(c) Starting from the final results of Exercise 5.3, establish Lagrange’s equations of motion for the pendulum system. In your solution, clearly distinguish the equations governing the motion of the particle and the equations giving the components ofc.
(d) Let us now establish some of the equations of (c) by using an equivalent approach. Impose the constraints on T to determine the constrained kinetic energy ˜T. In addition, determine the constrained potential energy ˜U. Verify that the following equations correspond to those you obtained from (c)∗:
d
(e) Suppose a nonintegrable constraint is imposed on the pendulum system discussed in (d):
f1· v1+ f2· v2+ e = 0.
Show that this constraint can be expressed as f· v + e = 0.
In addition, what are the equations governing the motion of the integrably constrained system? Illustrate your solution with an non-integrable constraint of your choice.
5.5. As shown in Figure 5.9, a model for an artificial satellite consists of two particles of mass m1and m2connected by a rigid massless rod of length L0. A third particle of mass m3is assumed to be stationary at the fixed point O. In addition to the constraint force in the rod, the system is subject to conservative forces whose potential energy function is given by (5.15).
(a) What are the four constraints on the motion of the system of particles?
(b) Using Lagrange’s prescription, what are the constraint forces acting on the particles of mass m1 and m2? You should also, if possible, verify that the components of these forces are physically realistic.
(c) Using a set of Cartesian coordinates to describe the location of the center of mass C of the satellite of mass m1+ m2 and a set of spherical polar coordinates to parameterize the position of m2 relative to C, establish an expression for the kinetic energy of the system.
(d) Establish the equations of motion for the system.
∗ It is crucial to note that∂∂ ˜T˙r
1= ∂ ˙∂ ˜RT
2 = ∂∂ ˜zT˙1 =∂r∂ ˜T1 =∂R∂ ˜T2 =∂z∂ ˜T1= 0.
Exercises 5.5–5.6 157
m1
m2
m3 O
E1
E2 E3
Figure 5.9. Schematic of a model for a satellite orbiting a fixed body of mass m3.
(e) Show that the solutions to the equations of motion for the system conserve the total energy of the system and the angular momentum of the system relative to O.
(f) Show that it is possible for C to execute a steady circular motion about O.
What are the possible orientations of the rigid massless rod of length L0 during such motions?
5.6. As shown in Figure 5.10, a particle of mass m1is connected by a linear spring of stiffness K1 and unstretched length L0 to a fixed point O. A second particle of mass m2 is attached by a rod of length L2 to the particle of mass m1 with a pin joint. For this system, which is a variation on the classical system of a planar double pendulum, we assume that the motions of m1 and m2 are constrained to move on the E1− E2plane.
m1
m2
O E1
E2
g
linear spring
rigid massless rod Figure 5.10. A system of two
parti-cles connected by a rigid massless rod of length L2.
To describe the kinematics of this system, a cylindrical polar coordinate system {r1, θ1,z1} is used to parameterize the motion of the particle of mass m1and another cylindrical polar coordinate system{r2, θ2,z2} is used to parameterize the motion of the particle of mass m2relative to m1:
r1= r1er1+ z1E3, r2= r1+ r2er2+ z2E3. We define six coordinates as follows:
q1 = θ1, q2= θ2, q3= r1, q4= r2, q5= z1, q6= z2. (5.16)
(a) With the help of (5.16), what are the 12 vectors ∂q∂r1K and ∂q∂r2K? Here, K= 1, . . . , 6.
(b) What are the three constraints on the motion of the system of particles?
Argue that the constraint forces Fc1 and Fc2 acting on the individual particles have the prescriptions
Fc1= µ1er2+ µ2E3, Fc2= −µ1er2+ µ3E3. (5.17) Compute the following six components:
cK= Fc1· ∂r1
∂qK + Fc2· ∂r2
∂qK. Comment on the values of the first three components.
(c) In terms of the coordinates q1, . . . ,q3 and their time derivatives, what are the kinetic energy ˜T and potential energy ˜U of the constrained system of particles?
(d) What are Lagrange’s equations of motion for the generalized coordinates of this system of particles?
(e) Suppose a nonintegrable constraint∗
r1θ˙1+ L2˙θ2= 0 (5.18) is imposed on the system of particles. After expressing this constraint in the form f1· v1+ f2· v2= 0, argue that
Fc1= µ1er2+ µ2E3+ µ4(eθ1− eθ2) , Fc2= −µ1er2+ µ3E3+ µ4eθ2. With the help of your results from (d), determine the equations of motion for the system of particles.
(f) Starting from the work–energy theorem ˙T= F1· v1+ F2· v2, show that the total energy E is conserved.
(g) Suppose that the spring is replaced with a rigid rod of length L1 and nonintegrable constraint (5.18) is removed. In this case, which is the classic
∗ Referring to the discussion of constraint (1.16) in Chapter 1, this constraint is arguably the simplest nonintegrable constraint that we can impose on this system.
Exercises 5.6–5.7 159
planar double pendulum, show that the equations governing the motion of the system are In writing (5.19), we have used the following dimensionless parameters and time variable:
The configuration manifold M for this system is a torus. What is the kinematical line-element forM?
(h) Numerically integrate (5.19) for a variety of initial conditions and illustrate your solutions on the configuration manifold for the planar double pendu-lum. You should verify that your solutions conserve the total energy of the system.
5.7. This problem is adapted from Section 156 of Whittaker [228] and the introduc-tion to [30]. Consider a system of N particles, and, following Lecture 4 from Jacobi [102], define the following function:
J = 1 2
N k=1
mk||rk||2.
The quantity 2J is often known as the moment of inertia of the system of particles.
(a) Assuming that the center of mass C of the system is stationary and located at the origin, show that J has the equivalent representation:
J= 1
(b) As in (a), assuming that the center of mass C of the system is station-ary, show that the kinetic energy of the system of particles has the representation
(c) Now suppose that the system of particles is in motion subject to a conservative Newtonian force field:
Un= −1 2
N j=1
N k=1,k=j
Gmkmj
rk− rj. (5.22) The presence of the 12in the expression for Unshould be noted: It is needed to ensure that the summations on k and j yield the correct expression for Un. With the help of (5.20)–(5.22) and balances of linear momenta for each particle, establish Jacobi’s equation:
¨J = 2T + Un. (5.23)
This equation is also known as the Lagrange–Jacobi equation (see, for example, [220]).
(d) For the orbits of the three-body problem shown in Figure 5.6(b), show that T= −12Un.
(e) Show that J is a measure of the distance squared from the origin of the con-figuration spaceE3Nfor the representative particle discussed in Section 4.7.