Hirsch M., Smale S., Devaney R., Differential Equations, Dynamical Systems, and an Introduction to Chaos

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Applications in Mechanics

We turn our attention in this chapter to the earliest important examples of

differential equations that, in fact, are connected with the origins of calculus.

These equations were used by Newton to derive and unify the three laws of

Kepler. In this chapter we give a brief derivation of two of Kepler’s laws and

then discuss more general problems in mechanics.

The equations of Newton, our starting point, have retained importance

throughout the history of modern physics and lie at the root of that part of

physics called classical mechanics. The examples here provide us with concrete

examples of historical and scientiﬁc importance. Furthermore, the case we

consider most thoroughly here, that of a particle moving in a central force

gravitational ﬁeld, is simple enough so that the differential equations can be

solved explicitly using exact, classical methods (just calculus!). However, with

an eye toward the more complicated mechanical systems that cannot be solved

in this way, we also describe a more geometric approach to this problem.

13.1 Newton’s Second Law

We will be working with a particle moving in a force ﬁeld F . Mathematically, F

is just a vector ﬁeld on the (conﬁguration) space of the particle, which in our

case will be

. From the physical point of view, F (X) is the force exerted on

a particle located at position X ∈

The example of a force ﬁeld we will be most concerned with is the gravita-

tional ﬁeld of the sun: F (X) is the force on a particle located at X that attracts

the particle to the sun. We go into details of this system in Section 13.3.

277

278 Chapter 13 Applications in Mechanics

The connection between the physical concept of a force ﬁeld and the math-

ematical concept of a differential equation is Newton’s second law: F = ma.

This law asserts that a particle in a force ﬁeld moves in such a way that the

force vector at the location X of the particle, at any instant, equals the acceler-

ation vector of the particle times the mass m. That is, Newton’s law gives the

second-order differential equation



= F(X).

As a system, this equation becomes



= V



F(X )

where V = V (t) is the velocity of the particle. This is a system of equations on

×R

. This type of system is often called a mechanical system with n degrees

of freedom.

A solution X(t ) ⊂

of the second-order equation is said to lie in conﬁg-

uration space. The solution of the system (X (t ), V (t )) ⊂

× R

lies in the

phase space or state space of the system.

Example. Recall the simple undamped harmonic oscillator from Chapter 2.

In this case the mass moves in one dimension and its position at time t is given

by a function x(t ), where x :

R → R. As we saw, the differential equation

governing this motion is



=−kx

for some constant k>0. That is, the force ﬁeld at the point x ∈

R is given

by −kx.



Example. The two-dimensional version of the harmonic oscillator allows

the mass to move in the plane, so the position is now given by the vector

X(t ) = (x

(t), x

(t)) ∈ R

. As in the one-dimensional case, the force ﬁeld is

F(X ) =−kX so the equations of motion are the same



=−kX

with solutions in conﬁguration space given by

(t) = c

cos(

√

k/mt) + c

sin(

√

k/mt)

(t) = c

cos(

√

k/mt) + c

sin(

√

k/mt)

13.1 Newton’s Second Law 279

for some choices of the c

as is easily checked using the methods of

Chapter 6.



Before dealing with more complicated cases of Newton’s law, we need to

recall a few concepts from multivariable calculus. Recall that the dot product

(or inner product ) of two vectors X, Y ∈

is denoted by X ·Y and deﬁned by

X · Y =



i=1

where X = (x

, ..., x

) and Y = (y

, ..., y

). Thus X ·X =|X |

.IfX, Y : I →

are smooth functions, then a version of the product rule yields

(X · Y )



= X



· Y + X · Y



as can be easily checked using the coordinate functions x

and y

Recall also that if g :

→ R, the gradient of g, denoted grad g ,is

deﬁned by

grad g (X) =



∂g

∂x

(X), ...,

∂g

∂x

(X)



As we saw in Chapter 9, grad g is a vector ﬁeld on

Next, consider the composition of two smooth functions g ◦ F where

F :

R → R

and g : R

→ R. The chain rule applied to g ◦ F yields

g (F(t )) = grad g (F(t )) · F



(t)



i=1

∂g

∂x

(F(t ))

(t).

We will also use the cross product (or vector product) U × V of vectors

U , V ∈

. By deﬁnition,

U × V =

(

− u

, u

− u

, u

− u

)

∈

Recall from multivariable calculus that we have

U × V =−V × U =|U ||V |N sin θ

where N is a unit vector perpendicular to U and V with the orientations of

the vectors U , V , and N given by the “right-hand rule.” Here θ is the angle

between U and V .

280 Chapter 13 Applications in Mechanics

Note that U ×V = 0 if and only if one vector is a scalar multiple of the other.

Also, if U × V = 0, then U × V is perpendicular to the plane containing U

and V .IfU and V are functions of t in

R, then another version of the product

rule asserts that

(U × V ) = U



× V + U × V



as one can again check by using coordinates.

13.2 Conservative Systems

Many force ﬁelds appearing in physics arise in the following way. There is a

smooth function U :

→ R such that

F(X ) =−



∂U

∂x

(X),

∂U

∂x

(X), ...,

∂U

∂x

(X)



=−grad U (X).

(The negative sign is traditional.) Such a force ﬁeld is called conservative. The

associated system of differential equations



= V



=−

grad U (X )

is called a conservative system. The function U is called the potential energy

of the system. [More properly, U should be called a potential energy since

adding a constant to it does not change the force ﬁeld −grad U (X).]

Example. The planar harmonic oscillator above corresponds to the force

ﬁeld F (X) =−kX . This ﬁeld is conservative, with potential energy

U (X ) =

k |X|

For any moving particle X (t) of mass m, the kinetic energy is deﬁned to be

K =

m|V (t )|

Note that the kinetic energy depends on velocity, while the potential energy

is a function of position. The total energy (or sometimes simply energy)is

deﬁned on phase space by E = K +U . The total energy function is important

13.3 Central Force Fields 281

in mechanics because it is constant along any solution curve of the system.

That is, in the language of Section 9.4, E is a constant of the motion or a ﬁrst

integral for the ﬂow.



Theorem. (Conservation of Energy) Let (X (t ), V (t )) be a solution curve

of a conservative system. Then the total energy E is constant along this solution

curve.

Proof: To show that E(X(t )) is constant in t , we compute

E =



m|V (t )|

+ U (X(t ))



= mV · V



+ (grad U ) · X



= V · (−grad U ) + (grad U ) · V

= 0.



We remark that we may also write this type of system in Hamiltonian form.

Recall from Chapter 9 that a Hamiltonian system on

× R

is a system of

the form



∂H

∂y



=−

∂H

∂x

where H : R

× R

→ R is the Hamiltonian function. As we have seen, the

function H is constant along solutions of such a system. To write the conser-

vative system in Hamiltonian form, we make a simple change of variables. We

introduce the momentum vector Y = mV and then set

H = K + U =



+ U (x

, ..., x

This puts the conservative system in Hamiltonian form, as is easily checked.

13.3 Central Force Fields

A force ﬁeld F is called central if F(X) points directly toward or away from the

origin for every X. In other words, the vector F(X ) is always a scalar multiple

of X :

F(X ) = λ(X )X

282 Chapter 13 Applications in Mechanics

where the coefﬁcient λ(X) depends on X. We often tacitly exclude from con-

sideration a particle at the origin; many central force ﬁelds are not deﬁned

(or are “inﬁnite”) at the origin. We deal with these types of singularities in

Section 13.7. Theoretically, the function λ(X) could vary for different values

of X on a sphere given by |X|=constant. However, if the force ﬁeld is

conservative, this is not the case.

Proposition. Let F be a conservative force ﬁeld. Then the following statements

are equivalent:

1. F is central;

2. F(X) = f (|X|)X;

3. F(X) =−grad U (X ) and U (X ) = g (|X |).

Proof: Suppose (3) is true. To prove (2) we ﬁnd, from the chain rule:

∂U

∂x

= g



(|X|)

∂

∂x



+ x



1/2



(|X|)

|X|

This proves (2) with f (|X|) =−g



(|X|)/|X |. It is clear that (2) implies (1). To

show that (1) implies (3) we must prove that U is constant on each sphere

={X ∈ R

||X|=α > 0}.

Because any two points in

can be connected by a curve in S

, it sufﬁces

to show that U is constant on any curve in

. Hence if J ⊂ R is an interval

and γ : J →

is a smooth curve, we must show that the derivative of the

composition U ◦ γ is identically 0. This derivative is

U (γ (t )) = grad U (γ (t )) · γ



(t)

as in Section 13.1. Now grad U (X) =−F(X) =−λ(X )X since F is central.

Thus we have

U (γ (t )) =−λ(γ (t ))γ (t ) · γ



(t)

=−

λ(γ (t ))

γ (t )

= 0

because |γ (t)|≡α.



13.3 Central Force Fields 283

Consider now a central force ﬁeld, not necessarily conservative, deﬁned on

. Suppose that, at some time t

, P ⊂ R

denotes the plane containing the

position vector X (t

), the velocity vector V (t

), and the origin (assuming, for

the moment, that the position and velocity vectors are not collinear). Note

that the force vector F(X (t

)) also lies in P. This makes it plausible that the

particle stays in the plane

P for all time. In fact, this is true:

Proposition. A particle moving in a central force ﬁeld in

always moves in

a ﬁxed plane containing the origin.

Proof: Suppose X (t ) is the path of a particle moving under the inﬂuence of

a central force ﬁeld. We have

(X × V ) = V × V + X × V



= X × X



= 0

because X



is a scalar multiple of X . Therefore Y = X(t) ×V (t ) is a constant

vector. If Y = 0, this means that X and V always lie in the plane orthogonal

to Y , as asserted. If Y = 0, then X



(t) = g (t )X(t) for some real function g (t).

This means that the velocity vector of the moving particle is always directed

along the line through the origin and the particle, as is the force on the particle.

This implies that the particle always moves along the same line through the

origin. To prove this, let (x

(t), x

(t)) be the coordinates of X(t). Then

we have three separable differential equations:

= g (t )x

(t), for k = 1, 2, 3.

Integrating, we ﬁnd

(t) = e

h(t)

(0), where h(t ) =



g (s) ds.

Therefore X(t) is always a scalar multiple of X(0) and so X (t ) moves in a ﬁxed

line and hence in a ﬁxed plane.



The vector m(X ×V ) is called the angular momentum of the system, where

m is the mass of the particle. By the proof of the preceding proposition, this

vector is also conserved by the system.

Corollary. (Conservation of Angular Momentum) Angular momentum

is constant along any solution curve in a central force ﬁeld.



284 Chapter 13 Applications in Mechanics

We now restrict attention to a conservative central force ﬁeld. Because of

the previous proposition, the particle remains for all time in a plane, which we

may take to be x

= 0. In this case angular momentum is given by the vector

(0, 0, m(x

− x

)). Let

 = m(x

− x

So the function  is also constant along solutions. In the planar case we also

call  the angular momentum. Introducing polar coordinates x

= r cos θ and

= r sin θ,weﬁnd

= x



= r



cos θ − r sin θθ



= x



= r



sin θ + r cos θθ



Then

− x

= r cos θ(r



sin θ + r cos θθ



) − r sin θ (r



cos θ − r sin θθ



)

= r

(cos

θ + sin

θ)θ



= r



Hence in polar coordinates,  = mr



We can now prove one of Kepler’s laws. Let A(t) denote the area swept

out by the vector X (t ) in the time from t

to t. In polar coordinates we have

dA =

dθ . We deﬁne the areal velocity to be



(t) =

(t)θ



(t),

the rate at which the position vector sweeps out area. Kepler observed that the

line segment joining a planet to the sun sweeps out equal areas in equal times,

which we interpret to mean A



= constant. We have therefore proved more

generally that this is true for any particle moving in a conservative central force

ﬁeld.

We now have found two constants of the motion or ﬁrst integrals for a

conservative system generated by a central force ﬁeld: total energy and angular

momentum. In the 19th century, the idea of solving a differential equation

was tied to the construction of a sufﬁcient number of such constants of the

motion. In the 20th century, it became apparent that ﬁrst integrals do not exist

for differential equations very generally; the culprit here is chaos, which we

will discuss in the next two chapters. Basically, chaotic behavior of solutions of

a differential equation in an open set precludes the existence of ﬁrst integrals

in that set.

13.4 The Newtonian Central Force System 285

13.4 The Newtonian Central Force

System

We now specialize the discussion to the Newtonian central force system. This

system deals with the motion of a single planet orbiting around the sun. We

assume that the sun is ﬁxed at the origin in

and that the relatively small

planet exerts no force on the sun. The sun exerts a force on a planet given by

Newton’s law of gravitation, which is also called the inverse square law. This

law states that the sun exerts a force on a planet located at X ∈

whose

magnitude is gm

, where m

is the mass of the sun, m

is the mass of the

planet, and g is the gravitational constant. The direction of the force is toward

the sun. Therefore Newton’s law yields the differential equation



=−gm

|X|

For clarity, we change units so that the constants are normalized to one and

so the equation becomes more simply



= F(X) =−

|X|

where F is now the force ﬁeld. As a system of differential equations, we have



= V



=−

|X|

This system is called the Newtonian central force system. Our goal in this section

is to describe the geometry of this system; in the next section we derive a

complete analytic solution of this system.

Clearly, this is a central force ﬁeld. Moreover, it is conservative, since

|X|

= grad U (X)

where the potential energy U is given by

U (X ) =−

|X|

Observe that F(X) is not deﬁned at 0; indeed, the force ﬁeld becomes inﬁnite

as the moving mass approaches collision with the stationary mass at the origin.

As in the previous section we may restrict attention to particles moving in the

plane

. Thus we look at solutions in the conﬁguration space C = R

−{0}.

We denote the phase space by

P = (R

−{0}) × R