**Chapter 8
**

**Central-Force
Motion
**

__Two-Body
Systems with a Central Force
__

Consider
the motion of two objects that are effected by a force acting along the line
connecting the centers of the objects. To specify the state of the system, we must specify six coordinates (for
example, the (*x*, *y*, *z*)
coordinates of their centers). The
Lagrangian for this system is given by

Note: here we have assumed that the
potential depends on the position vector between the two objects. This is not the only way to describe
the system; we can for example also specify the position of the center-of-mass, *R*, and the three components of the
relative position vector *r*. In this case, we choose a coordinate
system such that the center-of-mass is at rest, and located at the origin. This requires that

The relative position vector is defined as

The position vectors of the two masses can be expressed in terms of the relative position vector:

The Lagrangian can now be rewritten as

where *m* is the reduced mass of the system:

__Two-Body
Systems with a Central Force: Conserved Quantities__

Since
we have assumed that the potential *U* depends only on the relative position between the two objects, the system poses
spherical symmetry. As we have
seen in Chapter 7, this type of symmetry implies that the angular momentum of
the system is conserved. As a
result, the momentum and position vector will lay in a plane, perpendicular to
the angular momentum vector, which is fixed in space. The three-dimensional problem is thus reduced to a
two-dimensional problem. We can
express the Lagrangian in terms of the radial distance *r* and the polar angle *q*:

The generalized momenta for this Lagrangian are

The Lagrange equations can be used to determine the derivative of these momenta with respect to time:

The last equation tells us that the
generalized momentum *p*_{q} is constant:

The constant *l* is related to the areal velocity. Consider the situation in Figure
1. During the time interval *dt*, the radius vector sweeps an area *dA* where

Figure 1. Calculation of the areal velocity.

The areal velocity, *dA*/*dt*, is
thus equal to

This result is also known as **Kepler's
Second Law**.

The
Lagrangian for the two-body system does not depend explicitly on time. In Chapter 7 we showed that in that
case, the energy of the system is conserved. The total energy *E* of
the system is equal to

__Two-Body
Systems with a Central Force: Equations of Motion__

If
the potential energy is specified, we can use the expression for the total
energy *E* to determine *dr*/*dt*:

This equation can be used to find
the time *t* as function of *r*:

However, in many cases, the shape of
the trajectory, *q*(*r*), is more important than the time dependence. We can express the change in the polar
angle in terms of the change in the radial distance:

Integrating both sides we obtain the following orbital equation

The extremes of the orbit can be
found in general by requiring that *dr*/*dt* = 0, or

In general, this equation has two
solutions, and the orbit is confined between a minimum and maximum value of *r*. Under
certain conditions, there is only a single solution, and in that case the orbit
is circular. Using the orbital
equation we can determine the change in the polar angle when the radius changes
from *r*_{min} to *r*_{max}. During one period, the polar angle will change by

If the change in the polar angle is
a rational fraction of 2*p* then
after a number of complete orbits, the system will have returned to its
original position. In this case,
the orbit is closed. In all other
cases, the orbit is open.

The
orbital motion is specified above in terms of the potential *U*. Another approach to study the equations of motion is to start from the
Lagrange equations. In this case
we obtain an equation of motion that includes the force *F* instead of the potential *U*:

This version of the equations of motion is useful when we can measure the orbit and want to find the force that produces this orbit.

__Example:
Problem 8.8
__

Investigate
the motion of a particle repelled by a force center according to the law *F*(*r*) = *kr*. Show
that the orbit can only be hyperbolic.

The general expression for *q*(*r*) is [see Eq. (8.17) in the text book]

(8.8.1)

where

in the present case. Substituting *x* = *r*^{2} and *dx* = 2*rdr* into (8.8.1), we have

(8.8.2)

Using Eq. (E.10b), Appendix E,

(8.8.3)

and expressing again in terms of *r*, we find

(8.8.4)

or,

(8.8.5)

In order to interpret this result, we set

(8.8.6)

and specifying *q*_{0} = *p*/4, (8.8.5) becomes

(8.8.7)

or,

(8.8.8)

Rewriting (8.8.8) in *x*-*y* coordinates, we find

(8.8.9)

or,

(8.8.10)

Since *a* '
> 0, *e** *' > 1 from the definition, (8.8.10) is
equivalent to

(8.8.11)

which is the equation of a hyperbola.

__Solving
the Orbital Equation
__

The
orbital equation can only be solved analytically for certain force laws. Consider for example the gravitational
force. The corresponding potential
is -*k*/*r* and the polar angle *q* is thus equal to

Consider the change of variables
from *r* to *u* = *l*/*r*:

The integral can be solved using one of the integrals found in Appendix E (see E8.c):

This equation can be rewritten as

We can always choose our reference
position such that the constant is equal to *p*/2
and we thus find the following solution:

We can rewrite this expression such
that we can determine the distance *r* as
function of the polar angle:

Since cos*q* varies between -1 and +1, we see that the minimum (the **pericenter**) and the maximum (the **apocenter**) positions are

The equation for the orbit is in
general expressed in terms of the **eccentricity** *e* and the **latus
rectum** 2*a*:

The possible orbits are usually parameterized in terms of the eccentricity, and examples are shown Figure 2.

Figure 2. Possible orbits in the gravitational field.

The period of the orbital motion can
be found by integrating the expression for *dt* over one complete period:

When we take the square of this equation we get Kepler's third law:

__The
Centripetal Force and Potential
__

In
the previous discussion it appears as if the potential *U* is modified by the term *l*^{2}/(2*m**r*^{2}). This term depends only on the position *r* since *l* is constant, and it is interpreted as a potential energy. The force associated with this
potential energy is

This force is often called the **centripetal
force** (although it is not a real force),
and the potential is called the **centripetal potential**. This
potential is a fictitious potential and it represents the effect of the angular
momentum about the origin. Figure
3 shows an example of the real potential, due to the gravitational force in
this case, and the centripetal potential. The effective potential is the sum of these two potentials and has a
characteristic dip where the potential energy has a minimum. The result of this dip is that there
are certain energies for which the orbit is bound (has a minimum and maximum
distance). These turning points
are called the **apsidal distances** of the orbit.

Figure 3. The effective potential for the
gravitational force when the system has an angular momentum *l*.

We also note that at small distances the force becomes repulsive.

__Example:
Problem 8.22__

Discuss
the motion of a particle moving in an attractive central-force field described
by *F*(*r*) = –*k*/*r*^{3}. Sketch some of the orbits for different values of the total energy.

For the given force

the potential is

(8.22.1)

and the effective potential is

(8.22.2)

The equation of the orbit is [cf. Eq. (8.20) in the text book]

(8.22.3)

or,

(8.22.4)

Let us consider the motion for various values of l.

i) :

In this case the effective potential *V*(*r*)
vanishes and the orbit equation is

(8.22.5)

with the solution

(8.226)

and the particle spirals towards the force center.

ii) :

In this case the effective potential is positive and
decreases monotonically with increasing *r*. For any value of the total energy *E*, the particle will approach the force center and
will undergo a reversal of its motion at *r* = *r*_{0}; the
particle will then proceed again to an infinite distance. Setting

equation (8.22.4) becomes

(8.22.7)

with the solution

(8.22.8)

Since the minimum value of *u* is zero, this solution corresponds to unbounded motion, as expected
from the form of the effective potential *V*(*r*).

iii) :

For this case we set

and the orbit equation becomes

(8.22.9)

with the solution

(8.22.10)

so that the particle spirals in towards the force center.

__Orbital
Motion__

The understanding of orbital dynamics is very important for space travel. The orbit in which a spaceship travels is determined by the energy of the spaceship. When we change the energy of the ship, we will change the orbit from for example a spherical orbit to an elliptical orbit. By changing the velocity at the appropriate point, we can control the orientation of the new orbit.

The Hofman transfer represents the path of minimum energy expenditure to move from one solar-based orbit to another. Consider travel from earth to mars (see Figure 4). The goal is to get our spaceship in an orbit that has apsidal distances that correspond to the distance between the earth and the sun and between mars and the sun. This requires that

and

The eccentricity of such an orbit is thus equal to

The total energy of an orbit with a
major axis of *a* = (*r*_{1} + *r*_{2})/2 is equal to

Since the space ship starts from a
circular orbit with a major axis *a* = *r*_{1}, its initial energy is equal to

Figure 4. The Hofman transfer to travel from earth to mars.

The increase in the total energy is thus equal to

This energy must be provided by the thrust of the engines that increase the velocity of the space ship (note: the potential energy does not change at the moment of burn, assuming the thrusters are only fired for a short period of time).

The problem with the Hofman transfer mechanism is that the conditions have to be just right, and only of the planets are in the proper position will the transfer work. There are many other ways to travel between earth and mars. Many of these require less time than the time required for the Hofman transfer, but they require more fuel (see Figure 5).

Figure 5. Different ways to get from earth to mars.

**SECTIONS 8.9 AND 8.10 WILL BE SKIPPED!
**

**
**

In this Chapter we will use the theory we have discussed in Chapter 6 and 7 and apply it to very important problems in physics, in which we study the motion of two-body systems on which central force are acting. We will encounter important examples from astronomy and from nuclear physics