**Lab #5**

You will have two weeks to complete this lab, professionally write up the answers, and submit it for a grade.

The purpose of the lab is

- to understand gradient vector fields and their relationship with higher-dimensional surfaces and level sets of those surfaces;
- to explore the geometry and discover the relationships between a surface, gradients to that surface, and the ``gradient flow'' on that surface.
- to see how multivariate functions naturally give rise to gradient differential equations.

**Background:**
One of the greatest achievements of Newton (indeed, of all
mathematicians!) was formulating the
mathematics of gravitational attraction. Newton published his laws of
gravitation in his famous book, *Principia Mathematica*, in 1687. We
will not subject you to Newton's original formulation of gravitational
laws as it appeared in latin, but instead summarize in English: two
objects separated by a distance *r* feel a gravitational force due to
each other. The magnitude of this force is proportional to .

It turns out that the gravitational force field is a so-called ``gradient field.'' This means that the force at any point is actually the gradient of some function, called the ``gravitational potential function.'' An important way to visualize gravitational force fields is to visualize the graph of the gravitational potential function.

In order to simplify the mathematics, we assume that we are only concerned about the gravitational field in a plane. This is actually a close approximation to reality, since most of the objects in the solar system (with the exception of pluto and some comets) essentially lie in a plane (called the ``ecliptic plane''). Also, since we are concentrating on the geometry, we will scale the physical equations so that we do not need to worry about physical quantities such as the mass of the earth or the distance from the earth to a satellite or to the moon.

We begin by considering the gravitational force felt by a satellite in space. We place the earth at the center of our coordinate system. We then define the gravitational potential function to be

and the force felt by
an object located at position (*x*,*y*) to be

To get started, launch Maple and type in the following three commands:

`with(linalg): with(plots):`

`P := -1/sqrt(x^2 + y^2);`

`F := grad( -P, [x,y] ); `

**Note that we have defined the force to be the ``minus gradient'' of
the potential function. **
Notice also that the
the potential is a function from , whereas the force is vector-valued.

Now think about placing a ball on the graph of the potential function. When the ball is far away from the origin, the potential surface is only slightly tilted, so the ball feels only a small force towards the origin. If the ball is near the origin, however, the potential surface is very steep, so the ball feels a very strong pull towards the origin. Mathematically, we say that the force felt by an object is the negative of the gradient of the potential function. Let's see how this all works:

- Compute the force felt by a particle at location
(
*x*,*y*)=(1,1). This is a vector; call it*v*=*F*(1,1). - Think of
*v*as being based at the point (1,1). What direction does*v*point and what is its magnitude?

to visualize the ``vector field''

In the second part of this lab we will use Matlab to study the ``lines of force'' that are solutions to a certain differential equation related to the gradient vector field. For now:

For the next question, we will change the scenario under consideration and ask you to investigate the new situation.

Assume that instead of one gravitational mass at the origin, we have
two gravitational masses. The one mass (say, the earth) is located at
(*x*,*y*)=(-1,0) and is very massive; the other mass (say, the moon) is
located at (*x*,*y*)=(0,1) and is less massive. In actuality, the
moon has about 1/100 the mass of the earth; we will use a ``massive
moon'' in order to make the geometry more apparent. In particular, we
will model the geometry of this new situation by
the gravitational potential function and the new force
field (in Maple notation) by

`P := -1/sqrt((x+1)^2 + y^2) - 1/10/sqrt((x-1)^2 + y^2);`

`F := grad( -P, [x,y] ); `

At this point in the lab, we begin using Matlab to investigate the same problem. Caution: the trajectories you will see are NOT trajectories of an object in orbit around a gravitational body! It is possible to compute such trajectories (we will do this briefly at the end of the lab), but the computation requires looking at a differential equation in which the positions

If we write this differential equation in vector form, the right-hand side of the equation is just . The solution to this differential equation gives us the ``lines of gravitational force'' for this gravitational system. If instead of gravitational bodies we were considering charged bodies, then the solutions would be ``lines of electromagnetic force.'' We could also consider magnetically charged bodies to get lines of ``magnetic force.''

- questions here!

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Last modified: Oct 31 1996

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