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toward body one from body two. <p>
Converting to  a Cartesian form the acceleration of body one is:<p>
<!WA11><!WA11><!WA11><!WA11><img src="http://www.tc.cornell.edu/Visualization/Education/cs417/SECTIONS/dynamics.eqn.gravvector.gif"> 
 <p>
where the vectors<p>
<!WA12><!WA12><!WA12><!WA12><img src="http://www.tc.cornell.edu/Visualization/Education/cs417/SECTIONS/dynamics.eqn.gravra.gif"> 
<p>
The calculation procedure for each time step is to compute:
<ol>
<li>  the acceleration, a, based on the positions at time n-1
<li>  a new set of positions using the Verlet method
<li>  the acceleration at time n based on the newly computed positions 
<li>  a new velocity from the Verlet method
using the acclerations at times n-1 and n.
</ol>
For the gravitational animation given at the beginning of this page,
three masses were simulated. The accleration of each mass was determined
as the vector sum of the accelerations caused by each of the other two
bodies.
<hr>

<b> Water Waves </b> <p>
The water wave solver presented here is based on the derivation in Kass and 
Miller [2] for the case of shallow water, low amplitude, non-breaking
waves. The form of the resulting equation of motion for the waves looks
like the classical linear wave equation with a propagation velocity
proportional to the depth of the water.
<p>
<!WA13><!WA13><!WA13><!WA13><img src="http://www.tc.cornell.edu/Visualization/Education/cs417/SECTIONS/dynamics.eqn.water.gif">
<p>
where h is the height of the water surface and d is the depth, that is,
d(x,y) = h(x,y)-b(x,y)
where b is the vertical position of the containing vessel (or ocean
bottom) at (x,y). The costant g is poroportional to the force of
gravity. This partial differential equation can be discretized
in many ways. We will use a method which is stable enough to be
acceptable for computer graphics. 
<p>
First note that the partial with respect to time is the acceleration
of a small surface element of water, so
that if the right side of the equation can be put in a discrete form we
can apply the Verlet method to a 2D grid of "bodies", each representing
a small chunk of water.
<p>
To solve this equation numerically we need to discretize it both in
space, on a two dimensional grid, and in time. The discrete spatial
approximation is that the second derivitives (at time n) are <p>
<!WA14><!WA14><!WA14><!WA14><img src="http://www.tc.cornell.edu/Visualization/Education/cs417/SECTIONS/dynamics.eqn.waterpartials.gif">
<p>
were i and j are the grid indices in the x and y directions and n is the
time index. Note that at the edges of the array the discrete partial
derivitives depend on values outside the array. These boundary
conditions need to be specified for a solution. One easy boundary condition,
corresponding to no transport across the coundary, is to copy the value
at the edge of the array whenever a value outside the array is needed.
<p>It seems that the solution is more stable if the minimum depth of the
water is limited to about .001 of the average value, rather than letting
it go to zero. So
the depth at time n is given by <p>
<!WA15><!WA15><!WA15><!WA15><img src="http://www.tc.cornell.edu/Visualization/Education/cs417/SECTIONS/dynamics.eqn.watergetd.gif">
<p>
To get the surface motion, first assign an initial height and vertical
velocity to each grid point (i,j) then at each time step compute :
<ol>
<li>  the acceleration, a, based on the water heights at time n-1 <p>
<!WA16><!WA16><!WA16><!WA16><img src="http://www.tc.cornell.edu/Visualization/Education/cs417/SECTIONS/dynamics.eqn.watertm1.gif">
  <p>
<li>  a new set of positions using the Verlet method for each (i,j)
<li>  the acceleration at time n based on the newly computed heights <p>
<!WA17><!WA17><!WA17><!WA17><img src="http://www.tc.cornell.edu/Visualization/Education/cs417/SECTIONS/dynamics.eqn.watert.gif">
<p>
<li>  a new velocity from the Verlet method for each (i,j)
<li> Ensure conservation of volume of the total volume by adjusting the
average water height to a constant. That is, sum all the heights over the
whole grid, then correct the sum to equal the initial sum
by adding a small increment to each grid location (i,j).
</ol>
These steps are repeated to generate moving waves. 
 <p>Disclaimer: While this
integration method seems to give visually reasonable results, it should not
be used for analytical simulations without careful validation. The wave
equation is notoriously hard to integrate using explicit techniques, like
the one described here.

<p>

<hr>
<b> Billards </b> <p>
A billards, hard ball, system is different than the other two systems
just described because the forces between balls are zero until they
touch and become very large if the balls try to pass through each other.
This means that there is a large, impulsive force just when the balls
meet and at no other times. The Verlet integration scheme will fail
badly on this system because the accelerations are large for a short
time and thus not smooth enough to average. What we will do to step the
billards system forward in time is to calculate the total change 
in velocity from
a collision, without worrying exactly how forces change the velocity.
<p>
The method described here is a less exact version of that described in
[4]. The method described in the paper steps time at uneven intervals,
and is thus unsuitable for animation. My modification steps time
uniformly, at the expense of exact collision dynamics.
<p>
The change in velocity during impact can be derived for frictionless
balls of  equal mass by noting that the the impact force must act in a
direction parallel to the line connecting the centers of the
two impacting balls. The change in velocity must be parallel to 
the connecting line also, with the velocity component parallel to the 
line having its sign reversed by the collision and the velocity
component perpendicular to the line unchanged. Projecting the initial
velocity onto the line connecting the centers, negating the result, and
resolving it back into x and y velocity components gives the velocity
change. If i and j are the indices of the colliding balls define:<p>

<!WA18><!WA18><!WA18><!WA18><img src="http://www.tc.cornell.edu/Visualization/Education/cs417/SECTIONS/dynamics.eqn.billardsdiff.gif"> <p>
then delta v for ball i is given by the following where the right-most term
represents the projection of the velocity onto the line and the other
term converts the projection back to x,y coordinates. <p>
 <!WA19><!WA19><!WA19><!WA19><img src="http://www.tc.cornell.edu/Visualization/Education/cs417/SECTIONS/dynamics.eqn.billards.gif"> <p>
The calculation procedure for each time step is to compute:
<ol>
<li>  the delta v based on the positions of the balls
<li>  a new set of positions using <p>
<!WA20><!WA20><!WA20><!WA20><img src="http://www.tc.cornell.edu/Visualization/Education/cs417/SECTIONS/dynamics.eqn.billardsnew.gif">
<p>
<li> if walls are present detect collisions and modify velocities.
</ol>
The time step needs to be small enough so that the balls do not
penetrate each other too much during one time step.

<hr>


<b> References </b> <p>
<ol>
<li>  <i>An Introduction to Computer Simulation Methods, Part 1,</i>
 by Harvey
Gould and Jan Tobochnik, Addison-Wesley, 1988 <p>
<li> <i> Rapid, Stable Fluid dynamics for Computer Graphics, </i>
by Micheal Kass and Gavin Miller, Computer Graphics, Vol 24 #4, Aug 1990,
pp 49-55 <p>
<li> <i> Numerical Recipes,</i> by William Press, Saul Teukolsky,
William Vetterling and Brian Flannery, Cambrige, 2nd edition, 1992 <P>
<li><i>Studies in Molecular Dynamics. I. General Method,</i> by B. Alder
and T. Wainwright, Journal of Chemical Physics, Vol 31 #2, Aug 1959,
pp 459-466 <p>
<li> <i> Physics for students of science and engineering, Part 1,</i> by
Robert Resnick and David Halliday, Wiley, 1963. <p>
</ol>
<hr>
Comments about Theory Center online documents are welcome and may be sent to
<i>[e-mail]</i>.
 <p>

Last modified, 10/12/95 B. Land. 
<! Revision history:
	Original document: B Land.
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