100.08 - Fish Mathematics - Adult Level

Fish Mathematics:

Suppose we have n particles (fish) with positions x₁, ..., x_n and constant masses m₁, ..., m_n.

The Newtonian motion of particles satisfies m_i a_i = F_i, where F_i is the net force on particle i and a_i is the acceleration or second derivative of the x_i position with respect to time.

If the forces between particles are gravitational, then net force on i can be the sum of individual forces on i due to all of the other particles. The force of particle j on particle i, F_ij, in the gravitational sense, may be determined by the following formula (|| || is used to denote the Euclidean norm).

If you want a coulombic attraction, ascribe charges to the particles and change the constant a little:

Note, the last term correlates to the direction of the force. If you want to try van der Waals forces, they are a bit more complex:

where C is about 1/10. While particles far away are attracted, particles that become too close are repelled introducing some interesting dynamics.

The differential equations for the motion of the system then become:

.

Given initial positions and velocities, we can hypothetically integrate these equations to find the exact position as a function of time. In practice, there are several ways of numerically solving this system, and one common quick (yet innacurate) method we use is the Euler method.

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Baitman Note: To find the value of "fish" and to purchase further Study Guides, see Answer in the next Posting.

9/20/04