Least Action

Suppose our particle moves from its original place to another place in an interval. Then the particle does it again but follows a different path through the hemolymph, but it gets to the same place it went the first time in the same amount of time. If we calculate the particle's kinetic energy at every moment along the path and subtract the potential energy at every moment and integrate it over the time the particle needs for the whole trip, the number we get is larger than the particle's actual motion.

The principle of least action states that the average kinetic energy minus the average potential energy stays as small as possible while going from one point to another. The true shortest path is the path for which this integral is least.

Why is this possible? Because were our particle to take any other path than the one it takes, its velocities would sometimes exceed and sometimes be less than the average velocity. Average speed through the hemolymph is the total distance traversed over the time. This means that the mean square of something varying around an average exceeds the square of the mean, so that now the integral for the kinetic energy would be greater if the velocity were irregular than if the velocity were uniform. This is another way of saying that our integral is minimal if the velocity remains constant, and we have just a uniform push and no random forces. Such solutions are always balances between holding on to the most potential energy while expending the least extra kinetic energy to keep the difference between kinetic energy minus potential energy as small as possible.

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