Kinetic energy minus potential energy integrated over time is, of course, our particle's action. For each path there is a minimum action. Think of a circle as a locus of all points at a constant distance from a fixed point or as a curve of a specified length enclosing the largest area. Any shape for the perimeter other than a circle must enclose a smaller surface area.

So in considering paths through the hemocoel from one point to another, we imagine there is one true shortest path where the action is minimal, so that taking any other curve means taking a false path, a more energetically wasteful path, because if we calculate the action over the false path, the action is greater than if we took the shorter path. Or let's assume the particle takes a minimum path to start with. If we deviate from it in the first order, the function deviates from its minimum only by the second order. At any place along the curve, if we move a small distance away, the value of the function changes also to the first order but at a minimum. Taking just a tiny step to the side makes no difference at all in the first approximation. If there is a change in the first order when a particle deviates, the change in the action is proportional to this deviation. Reversing the sign of the deviation makes the action less so. At this point we can have the action increase going one way and have it decrease going the other.

It turns out that the path having the least action is the path satisfying Newton's law. However, in the hemocoel, we cannot forget friction. The principle of least action only works for conservative systems in which we obtain all our forces from a single potential function. But at the microscopic level of organization there are no conservative functions. The Lagrangian is the function integrated over time to give the action. The Lagrangian is a function only of the velocities and positions of particles.

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