A body of mass m moves under the influence of force F(r). Given it's trayectory r(t), find F(r).

To find the force function F(r) given a trajectory r(t) for a body of mass m, we need to make use of Newton's second law of motion.

Newton's second law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. Mathematically, this can be represented as F = m * a, where F is the force, m is the mass, and a is the acceleration.

Since we are given the trajectory r(t), we can find the acceleration by taking the second derivative of the position function with respect to time. In other words, we need to find d²r/dt².

Once we have the acceleration, we can substitute it into Newton's second law to obtain the force function F(r) = m * d²r/dt².

So, to summarize:

1. Given the trajectory r(t) for a body, differentiate it twice with respect to time to find d²r/dt², which represents the acceleration.

2. Substitute the acceleration into Newton's second law, F = m * d²r/dt², to obtain the force function F(r).