In order to determine how a force, F, acting on a particle with a fixed mass, m, affects the position of that particle in space, one needs to solve one of Newton's laws:
where a is the resulting acceleration for the particle. The force itself is determined by the environment of the particle. This equation is also known as Newton's equation of motion.
That this equation is really a second order differential equation in time, t, can be understood from the fact that acceleration is the derivative of the velocity, v, to time and velocity is the derivative of position, u, to time, viz.
The unit of force is called the Newton, N, where 1 N = 1 kg m s-2.
If the force acting on a particle is known, one can determine the position as a function of time by solving the second order differential equation. This can be done by integrating the equation twice. The first integration will yield the velocity as a function of time, while the second one gives the position as a function of time. For some simple cases, specifically when the force is not a function of time, this is easily done and a mathematical function or expression, also called an analytical solution, may be found.
In some cases it is not possible to find such an analytical solution. What mathematicians sometimes do in this kind of situation is to define a new function that has the property that it is a solution to that particular differential equation. This allows them to use a simple notation in their further investigation of the behavior of the solutions, and investigate the effect of variable occuring in the expression on the solutions.
When one is interested in the effect of initial conditions, the solution of that particular case is needed. To find this solution in the case that an analytical solution is difficult, one can solve the second order differential equation in a numerical fashion by numerical integration . For a second order differential equation two integrations need to be performed. The first integration builds the velocity function, based on an initially know velocity. The second integration integrates the velocity function to calculated the positions point by point, starting with an initial position of the particle. Thus armed with these two values and the equation of motion, one may numerically determine the trajectory of particles.
When the force is proportional to the displacement
the force to restore the system to equilibrium is stronger when the system is farther away from equilibrium, u = 0. In this case, one calls such a force a strong focusing force. Two examples of this kind of systems are:
Note that the constant force in the harmonic oscillator is a special instance of the dynamic force case, where V = 0 and U = k/k. This means that the solutions to the second example should encompass those of the first. Also, the second example is much more complicated by the fact that the force is also a function of time.