How does Newton's law of gravity lead to problems that require approximate solutions?

Newton’s law of gravitation provides a fundamental model for physical reality, but it frequently leads to problems that are too complex to be solved explicitly, necessitating the use of numerical analysis.1

In the context of mathematics, these problems typically involve ordinary differential equations. Because Newton’s second law relates force to acceleration—the second derivative of position with respect to time—calculating the motion of a body requires solving for position within an equation that involves its derivatives. While simple cases with constant forces can be solved through integration, more realistic scenarios, such as the n-body problem, involve multiple interacting masses that lack closed-form analytic solutions.1234

To address these complexities, mathematicians and scientists employ numerical techniques to obtain highly useful approximate solutions. These methods often involve:1
* Approximation theory: Replacing unknown functions with simpler ones, such as polynomials or splines.5
* Finite difference methods: Approximating derivatives or integrals to solve equations at a discrete set of points.5
* Finite element methods: Breaking down complex continuous processes or structures into a finite number of simple elements.15