Physics education for the 21st century, solving analytically unsolvable differential equations

For nearly three centuries, Newton’s paradigm—modeling physical systems with differential equations and analyzing them with analytic calculus—defined classical physics. However, analytic calculus has a fundamental limitation: most differential equation models of real physical systems are analytically unsolvable. Before the mid-20th century this was the central problem in classical physics, it was resolved when the computer and computational calculus revolutionized the practice of physics by making it possible to solve analytically unsolvable systems. Simulation, based on computational calculus, quickly became the norm for analyzing physical systems in science and industry. Unlike analytic calculus, computational calculus is conceptually simple and intuitively transparent; its basic method can be taught to students in a single one-hour lecture. Yet, physics education has not incorporated this fundamental shift in the mathematics of solving differential equations. Simulation, if it appears at all, is typically limited to a cursory introduction in an upper-division elective in computational physics. Less frequently, it appears in courses in computational fluid dynamics or computational electrodynamics, which are usually offered at the graduate level. This paper, together with The Coming Revolution in Physics Education (The Physics Teacher), contains complete computational analyses of analytically unsolvable prototypical systems spanning classical physics. The differential equation models of the systems are derived from the laws of physics, translated 1-to-1 to computational equations, and striking visual simulations are generated. Incorporating computational calculus will fundamentally transform every course in the classical physics curriculum. The range and complexity of the systems that can be analyzed explodes. This will align physics education with current practice and prepare students for careers in 21st‑century science and engineering.