Newton's Laws of Motion

In 1687, after twenty years of work and at the urging of his friend Edmond Halley, Isaac Newton finally allowed his magnum opus to go to press. Philosophiae Naturalis Principia Mathematica — Mathematical Principles of Natural Philosophy — was published in three books, in Latin, at Halley's personal expense. Inside were three concise statements that, between them, unified terrestrial and celestial mechanics under a single mathematical framework and founded modern physics. The same equations that describe an apple falling from a tree describe the moon falling around the Earth. Before Newton, this was not obvious. After Newton, it became the entire premise of physics for the next two and a half centuries.

Newton's first law: an object in motion stays in motion at constant velocity unless a force acts on it. This is the law of inertia, dispelling the Aristotelian intuition that things naturally come to rest. Rest is not the natural state — it is one particular state of constant velocity. Newton's second law: the rate of change of an object's momentum equals the net force acting on it, F = ma in the simplest form (force equals mass times acceleration). Mathematically, this is a second-order differential equation: position is the unknown, force determines acceleration (the second derivative of position), and the equation can be integrated given initial position and velocity. Newton's third law: for every action there is an equal and opposite reaction — forces come in pairs. From these three laws plus a force law (gravity, friction, springs, electric forces, …), the motion of every macroscopic non-relativistic non-quantum system can in principle be computed. The framework is deterministic: given exact initial conditions, the future is exactly determined. Lagrangian and Hamiltonian mechanics — reformulations developed by Lagrange (1788) and Hamilton (1830s) — recast Newton's laws in terms of generalized coordinates and energy functions, more elegant and more naturally extending to fields and quantum mechanics. Conservation laws — momentum, energy, angular momentum — fall out as theorems within the framework, later illuminated by Noether's theorem as consequences of symmetries of the Lagrangian. The deep observation, however, is that most physical phenomena look the way they do because Newton's laws are nearly correct: the relativistic corrections are small at everyday speeds, the quantum corrections are small at everyday scales, and Newton's three sentences plus a few force laws are what every engineer still uses to design every macroscopic mechanical thing.