Classical Mechanics — Newtonian

Newton’s mechanics begins with three deceptively simple statements, each of which seems obvious today but represented a radical shift in thinking at the time. The first law (the law of inertia) declares that a body in motion remains in uniform motion, and a body at rest remains at rest, unless acted upon by a net external force. It sounds trivial, but it shattered centuries of Aristotelian belief that motion required constant pushing. The second law provides the quantitative heart: the acceleration of a body is directly proportional to the net force and inversely proportional to its mass, written today as \( \vec{F} = m \vec{a} \). Finally, the third law states that every action has an equal and opposite reaction, a principle that underpins momentum conservation and the very possibility of rockets leaving Earth. These three rules form a closed loop of logic: inertia provides the baseline, force defines the change, and interaction ensures balance in the universe.

What makes Newton’s laws powerful is not their poetry but their predictive universality. Whether you drop an apple from a tree, analyze a pendulum swinging in a classroom, or compute the orbit of Jupiter’s moons, the same compact expressions apply. Newton unified terrestrial and celestial motion under one framework: the same law that makes a ball arc through the air explains why the Moon does not fall into Earth. By marrying force, mass, and acceleration, Newton built the first truly general engine for prediction in physics.

Forces in Newtonian mechanics are vectors, which means they have both magnitude and direction. Gravity pulls toward Earth’s center, tension pulls along a rope, friction resists motion parallel to a surface, and the normal force pushes perpendicular to contact. Each of these forces can be decomposed and added, and their sum determines acceleration. A free-body diagram becomes the language in which Newton’s second law is written: sketch the object, identify all forces, assign coordinates, and compute. For a block sliding on an incline, the gravitational force \( mg \) splits into parallel and perpendicular components. The parallel part, \( mg \sin\theta \), drives the motion down, while the perpendicular part, \( mg \cos\theta \), is balanced by the normal force. Already, a full predictive equation emerges: \( a = g \sin\theta \).

More elaborate systems reveal the reach of the method. In circular motion, a constant inward force produces acceleration toward the center, not along the tangent. This is why planets orbit stars in ellipses under the inverse-square gravitational law. The required centripetal acceleration is \( v^2 / r \), which ties velocity, radius, and gravitational force into a single balance: \( \frac{G M m}{r^2} = \frac{mv^2}{r} \). Canceling mass \( m \) from both sides gives the orbital velocity formula, showing that every planet, regardless of its own weight, dances to the tune of its star’s mass. In this way Newton’s framework not only solves classroom puzzles but opens astronomy itself to calculation.

Although Newton wrote laws in terms of forces and accelerations, another language soon emerged: energy. Work is defined as force applied through a displacement, \( W = \int \vec{F} \cdot d\vec{r} \). This integral captures not only the amount of effort applied but also the geometry of motion. A force perfectly perpendicular to motion, like the centripetal pull in uniform circular motion, does no work even though it continuously redirects the path. Potential energy provides a way to store the effect of forces: lifting a mass in a gravitational field builds energy that can later be released. Kinetic energy measures the motion itself, \( T = \tfrac{1}{2} m v^2 \). The principle of work and energy then declares that the net work done equals the change in kinetic energy, a statement equivalent to Newton’s second law but often easier to apply.

Conservation of energy elevates this viewpoint: in a closed system where only conservative forces act, the total mechanical energy (kinetic plus potential) remains constant. A pendulum illustrates this vividly. At its peak, energy is stored entirely as potential \( mgh \). As it swings downward, potential converts to kinetic until, at the bottom, velocity is maximum and potential minimal. Yet the sum remains the same, ignoring friction. This way of thinking is not only efficient but profound, because it reveals quantities that remain unchanged across time — invariants that define the identity of the system.

Newton’s third law implies conservation of momentum. Momentum, defined as \( \vec{p} = m \vec{v} \), is exchanged but not destroyed in interactions. When two ice skaters push off each other, one glides left, the other right, their total momentum balancing to zero just as before. Elastic collisions, where kinetic energy is also conserved, provide clean textbook problems: two billiard balls colliding obey simple algebraic relations. Inelastic collisions, where kinetic energy is not preserved but momentum still is, describe car crashes, meteor impacts, or particles sticking together. In all cases, momentum conservation provides a reliable compass through complexity. In continuous systems, this principle becomes even richer, leading to the momentum flux terms in fluid mechanics and stress tensors in solid mechanics.

For all its triumphs, Newton’s mechanics has boundaries. At velocities approaching the speed of light, relativistic corrections become essential, replacing \( F = ma \) with dynamics drawn from Einstein’s theory. At atomic scales, quantum mechanics replaces precise trajectories with probabilities. Still, Newton’s framework endures in its realm: human-scale motion, engineering problems, orbital mechanics, fluid flow approximations. Its blend of clarity, predictability, and geometric intuition makes it the entry point for every student and the workhorse for countless applications. To learn Newtonian mechanics deeply is to learn how to think like a physicist: to reduce the world to its essentials, to state assumptions plainly, and to follow consequences through.