Newton's laws of motion

• 1. Introduction

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  Newton's laws of motion are three physical laws describing how an object moves in relation to the forces acting on it. A body left alone stays at rest or glides in a straight line at constant speed. Push or pull it, and its acceleration follows the force divided by its mass. Press on another body, and that body presses back with equal strength in the opposite direction. Isaac Newton first stated these three laws in his Philosophiae Naturalis Principia Mathematica, published in 1687. With them, he explained the motion of countless physical objects and systems. But how does a single rule connect a falling apple to the Moon's orbit? Why does the third law confuse generations of students who think the support of a table is gravity's reaction? And what happens when the laws meet light, or the very small, where they begin to break down? The answers stretch from a Byzantine thinker mocking armies with bellows to a proverb about spacetime.

• 2. Three Laws In Words

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  Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it. That is Newton's first law, and it expresses the principle of inertia. The natural behavior of a body is to keep moving in a straight line at constant speed, preserving the status quo until an external force perturbs it.

  By "motion" in the second law, Newton meant the quantity now called momentum, which depends on the amount of matter in a body, its speed, and its direction. The change of motion of an object is proportional to the force impressed, and made in the direction of the straight line in which the force is impressed. In modern notation momentum is the product of mass and velocity, and the second law states that the time derivative of momentum equals the force. When the net force on a body is zero, the body does not accelerate and is said to be in mechanical equilibrium.

  "To every action there is always opposed an equal reaction," Newton wrote of the third law, "or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts." If one body pushes a second, the second pushes back with equal magnitude in the opposite direction. The brief paraphrase "action equals reaction" has caused confusion: the two forces apply to different bodies. Consider a book resting on a table. The Earth's gravity pulls the book down, and the reaction to that pull is not the table's support but the book's own gravity tugging on the Earth. This relation connects to a deeper principle, the conservation of momentum, which holds even where Newton's statement does not.

• 3. The Language Of Motion

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  Kinematics, the mathematical description of motion, begins with specifying position using numerical coordinates. A body's trajectory becomes a function assigning position values to each moment of time. In the simplest one-dimensional case, a body slides along a track, its location given by distance from a chosen origin, with negative numbers to the left and positive numbers to the right.

  The Greek letter delta, used by tradition to mean "change in," defines average velocity over a time interval. Calculus sharpens this into instantaneous velocity, the derivative of position with respect to time, the speed and direction at a single moment. Acceleration is to velocity as velocity is to position: the derivative of velocity, and so the second derivative of position.

  Position, velocity, and acceleration are vector quantities, each carrying both magnitude and direction. A velocity vector of 3 metres per second along the horizontal axis and 4 metres per second along the vertical describes the same motion that a different coordinate system would record with different numbers. Force, too, is a vector. The study of mechanics is complicated because household words carry technical meanings: force is not power or pressure, and mass differs from weight. These distinctions matter once forces from strings, friction, muscle, and gravity enter the equations.

• 4. Inertial Observers

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  A person standing on the ground watching a train pass is an inertial observer, feeling no effects of motion. The modern reading of Newton's first law is that no inertial observer is privileged over any other. If the train moves smoothly in a straight line at constant speed, the passenger seated inside is equally an inertial observer, feeling no motion at all.

  There is no way to say which observer is "really" moving and which is "really" standing still. One observer's state of rest is another's state of uniform motion, and no experiment can declare either point of view correct. There is no absolute standard of rest. Newton himself believed absolute space and time existed, yet he held that only relative measures of space and time are accessible to experiment. That tension would return when light entered the picture.

• 5. Solved Problems

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  A body falling from rest near the Earth's surface, with no air resistance, accelerates at a constant rate called free fall. The speed grows in proportion to the elapsed time, and the distance falls in proportion to the square of the elapsed time. The acceleration is the same for all bodies regardless of mass, because combining the second law with the law of universal gravitation makes the falling body's mass cancel from both sides. Launch the body upward or sideways instead, and free fall becomes projectile motion, tracing a parabola because gravity affects vertical motion but not horizontal. At the peak the vertical velocity is zero, yet the acceleration still points downward, and setting the wrong vector to zero is a common student error.

  Uniform circular motion changes a body's direction but not its speed, with acceleration pointing toward the center of the circle. The centripetal force that sustains it points inward too. Many orbits, such as the Moon around the Earth, can be approximated this way, with gravity supplying the centripetal force. This lets the mass of a body be calculated from observations of another body orbiting it.

  Newton's cannonball is a thought experiment bridging projectile motion and orbit. A cannonball lobbed weakly off a tall cliff hits the ground in the same time as one dropped from rest, because gravity acts only downward. Fire it faster and it travels farther but lands in the same time. Fire it faster still and the curvature of the Earth matters: the ground curves away beneath it. A very fast cannonball falls away from a straight line at the same rate the Earth curves, and so it is in orbit. The harmonic oscillator offers another key example, approximating any system near a stable equilibrium, such as a pendulum that swings back and forth when nudged from vertical.

• 6. Beyond Particles

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  A rigid body keeps its shape and is too large to treat as a point. Its motion separates into the movement of its center of mass and movement around that center. The center of mass behaves as if all the body's mass were concentrated at one location, and in the absence of a net external force it moves at constant speed in a straight line. When two bodies collide, their internal forces occur in balanced pairs by the third law, so the center of mass holds its course.

  Apply Newton's laws to rotation and new quantities appear. The analogue of mass is the moment of inertia, the counterpart of momentum is angular momentum, and the counterpart of force is torque. When the torque is zero, angular momentum stays constant, just as zero force keeps momentum constant. Torque can vanish even with a nonzero force if the force and the displacement vector lie along the same line.

  Matter that enters or leaves a body raises the problem of variable mass. A rocket of mass moving at some velocity ejects matter at a velocity relative to the rocket, and the laws apply by tracking which pieces of matter belong to the object over time. The fan and sail example tests the third law: a fan attached to a cart blows on its own sail. Naively the forces would cancel and leave the vessel still, yet because the system is not enclosed, a sail that redirects most of the airflow back toward the fan can drive the vessel forward.

• 7. Where The Laws Break

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  Chaos lives inside Newton's laws. Physical systems obeying them can show sensitive dependence on initial conditions, where a slight change in position or velocity makes the whole system behave radically differently within a short time. The three-body problem, the double pendulum, dynamical billiards, and the Fermi-Pasta-Ulam-Tsingou problem are noteworthy examples. Adding a third mass to the Kepler problem yields the three-body problem, which in general has no exact closed-form solution and must be approximated by numerical methods looped over short time intervals.

  The Euler momentum equation carries the second law into fluid dynamics, treating a fluid as infinitesimal pieces pushing on their neighbors. Add viscosity and the Euler equation becomes the Navier-Stokes equation. Whether initially smooth solutions of the Euler and Navier-Stokes equations can "blow up" in finite time is not yet known, and the existence and smoothness of Navier-Stokes solutions is one of the Millennium Prize Problems. Point masses can even fling some of themselves to infinity in a finite time, an unphysical "noncollision singularity" that relies on the absence of a relativistic speed limit.

  That missing speed limit points to the deeper limits. Maxwell's theory predicts that electromagnetic waves travel through empty space at a constant, definite speed, seeming to grant a privileged status to observers who measure that speed. Light provides an absolute standard for speed, clashing with the principle of inertia that says no such standard exists. Special relativity resolves the tension by revising space and time so all inertial observers agree on the speed of light in vacuum. In relativity the mass of a composite object is no longer the sum of its parts, no force can push a body to the speed of light, and the third law must be modified because simultaneity is relative. General relativity goes further, reimagining gravity as curvature of spacetime, captured in John Archibald Wheeler's proverb: "Spacetime tells matter how to move; matter tells spacetime how to curve." At the scale of atoms, quantum mechanics takes over, where the Ehrenfest theorem links quantum expectation values to the second law in a connection that is necessarily inexact.

• 8. Predecessors And Heirs

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  Aristotle divided motion into "natural" and "violent," the natural motion of solid matter being to fall downward while a violent motion pushed a body sideways. A violent motion required an immediate cause, yet a javelin keeps flying after it leaves the hand. Aristotle concluded the air around the javelin must be given the power to push it forward. John Philoponus, a Byzantine Greek thinker of the sixth century, found this absurd, saying that if it were true, armies would launch weapons by blowing on them with bellows. He argued instead that setting a body in motion imparts a quality, impetus, carried within the body itself, a forerunner of momentum later developed by Avicenna, John Buridan, Albert of Saxony, and others.

  Rene Descartes introduced inertia through his "laws of nature" in The World, written between 1629 and 1633, but withheld it after Galileo's 1633 conflict with the Roman Catholic Inquisition, and it was published only in 1664. The modern concept of inertia is credited to Galileo, who wrote in Two New Sciences in 1638 that a particle projected along a frictionless horizontal plane "will move along this same plane with a motion which is uniform and perpetual." Galileo thought such a body would follow the curve of the Earth, an idea corrected toward straight-line motion by Isaac Beeckman, Descartes, and Pierre Gassendi. Descartes published the corrected laws in Principles of Philosophy in 1644, and Christiaan Huygens worked out his own concise version in 1656, printed only in 1703.

  Newton built his three laws incrementally. In a 1684 manuscript to Huygens he listed four laws, later adding a law of action and reaction, and he probably settled on the Principia's three primary laws during 1685. Though he had already developed calculus, which he called "the science of fluxions," he made no explicit use of it in the Principia, preferring geometric arguments in the tradition of Euclid. He cited Huygens, Christopher Wren, and John Wallis to support his third law, and on gravity he declared, "I feign no hypotheses." The work continued after him: Leonhard Euler pioneered the study of rigid bodies and fluid dynamics, Pierre-Simon Laplace's five-volume Traite de mecanique celeste of 1798 to 1825 rebuilt mechanics through algebra, and Emilie du Chatelet wrote in 1742 of "dead force" and "living force," the terms we now call potential and kinetic energy. The vector mathematics of every modern textbook came last, pioneered by Josiah Willard Gibbs and Oliver Heaviside out of the quaternions of William Rowan Hamilton.