Classical mechanics
Classical mechanics begins with a simple promise. If you know exactly where an object is right now, and how it is moving, you can calculate where it will be tomorrow, and where it was yesterday. This is the physics of large things. Projectiles in flight, the parts of machinery, spacecraft, planets, stars, and whole galaxies. It works on deformable solids, on fluids, on macromolecules. It describes the world we can see and touch, without reaching for quantum effects, and often without relativity either.
The qualifier in the name carries weight. The word classical sets this theory apart from methods developed after the revolutions in physics of the early 20th century. Those revolutions exposed limits. Where do those limits fall, and what waits beyond them? And how did a description of falling cannonballs and orbiting planets become a foundation that all of modern physics still rests upon?
Statics studies what does not move. It analyzes the force and torque acting on a physical system that feels no acceleration, sitting instead in equilibrium with its environment. A bridge that holds, a structure that stands, belongs to this branch.
Kinematics describes motion while ignoring its causes. It tracks the movement of points, of bodies, and of systems of bodies, without asking which forces drive them. The discipline is often called the geometry of motion, and it is sometimes treated as a branch of mathematics rather than physics.
Dynamics asks the question kinematics refuses. It goes beyond describing how objects behave and considers the forces that explain that behavior. These three branches formed the traditional division of the subject, though some authors, such as Greenwood in 1997, fold special relativity into classical dynamics as well.
Newtonian mechanics treats force as a vector quantity. A force in physics is any action that causes an object's velocity to change, that is, to accelerate. It originates within a field. An electro-static field from static charges, an electro-magnetic field from moving charges, a gravitational field from mass. Newton was the first to express the relationship between force and momentum mathematically.
Analytical mechanics takes a different path entirely. Instead of vectors, it uses scalar properties of the whole system, usually its kinetic energy and potential energy. The equations of motion are then derived from how that scalar quantity varies. The physical content matches the Newtonian picture exactly, but the insights and the calculations differ.
Lagrangian mechanics uses generalized coordinates and generalized velocities in what is called the tangent bundle of the configuration space. Hamiltonian mechanics swaps velocities for momenta, working in phase space, the cotangent bundle. A Legendre transformation links the two, so both carry the same information about a system's dynamics. Other formulations exist too, including Hamilton-Jacobi theory, Routhian mechanics, and Appell's equation of motion. All of them, in any formalism, trace back to a single result called the principle of least action.
A point particle is the working fiction at the heart of the theory. Classical mechanics models real objects as particles with negligible size, then handles extended objects by treating them as aggregates of rigidly connected particles. The center of mass of a composite object behaves, conveniently, like a single point particle. A baseball can spin while it moves, which a true point cannot, but the math of points still describes it.
Inertial frames give the laws their simplest form. An inertial frame is an idealized reference frame in which an object with zero net force moves at constant velocity, either at rest or gliding in a straight line. Step into a rotating or accelerating frame and strange forces appear, forces explained by no field. These are the fictitious forces, also called inertia forces or pseudo-forces, among them the centrifugal force and the Coriolis force.
Velocities simply add and subtract. If one car travels east at 60 km/h and passes another going the same way at 50 km/h, the slower car sees the faster one moving east at 10 km/h. From the faster car, the slower one drifts backward at 10 km/h to the west. The rule connecting two frames moving at a steady relative velocity is the Galilean transformation. Under it acceleration stays the same in every inertial frame, and so does force. Time, in this picture, is absolute, ticking identically for all observers, and space follows Euclidean geometry.
Newton's second law reads force as the rate of change of momentum. The quantity mass times velocity is the canonical momentum, and the net force on a particle equals how fast that momentum changes. Once the forces are known, this single law suffices to describe a particle's motion, yielding an equation of motion. Take friction modeled as proportional to velocity. The motion that results decays exponentially toward zero, the kinetic energy absorbed and converted into heat in keeping with the conservation of energy.
Newton's third law pairs every force with an equal and opposite reaction. If particle A pushes on particle B with a force, then B pushes back on A with exactly the negative of that force. The strong form demands these act along the line connecting A and B, defining central forces. But objects at rest are only at rest relative to one another, so central forces are an approximation. For non-central forces like the Lorentz force, the weak form of the law steps in, holding the line through conservation of momentum.
Conservative forces are the ones that forgive your path. If the work to move a particle from one point to another is the same no matter the route taken, the force is conservative. Gravity qualifies, and so does an idealized spring obeying Hooke's law. Friction does not. The work-energy theorem ties the total work done on a particle to its change in kinetic energy. When every force is conservative, the decrease in potential energy equals the increase in kinetic energy, and the total energy stays constant in time.
Aristotle may have been among the first to insist that everything happens for a reason, and that theory could help in understanding nature. Yet the preserved ideas of antiquity lacked mathematical theory and controlled experiment, the very factors that would later define modern science. In his Elementa super demonstrationem ponderum, the medieval mathematician Jordanus de Nemore introduced positional gravity and the use of component forces.
Johannes Kepler's Astronomia nova, published in 1609, gave the first modern description of planetary motion. Working from Tycho Brahe's observations of the orbit of Mars, Kepler concluded that orbits are ellipses, breaking with ancient thought. Around the same time, Galileo proposed abstract mathematical laws for moving objects. The famous test of dropping two cannonballs from the tower of Pisa is disputed, but he did roll balls down an inclined plane, and from those experiments built his theory of accelerated motion. His ideas grew out of earlier medieval work, especially that of Avicenna, Ibn Bajjah, and Jean Buridan. In 1673, Christiaan Huygens set down the first two laws of motion in his Horologium Oscillatorium.
Newton built classical mechanics on three laws: inertia, acceleration, and action and reaction. His Philosophiae Naturalis Principia Mathematica gave the second and third laws proper mathematical form, and stated the conservation of momentum and angular momentum. In his law of universal gravitation he produced the first correct formulation of gravity, showing the same laws govern a falling apple and a distant planet, and explaining Kepler's laws in the process. He had invented the calculus, yet wrote the Principia in long-established geometric methods, in emulation of Euclid.
Joseph-Louis Lagrange reshaped the subject in 1788. The Italian-French mathematician and astronomer first presented his ideas to the Turin Academy of Science in 1760, work that culminated in his grand opus, Mecanique analytique. His formulation rests on the stationary-action principle, also known as the principle of least action, and describes a mechanical system through a configuration space and a function called the Lagrangian.
William Rowan Hamilton refactored Lagrange's work in 1833. Hamiltonian mechanics trades generalized velocities for generalized momenta, and the Hamiltonian itself is the Legendre transform of the Lagrangian, often equal to the total energy. The formulation has a close bond with geometry, with symplectic geometry and Poisson structures, and it became a bridge between classical and quantum mechanics.
The cracks appeared late in the 19th century. One problem was compatibility with electromagnetic theory and the famous Michelson-Morley experiment, which led to special relativity. Another came from thermodynamics, where classical mechanics produced the Gibbs paradox of statistical mechanics, leaving entropy ill-defined. Black-body radiation resisted explanation without quanta. As experiments reached the atomic level, the theory could not account even approximately for the energy levels and sizes of atoms or for the photo-electric effect.
An atom's diameter marks one boundary. For objects about that small, quantum mechanics becomes necessary, and the electron in particular is described far more accurately there than by any point particle. The classical picture breaks down when the de Broglie wavelength is no longer much smaller than the system's other dimensions. The electrons used by Clinton Davisson and Lester Germer in 1927, accelerated by 54 V, carried a wavelength of 0.167 nm, long enough to show a single diffraction side lobe off a nickel crystal whose atomic spacing was 0.215 nm.
The speed of light marks another. As velocities approach it, special relativity takes over. In special relativity, momentum depends on the rest mass and on how the speed compares to the speed of light. When the speed is very small against light, the relativistic equation collapses back into the Newtonian one. The rest mass of an electron is 511 keV, so the frequency correction for a magnetic vacuum tube with a 5.11 kV accelerating voltage comes to just 1%.
Mass marks the last frontier. When objects grow extremely massive, so that the Schwarzschild radius is no longer negligible, deviations from Newtonian mechanics appear, measured through the parameterized post-Newtonian formalism, and general relativity becomes applicable. Since the end of the 20th century, classical mechanics has not stood as an independent theory. It is now treated as an approximation to the more general quantum mechanics, useful for the motion of non-quantum, low-energy particles in weak gravitational fields. No theory of quantum gravity yet unites general relativity with quantum field theory, leaving the case of objects both extremely small and extremely heavy unresolved.
Common questions
What is classical mechanics in physics?
Classical mechanics is a theory that describes the effect of forces on the motion of macroscopic objects and bulk matter, without considering quantum effects and often without relativistic effects. It is used to describe objects such as projectiles, machinery, spacecraft, planets, stars, galaxies, deformable solids, fluids, and macromolecules.
What are the three main branches of classical mechanics?
Classical mechanics was traditionally divided into statics, kinematics, and dynamics. Statics analyzes force and torque on a system in equilibrium, kinematics describes the motion of bodies without considering the forces that cause it, and dynamics considers the forces that explain that motion.
Who developed classical mechanics?
Isaac Newton laid the foundations of classical mechanics with three laws of motion and his law of universal gravitation, set out in his Philosophiae Naturalis Principia Mathematica. Energy-based methods were later developed by Euler, Joseph-Louis Lagrange, and William Rowan Hamilton, and earlier contributors included Johannes Kepler, Galileo, and Christiaan Huygens.
What is the difference between Newtonian, Lagrangian, and Hamiltonian mechanics?
Newtonian mechanics emphasizes force as a vector quantity, while Lagrangian and Hamiltonian mechanics use scalar properties such as kinetic and potential energy. Lagrangian mechanics uses generalized coordinates and velocities, Hamiltonian mechanics uses coordinates and momenta, and the two are linked by a Legendre transformation so they contain the same information.
When does classical mechanics stop working?
Classical mechanics provides accurate results for objects that are not extremely massive and move at speeds far below the speed of light. For objects about the size of an atom's diameter it gives way to quantum mechanics, for speeds approaching light it requires special relativity, and for extremely massive objects general relativity becomes applicable.
When were Lagrangian and Hamiltonian mechanics introduced?
Joseph-Louis Lagrange presented his work to the Turin Academy of Science in 1760, culminating in his 1788 grand opus Mecanique analytique. William Rowan Hamilton reformulated Lagrangian mechanics in 1833 by replacing generalized velocities with generalized momenta.