Classical mechanics is the branch of physics that describes the motion of macroscopic objects - from projectiles and machinery to planets and galaxies - under the influence of forces. It is grounded in the work of Isaac Newton, Galileo Galilei, and their contemporaries, and was later reformulated by Joseph-Louis Lagrange, William Rowan Hamilton, and others into more general and powerful mathematical frameworks. Classical mechanics applies reliably at scales and velocities well below those where quantum effects or relativistic corrections become significant.

The field encompasses several equivalent but conceptually distinct formulations. Newtonian mechanics, the oldest and most intuitive, treats force as the central quantity and describes motion through differential equations relating force, mass, and acceleration. Lagrangian mechanics reformulates the problem in terms of generalized coordinates and a scalar quantity called the Lagrangian, making it easier to handle constrained systems. Hamiltonian mechanics further abstracts the description using phase space and canonical variables, and serves as a direct conceptual bridge to quantum mechanics and statistical mechanics.

Classical mechanics is subdivided by the type of motion or system under study: particle mechanics deals with idealized point masses; rigid body dynamics treats extended objects with fixed shape; continuum mechanics addresses deformable bodies and fluids; and celestial mechanics applies classical principles to gravitational systems at astronomical scales.

The foundations were laid in the 17th century, with Galileo's experimental work on motion and Newton's synthesis in the Philosophiæ Naturalis Principia Mathematica (1687). The 18th and 19th centuries saw the development of the Lagrangian and Hamiltonian formulations, variational principles, and the mathematical apparatus of analytical mechanics. The late 19th and early 20th centuries revealed the limits of classical mechanics: it fails to describe motion at velocities approaching the speed of light (addressed by special relativity) and at atomic and subatomic scales (addressed by quantum mechanics). For a fuller treatment, see Classical Mechanics - History.

There is broad consensus in physics and engineering that classical mechanics accurately describes motion within its domain of applicability - macroscopic objects at non-relativistic velocities in weak gravitational fields. This consensus extends to the predictive power of Newtonian, Lagrangian, and Hamiltonian formulations as equivalent descriptions of the same physical reality. See Classical Mechanics - Physics Consensus.

While the empirical adequacy of classical mechanics within its domain is not seriously disputed, interpretive and philosophical questions remain active areas of discussion.

• Determinism - Classical mechanics is formally deterministic: given exact initial conditions, the future state of a system is in principle fully determined. Some interpret this as implying a deterministic universe; others argue that idealized initial conditions are physically unrealizable, or point to chaotic systems as practically indistinguishable from non-deterministic ones. See Determinism in Classical Mechanics - Viewpoint.

• Relationship to quantum mechanics - Whether classical mechanics is a limiting case of quantum mechanics, an independent framework, or a phenomenological approximation is a matter of ongoing foundational debate. See Quantum-Classical Limit - Viewpoint.

• Mathematical Platonism and physical law - The unreasonable effectiveness of mathematics in describing classical mechanics has prompted philosophical debate about whether mathematical structures are discovered or invented, and what this implies about the nature of physical law. See Mathematical Platonism - Viewpoint.

• Classical Mechanics - History
• Classical Mechanics - Physics Consensus
• Classical Mechanics - Debate
• Determinism in Classical Mechanics - Viewpoint
• Quantum-Classical Limit - Viewpoint
• Quantum Mechanics
• Special Relativity
• Thermodynamics
• Philosophy of Physics
• Philosophy of Mathematics

1. Newton, Isaac. Philosophiæ Naturalis Principia Mathematica. London: Royal Society, 1687.
2. Lagrange, Joseph-Louis. Mécanique Analytique. Paris, 1788.
3. Hamilton, William Rowan. “On a General Method in Dynamics.” Philosophical Transactions of the Royal Society, 1834-1835.
4. Goldstein, Herbert, Charles Poole, and John Safko. Classical Mechanics. 3rd ed. San Francisco: Addison-Wesley, 2002.
5. Wigner, Eugene. “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Communications on Pure and Applied Mathematics 13, no. 1 (1960): 1-14.