Classical Mechanics
Classical mechanics is the branch of physics that describes the motion of macroscopic bodies under the action of forces. Developed by Galileo Galilei and Sir Isaac Newton in the 17th century and reformulated analytically by Lagrange and Hamilton in the 18th–19th centuries, it remains the foundation for engineering, astrodynamics, and continuum physics whenever speeds are much smaller than the speed of light and quantum effects are negligible.
A reference frame in which an isolated body, free of external force, moves with constant velocity. Newton's laws hold in their simplest form in such frames; all inertial frames move uniformly relative to one another.
Newton's three laws of motion
- First law (law of inertia) — a body at rest stays at rest, and a body in uniform motion stays in uniform motion, unless acted upon by a net external force.
- Second law — the net force on a body equals the time-rate of change of its linear momentum: F = dp/dt = ma (for constant mass).
- Third law — for every action there is an equal and opposite reaction. Forces always occur in pairs acting on different bodies.
These three statements, combined with a force law (such as gravitation or Hooke's law), determine the trajectory of any classical particle.
Work, energy and conservation principles
The work-energy theorem states that the net work done on a body equals its change in kinetic energy:
W_net = ΔKE = ½mv² − ½mv₀²
For a conservative force (gravity, spring force, electrostatic), the work can be written as the negative gradient of a potential, F = −∇U, and the total mechanical energy E = KE + U is conserved.
- Linear momentum p = mv is conserved when external force is zero.
- Angular momentum L = r × p is conserved when external torque is zero.
- Mechanical energy is conserved in the absence of non-conservative forces (friction, drag).
- Centre-of-mass of an isolated system moves with constant velocity.
Rotational dynamics
For a rigid body rotating about a fixed axis, the rotational analogues of Newton's laws apply:
| Linear quantity | Rotational analogue |
|---|---|
| Force F | Torque τ = r × F |
| Mass m | Moment of inertia I = ∫r² dm |
| Momentum p = mv | Angular momentum L = Iω |
| F = ma | τ = Iα |
| KE = ½mv² | KE_rot = ½Iω² |
The parallel-axis theorem relates the moment of inertia about any axis to that about a parallel axis through the centre of mass: I = I_cm + Md².
Gravitation
Newton's law of universal gravitation states that every particle attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them:
F = G m₁m₂ / r², with G = 6.674 × 10⁻¹¹ N·m²/kg²
Combined with the second law, this yields Kepler's three laws of planetary motion and explains orbital mechanics, tides, and the precession of equinoxes. The gravitational potential energy of two point masses is U = −Gm₁m₂/r.
Escape velocity
Setting total energy to zero gives the escape velocity from a spherical body of mass M and radius R:
v_esc = √(2GM/R)
For Earth this is about 11.2 km/s; for the Moon, 2.4 km/s; for the Sun's surface, 617.5 km/s.
Oscillations and simple harmonic motion
A body subject to a restoring force proportional to displacement, F = −kx, executes simple harmonic motion (SHM) with angular frequency ω = √(k/m) and period T = 2π√(m/k). Examples include the simple pendulum (for small angles, T = 2π√(L/g)) and mass-on-spring systems.
In the exam, remember that period of a simple pendulum is independent of its mass and amplitude (small-angle approximation). Galileo discovered this isochronism while watching a swinging chandelier in Pisa cathedral.
Lagrangian and Hamiltonian formulations
For systems with constraints or many degrees of freedom, the Newtonian approach becomes cumbersome. Joseph-Louis Lagrange (1788) reformulated mechanics around the Lagrangian L = T − U (kinetic minus potential energy). The Euler–Lagrange equations:
d/dt (∂L/∂q̇) − ∂L/∂q = 0
yield the equations of motion in any chosen generalised coordinates. William Rowan Hamilton (1833) further transformed the theory using the Hamiltonian H = T + U and canonical momenta. Hamiltonian mechanics later provided the bridge to quantum theory through Poisson brackets and the correspondence principle.
Limits of classical mechanics
Classical mechanics breaks down in three regimes:
- High speeds (v approaching c): replaced by special relativity.
- Strong gravitational fields: replaced by general relativity.
- Atomic and subatomic scales: replaced by quantum mechanics.
Nevertheless, within its domain of validity classical mechanics agrees with experiment to extraordinary precision — spacecraft trajectories, bridge stresses, and satellite orbits are all computed using Newton's laws.