Lagrangian mechanics (Joseph-Louis Lagrange, 1788) and Hamiltonian mechanics (William Rowan Hamilton, 1833) are two mathematically equivalent but conceptually different reformulations of classical (Newtonian) mechanics. Instead of Newton's force-based approach, they describe motion using energy — the Lagrangian uses the difference between kinetic and potential energy, the Hamiltonian uses their sum — and derive equations of motion from elegant variational and geometric principles. Both formulations proved essential to the later development of quantum mechanics, statistical mechanics, and general relativity.
Newton's approach to mechanics centres on forces: F=ma, and to find how a system moves, identify all the forces acting on each part of it and solve the resulting equations. For simple systems — one ball, one spring — this works well. For complex systems — a double pendulum, a molecule, a planet in a non-spherical gravitational field — identifying and computing all the forces becomes unwieldy. Lagrangian and Hamiltonian mechanics offer a completely different starting point: energy.
The real power of the Lagrangian and Hamiltonian approaches emerged long after classical mechanics — they turned out to be exactly the right mathematical language for quantum mechanics (where the Hamiltonian becomes an operator governing the evolution of quantum states) and for general relativity (where action principles describe spacetime geometry itself). A formalism invented to simplify planetary motion calculations in the 1700s became the foundation of modern theoretical physics.
A key advantage of the Lagrangian approach is its flexibility in choice of coordinates. Instead of being tied to x, y, z positions, you can use any convenient set of independent quantities — angles of a pendulum, relative distances in a molecule — called generalised coordinates q₁, q₂, ..., qₙ. The Lagrangian formalism works equally well in any such coordinate system, automatically handling constraints (like a bead constrained to a wire) without Lagrange multipliers.
The Lagrangian L is defined as L = T − V, where T is kinetic energy and V is potential energy, both expressed in terms of the generalised coordinates and their time derivatives. The equations of motion follow from the Euler-Lagrange equations:
One such equation for each generalised coordinate — exactly as many equations as degrees of freedom, automatically, with no forces to compute.
The Hamiltonian H = T + V is the total energy of the system, expressed in terms of generalised coordinates q and their conjugate momenta p (defined as p = ∂L/∂q̇). Hamilton's equations are:
These are twice as many equations (one pair per coordinate) but each is first-order rather than second-order — a form particularly suited to geometric analysis and to the passage to quantum mechanics.
The Lagrangian formalism makes Noether's theorem (see that entry) particularly transparent: every continuous symmetry of the Lagrangian corresponds to a conserved quantity. Time-translation symmetry gives conservation of energy; spatial-translation symmetry gives conservation of momentum; rotational symmetry gives conservation of angular momentum. In the Lagrangian framework, these deep connections between symmetry and conservation become simple, automatic consequences of the mathematics.
The Hamiltonian formalism naturally lives in phase space — a 2n-dimensional space whose coordinates are the n generalised positions q and n conjugate momenta p. The geometry of phase space is not Euclidean but symplectic: it is equipped with a special 2-form ω = Σ dpᵢ ∧ dqᵢ that is preserved by Hamiltonian flow (Liouville's theorem). Symplectic geometry — the study of manifolds equipped with such a structure — has grown into a rich and active field of pure mathematics, with connections to low-dimensional topology, string theory, and mirror symmetry.
The Poisson bracket {f, g} = Σ (∂f/∂qᵢ ∂g/∂pᵢ − ∂f/∂pᵢ ∂g/∂qᵢ) provides an algebraic structure on functions on phase space. Hamilton's equations can be written as dA/dt = {A, H} for any observable A. A quantity is conserved if and only if its Poisson bracket with H vanishes. The Poisson bracket algebra of classical observables is the direct classical antecedent of the commutator algebra in quantum mechanics.
The passage from classical Hamiltonian mechanics to quantum mechanics via canonical quantisation replaces the classical Poisson bracket {q, p} = 1 with the quantum commutator [q̂, p̂] = iℏ (Dirac's quantisation condition). The classical Hamiltonian H(q, p) becomes a quantum operator Ĥ acting on a Hilbert space of state vectors, and Hamilton's equations become the Schrödinger equation iℏ ∂ψ/∂t = Ĥψ. This deep structural connection — classical Poisson brackets becoming quantum commutators — makes the Hamiltonian formulation the natural classical limit of quantum mechanics and the natural starting point for quantum field theory.
The Lagrangian action principle generalises from mechanics (finite degrees of freedom) to field theory (infinite degrees of freedom) by replacing the Lagrangian with a Lagrangian density ℒ(φ, ∂μφ) integrated over spacetime. The resulting Euler-Lagrange equations are the field equations — Maxwell's equations for electromagnetism, the Klein-Gordon equation for a scalar field, the Einstein field equations for gravity (derived from the Einstein-Hilbert action). Every fundamental theory of modern physics is formulated as an action principle with a specific Lagrangian density, making the Lagrangian framework the universal language of theoretical physics.