lagrangiano

Lagrangiano, in physics, is a function L that encodes the dynamics of a system, typically written as L(q1, ..., qn, q̇1, ..., q̇n, t) where qi are generalized coordinates and q̇i their time derivatives. In many mechanical problems, L takes the form L = T − V, with T representing kinetic energy and V potential energy, though more general dependences on q, q̇, and t are possible.

The equations of motion follow from the principle of stationary action: S = ∫ L dt, with variations of the trajectory that keep endpoints fixed yielding the Euler–Lagrange equations, one for each generalized coordinate qi: d/dt (∂L/∂q̇i) − ∂L/∂qi = 0. These equations generalize Newton’s laws and remain valid under coordinate transformations, making the lagrangiano especially powerful for systems with constraints or non-Cartesian coordinates.

In field theory, the concept extends to a Lagrangian density ℒ(φ, ∂μφ, x), defined so that the action is S = ∫ ℒ d^4x. The corresponding Euler–Lagrange equations become ∂μ (∂ℒ/∂(∂μφ)) − ∂ℒ/∂φ = 0, yielding the field equations for the dynamical fields φ.

The Lagrangian formalism relates to the Hamiltonian formalism via a Legendre transform, with canonical momenta pi = ∂L/∂q̇i and H = Σ pi q̇i − L. Noether’s theorem connects symmetries of the lagrangiano to conserved quantities, underscoring its foundational role across physics, from classical mechanics to quantum theory and relativity. The lagrangiano concept is named after Joseph-Louis Lagrange.