Explore: Lagrangian Equations

Lagrange's equations and defining the Lagrangian as L = T − V obtains Lagrange's equations of the second kind or the Euler–Lagrange equations of motion

the Routhian equations are exactly the Hamiltonian equations for some coordinates and corresponding momenta, and the Lagrangian equations for the rest

equations are closely related to the Yang–Mills–Higgs equations. Another closely related Lagrangian is found in Seiberg–Witten theory. The Lagrangian

Hamiltonian equations and those which enter the Lagrangian equations is arbitrary. It is simply convenient to let the Hamiltonian equations remove the

and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the

_{0}}}F^{ab}F_{ab}-j^{a}A_{a}\,.} To obtain the field equations, the electromagnetic tensor in the Lagrangian density needs to be replaced by its definition

Hamilton–Jacobi equation Hamilton–Jacobi–Einstein equation Lagrangian mechanics Maxwell's equations Hamiltonian (quantum mechanics) Quantum Hamilton's equations Quantum

free equations reduce to Maxwell's equations without charge or current, and the above reduces to Maxwell's charge equation. This Proca field equation is

are considered later). If a system is described by a Lagrangian L, the Euler–Lagrange equations d d t ∂ L ∂ r ˙ = ∂ L ∂ r {\displaystyle {\frac {d}{dt}}{\frac

relativity. Action principles start with an energy function called a Lagrangian describing the physical system. The accumulated value of this energy function