Noun
Euler-Lagrange equation (plural Euler-Lagrange equations)
(mechanics, analytical mechanics) A differential equation which describes a function
q
(
t
)
{\displaystyle \mathbf {q} (t)}
which describes a stationary point of a functional,
S
(
q
)
=
∫
L
(
t
,
q
(
t
)
,
q
˙
(
t
)
)
d
t
{\displaystyle S(\mathbf {q} )=\int L(t,\mathbf {q} (t),\mathbf {\dot {q}} (t))\,dt}
, which represents the action of
q
(
t
)
{\displaystyle \mathbf {q} (t)}
, with
L
{\displaystyle L}
representing the Lagrangian. The said equation (found through the calculus of variations) is
∂
L
∂
q
=
d
d
t
∂
L
∂
q
˙
{\displaystyle {\partial L \over \partial \mathbf {q} }={d \over dt}{\partial L \over \partial \mathbf {\dot {q}} }}
and its solution for
q
(
t
)
{\displaystyle \mathbf {q} (t)}
represents the trajectory of a particle or object, and such trajectory should satisfy the principle of least action.