Introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788, Lagrangian mechanics is a formulation of classical mechanics and is founded on the stationary action principle.
Lagrangian mechanics defines a mechanical system to be a pair
(
M
,
L
)
{\displaystyle (M,L)}
of a configuration space
M
{\displaystyle M}
and a smooth function
L
=
L
(
q
,
v
,
t
)
{\displaystyle L=L(q,v,t)}
called Lagrangian. By convention,
L
=
T
−
V
,
{\displaystyle L=T-V,}
where
T
{\displaystyle T}
and
V
{\displaystyle V}
are the kinetic and potential energy of the system, respectively. Here
q
∈
M
,
{\displaystyle q\in M,}
and
v
{\displaystyle v}
is the velocity vector at
q
{\displaystyle q}
(
v
{\displaystyle (v}
is tangential to
M
)
.
{\displaystyle M).}
(For those familiar with tangent bundles,
L
:
T
M
×
R
t
→
R
,
{\displaystyle L:TM\times \mathbb {R} _{t}\to \mathbb {R} ,}
and
v
∈
T
q
M
)
.
{\displaystyle v\in T_{q}M).}
Given the time instants
t
1
{\displaystyle t_{1}}
and
t
2
,
{\displaystyle t_{2},}
Lagrangian mechanics postulates that a smooth path
x
0
:
[
t
1
,
t
2
]
→
M
{\displaystyle x_{0}:[t_{1},t_{2}]\to M}
describes the time evolution of the given system if and only if
x
0
{\displaystyle x_{0}}
is a stationary point of the action functional
S
[
x
]
=
def
∫
t
1
t
2
L
(
x
(
t
)
,
x
˙
(
t
)
,
t
)
d
t
.
{\displaystyle {\cal {S}}[x]\,{\stackrel {\text{def}}{=}}\,\int _{t_{1}}^{t_{2}}L(x(t),{\dot {x}}(t),t)\,dt.}
If
M
{\displaystyle M}
is an open subset of
R
n
{\displaystyle \mathbb {R} ^{n}}
and
t
1
,
{\displaystyle t_{1},}
t
2
{\displaystyle t_{2}}
are finite, then the smooth path
x
0
{\displaystyle x_{0}}
is a stationary point of
S
{\displaystyle {\cal {S}}}
if all its directional derivatives at
x
0
{\displaystyle x_{0}}
vanish, i.e., for every smooth
{\displaystyle \delta {\cal {S}}\ {\stackrel {\text{def}}{=}}\ {\frac {d}{d\varepsilon }}{\Biggl |}_{\varepsilon =0}{\cal {S}}\left[x_{0}+\varepsilon \delta \right]=0.}
The function
δ
(
t
)
{\displaystyle \delta (t)}
on the right-hand side is called perturbation or virtual displacement. The directional derivative
δ
S
{\displaystyle \delta {\cal {S}}}
on the left is known as variation in physics and Gateaux derivative in Mathematics.
Lagrangian mechanics has been extended to allow for non-conservative forces.