# Lagrangian Definition and 118 Discussions

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

δ
:
[

t

1

,

t

2

]

R

n

,

{\displaystyle \delta :[t_{1},t_{2}]\to \mathbb {R} ^{n},}

δ

S

=

def

d

d
ε

|

ε
=
0

S

[

x

0

+
ε
δ

]

=
0.

{\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.

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