Lagrangians

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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