# What is Lagrange equation: Definition and 43 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.

View More On Wikipedia.org
1. ### Mass on a wedge, both can move

Consider the above system, where both the wedge and the mass can move without friction. We want to get the equations of motion for the both of them using Lagrangian formalism, where the constraints in the solution sheet are given as: $$y_2=0$$ $$\tan \alpha=\frac {y_1}{x_1-x_2}$$ However...
2. ### Why is Hamilton's Principle assumed to be valid for non-holonomic systems?

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7. ### I Lagrangian and the Euler Lagrange equation

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8. ### Euler Lagrange equation and a varying Lagrangian

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14. ### I Delta x in the derivation of Lagrange equation

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15. ### Alternative Lagrange equation

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16. ### Euler Lagrange equation issue with answers final form

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17. ### I Deriving Lagrange's Equation: Help Understanding Chain Rule

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18. ### Line element and derivation of lagrange equation

With coordinates q en basis e ,textbooks use as line element : ds=∑ ei*dqi But ei is a function of place, as one can see in deriving formulas for covariant derivative. Why don't they use as line element the correct: ds=∑ (ei*dqi+dei*qi) Same question in deriving covariant derivative,
19. ### Lagrange equation of motion for tensegrity

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21. ### Euler Lagrange equation of motion

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22. ### Lagrange equation particle on an inverted cone

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24. ### Lagrange equation: when exactly does it apply?

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37. ### Euler lagrange equation and Einstein lagrangian

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38. ### Lagrange equation for mass-spring-damper-pendulum

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39. ### Lagrange equation problem involving disk

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40. ### Euler lagrange equation, mechanics,

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41. ### Euler Lagrange Equation

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42. ### Lagrange equation of motion

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