- Source: Beltrami identity
The Beltrami identity, named after Eugenio Beltrami, is a special case of the Euler–Lagrange equation in the calculus of variations.
The Euler–Lagrange equation serves to extremize action functionals of the form
I
[
u
]
=
∫
a
b
L
[
x
,
u
(
x
)
,
u
′
(
x
)
]
d
x
,
{\displaystyle I[u]=\int _{a}^{b}L[x,u(x),u'(x)]\,dx\,,}
where
a
{\displaystyle a}
and
b
{\displaystyle b}
are constants and
u
′
(
x
)
=
d
u
d
x
{\displaystyle u'(x)={\frac {du}{dx}}}
.
If
∂
L
∂
x
=
0
{\displaystyle {\frac {\partial L}{\partial x}}=0}
, then the Euler–Lagrange equation reduces to the Beltrami identity,
where C is a constant.
Derivation
By the chain rule, the derivative of L is
d
L
d
x
=
∂
L
∂
x
d
x
d
x
+
∂
L
∂
u
d
u
d
x
+
∂
L
∂
u
′
d
u
′
d
x
.
{\displaystyle {\frac {dL}{dx}}={\frac {\partial L}{\partial x}}{\frac {dx}{dx}}+{\frac {\partial L}{\partial u}}{\frac {du}{dx}}+{\frac {\partial L}{\partial u'}}{\frac {du'}{dx}}\,.}
Because
∂
L
∂
x
=
0
{\displaystyle {\frac {\partial L}{\partial x}}=0}
, we write
d
L
d
x
=
∂
L
∂
u
u
′
+
∂
L
∂
u
′
u
″
.
{\displaystyle {\frac {dL}{dx}}={\frac {\partial L}{\partial u}}u'+{\frac {\partial L}{\partial u'}}u''\,.}
We have an expression for
∂
L
∂
u
{\displaystyle {\frac {\partial L}{\partial u}}}
from the Euler–Lagrange equation,
∂
L
∂
u
=
d
d
x
∂
L
∂
u
′
{\displaystyle {\frac {\partial L}{\partial u}}={\frac {d}{dx}}{\frac {\partial L}{\partial u'}}\,}
that we can substitute in the above expression for
d
L
d
x
{\displaystyle {\frac {dL}{dx}}}
to obtain
d
L
d
x
=
u
′
d
d
x
∂
L
∂
u
′
+
u
″
∂
L
∂
u
′
.
{\displaystyle {\frac {dL}{dx}}=u'{\frac {d}{dx}}{\frac {\partial L}{\partial u'}}+u''{\frac {\partial L}{\partial u'}}\,.}
By the product rule, the right side is equivalent to
d
L
d
x
=
d
d
x
(
u
′
∂
L
∂
u
′
)
.
{\displaystyle {\frac {dL}{dx}}={\frac {d}{dx}}\left(u'{\frac {\partial L}{\partial u'}}\right)\,.}
By integrating both sides and putting both terms on one side, we get the Beltrami identity,
L
−
u
′
∂
L
∂
u
′
=
C
.
{\displaystyle L-u'{\frac {\partial L}{\partial u'}}=C\,.}
Applications
= Solution to the brachistochrone problem
=An example of an application of the Beltrami identity is the brachistochrone problem, which involves finding the curve
y
=
y
(
x
)
{\displaystyle y=y(x)}
that minimizes the integral
I
[
y
]
=
∫
0
a
1
+
y
′
2
y
d
x
.
{\displaystyle I[y]=\int _{0}^{a}{\sqrt {{1+y'^{\,2}} \over y}}dx\,.}
The integrand
L
(
y
,
y
′
)
=
1
+
y
′
2
y
{\displaystyle L(y,y')={\sqrt {{1+y'^{\,2}} \over y}}}
does not depend explicitly on the variable of integration
x
{\displaystyle x}
, so the Beltrami identity applies,
L
−
y
′
∂
L
∂
y
′
=
C
.
{\displaystyle L-y'{\frac {\partial L}{\partial y'}}=C\,.}
Substituting for
L
{\displaystyle L}
and simplifying,
y
(
1
+
y
′
2
)
=
1
/
C
2
(constant)
,
{\displaystyle y(1+y'^{\,2})=1/C^{2}~~{\text{(constant)}}\,,}
which can be solved with the result put in the form of parametric equations
x
=
A
(
ϕ
−
sin
ϕ
)
{\displaystyle x=A(\phi -\sin \phi )}
y
=
A
(
1
−
cos
ϕ
)
{\displaystyle y=A(1-\cos \phi )}
with
A
{\displaystyle A}
being half the above constant,
1
2
C
2
{\displaystyle {\frac {1}{2C^{2}}}}
, and
ϕ
{\displaystyle \phi }
being a variable. These are the parametric equations for a cycloid.
= Solution to the catenary problem
=Consider a string with uniform density
μ
{\displaystyle \mu }
of length
l
{\displaystyle l}
suspended from two points of equal height and at distance
D
{\displaystyle D}
. By the formula for arc length,
l
=
∫
S
d
S
=
∫
s
1
s
2
1
+
y
′
2
d
x
,
{\displaystyle l=\int _{S}dS=\int _{s_{1}}^{s_{2}}{\sqrt {1+y'^{2}}}dx,}
where
S
{\displaystyle S}
is the path of the string, and
s
1
{\displaystyle s_{1}}
and
s
2
{\displaystyle s_{2}}
are the boundary conditions.
The curve has to minimize its potential energy
U
=
∫
S
g
μ
y
⋅
d
S
=
∫
s
1
s
2
g
μ
y
1
+
y
′
2
d
x
,
{\displaystyle U=\int _{S}g\mu y\cdot dS=\int _{s_{1}}^{s_{2}}g\mu y{\sqrt {1+y'^{2}}}dx,}
and is subject to the constraint
∫
s
1
s
2
1
+
y
′
2
d
x
=
l
,
{\displaystyle \int _{s_{1}}^{s_{2}}{\sqrt {1+y'^{2}}}dx=l,}
where
g
{\displaystyle g}
is the force of gravity.
Because the independent variable
x
{\displaystyle x}
does not appear in the integrand, the Beltrami identity may be used to express the path of the string as a separable first order differential equation
L
−
y
′
∂
L
∂
y
′
=
μ
g
y
1
+
y
′
2
+
λ
1
+
y
′
2
−
[
μ
g
y
y
′
2
1
+
y
′
2
+
λ
y
′
2
1
+
y
′
2
]
=
C
,
{\displaystyle L-y\prime {\frac {\partial L}{\partial y\prime }}=\mu gy{\sqrt {1+y\prime ^{2}}}+\lambda {\sqrt {1+y\prime ^{2}}}-\left[\mu gy{\frac {y\prime ^{2}}{\sqrt {1+y\prime ^{2}}}}+\lambda {\frac {y\prime ^{2}}{\sqrt {1+y\prime ^{2}}}}\right]=C,}
where
λ
{\displaystyle \lambda }
is the Lagrange multiplier.
It is possible to simplify the differential equation as such:
g
ρ
y
−
λ
1
+
y
′
2
=
C
.
{\displaystyle {\frac {g\rho y-\lambda }{\sqrt {1+y'^{2}}}}=C.}
Solving this equation gives the hyperbolic cosine, where
C
0
{\displaystyle C_{0}}
is a second constant obtained from integration
y
=
C
μ
g
cosh
[
μ
g
C
(
x
+
C
0
)
]
−
λ
μ
g
.
{\displaystyle y={\frac {C}{\mu g}}\cosh \left[{\frac {\mu g}{C}}(x+C_{0})\right]-{\frac {\lambda }{\mu g}}.}
The three unknowns
C
{\displaystyle C}
,
C
0
{\displaystyle C_{0}}
, and
λ
{\displaystyle \lambda }
can be solved for using the constraints for the string's endpoints and arc length
l
{\displaystyle l}
, though a closed-form solution is often very difficult to obtain.
Notes
References
Kata Kunci Pencarian:
- Primetime Creative Arts Emmy Awards ke-71
- Gereja di Timur
- Beltrami identity
- Calculus of variations
- Eugenio Beltrami
- Laplace–Beltrami operator
- Luneburg lens
- Euler–Lagrange equation
- Tautochrone curve
- Brachistochrone curve
- Catenary
- List of named differential equations
