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Beltrami identity

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The Beltrami identity, named after Eugenio Beltrami, is a simplified and less general version of the Euler–Lagrange equation in the calculus of variations.

Contents

The Euler–Lagrange equation serves to extremize action functionals of the form

I [ u ] = ∫ a b L [ x , u ( x ) , u ′ ( x ) ] d x ,

where a, b are constants and u′(x) = du / dx.

For the special case of ∂L / ∂x = 0, the Euler–Lagrange equation reduces to the Beltrami identity,

where C is a constant.

Derivation

The following derivation of the Beltrami identity starts with the Euler–Lagrange equation,

∂ L ∂ u = d d x ∂ L ∂ u ′ .

Multiplying both sides by u′,

u ′ ∂ L ∂ u = u ′ d d x ∂ L ∂ u ′ .

According to the chain rule,

d L d x = ∂ L ∂ u u ′ + ∂ L ∂ u ′ u ″ + ∂ L ∂ x ,

where u′′ = du′/dx = d2u / dx2.

Rearranging this yields

u ′ ∂ L ∂ u = d L d x − ∂ L ∂ u ′ u ″ − ∂ L ∂ x .

Thus, substituting this expression for u′ ∂L/∂u into the second equation of this derivation,

d L d x − ∂ L ∂ u ′ u ″ − ∂ L ∂ x − u ′ d d x ∂ L ∂ u ′ = 0 .

By the product rule, the last term is re-expressed as

u ′ d d x ∂ L ∂ u ′ = d d x ( ∂ L ∂ u ′ u ′ ) − ∂ L ∂ u ′ u ″ ,

and rearranging,

d d x ( L − u ′ ∂ L ∂ u ′ ) = ∂ L ∂ x .

For the case of ∂L / ∂x = 0, this reduces to

d d x ( L − u ′ ∂ L ∂ u ′ ) = 0 ,

so that taking the antiderivative results in the Beltrami identity,

L − u ′ ∂ L ∂ u ′ = C ,

where C is a constant.

Application

An example of an application of the Beltrami identity is the Brachistochrone problem, which involves finding the curve y = y(x) that minimizes the integral

I [ y ] = ∫ 0 a 1 + y ′ 2 y d x .

The integrand

L ( y , y ′ ) = 1 + y ′ 2 y

does not depend explicitly on the variable of integration x, so the Beltrami identity applies,

L − y ′ ∂ L ∂ y ′ = C .

Substituting for L and simplifying,

y ( 1 + y ′ 2 ) = 1 / C 2     (constant) ,

which can be solved with the result put in the form of parametric equations

x = A ( ϕ − sin ⁡ ϕ ) y = A ( 1 − cos ⁡ ϕ )

with A being half the above constant, 1/(2C ²), and φ being a variable. These are the parametric equations for a cycloid.

References

Beltrami identity Wikipedia
(Text) CC BY-SA


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