Beltrami identity

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The Beltrami identity is a simplified and less general version of the Euler-Lagrange equation in the calculus of variations.

The Euler-Lagrange equation serves to extremize action functionals of the form^[1]

$I[u]=\int _{a}^{b}L[x,u(x),u'(x)]\,dx\,,$

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

For the special case of $\partial L / \partial x = 0$ , the Euler-Lagrange equation reduces to the Beltrami identity,^[2]

$L-u'{\frac {\partial L}{\partial u'}}=C\,,$

where $C$ is a constant. Thus, the Legendre transform of the Lagrangian, the Hamiltonian, is constant on the dynamical path.

Derivation

The following derivation of the Beltrami identity^[3] starts with the Euler-Lagrange equation,

${\frac {\partial L}{\partial u}}={\frac {d}{dx}}{\frac {\partial L}{\partial u'}}\,.$

Multiplying both sides by $u'$ ,

$u'{\frac {\partial L}{\partial u}}=u'{\frac {d}{dx}}{\frac {\partial L}{\partial u'}}\,.$

According to the chain rule,

${dL \over dx}={\partial L \over \partial u}u'+{\partial L \over \partial u'}u''+{\partial L \over \partial x}\,,$

where $u'' = du' / dx = d 2 u / dx 2$ .

Rearranging this yields

$u'{\partial L \over \partial u}={dL \over dx}-{\partial L \over \partial u'}u''-{\partial L \over \partial x}\,.$

Thus, substituting this expression for $u' \partial L /\partial u$ into the second equation of this derivation,

${dL \over dx}-{\partial L \over \partial u'}u''-{\partial L \over \partial x}-u'{\frac {d}{dx}}{\frac {\partial L}{\partial u'}}=0\,.$

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

$u'{\frac {d}{dx}}{\frac {\partial L}{\partial u'}}={\frac {d}{dx}}\left({\frac {\partial L}{\partial u'}}u'\right)-{\frac {\partial L}{\partial u'}}u''\,,$

and rearranging,

${d \over dx}\left({L-u'{\frac {\partial L}{\partial u'}}}\right)={\partial L \over \partial x}\,.$

For the case of $\partial L / \partial x = 0$ , this reduces to

${d \over dx}\left({L-u'{\frac {\partial L}{\partial u'}}}\right)=0\,,$

so that taking the antiderivative results in the Beltrami identity,

$L-u'{\frac {\partial L}{\partial u'}}=C\,,$

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

$I[y]=\int _{0}^{a}{\sqrt {{1+y'^{{\,2}}} \over y}}dx\,.$

The integrand

$L(y,y')={\sqrt {{1+y'^{{\,2}}} \over y}}$

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

$L-y'{\frac {\partial L}{\partial y'}}=C\,.$

Substituting for $L$ and simplifying,

$y(1+y'^{{\,2}})=1/C^{2}~~{\text{(constant)}}\,,$

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

$x=A(\phi -\sin \phi )$

$y=A(1-\cos \phi )$

with $A$ being half the above constant, 1/(2C ²), and $φ$ being a variable. These are the parametric equations for a cycloid.^[4]

References

↑ Courant R, Hilbert D (1953). Methods of Mathematical Physics. Vol. I (First English ed.). New York, New York: Interscience Publishers, Inc. p. 184. ISBN 978-0471504474. |accessdate= requires |url= (help)
↑ Weisstein, Eric W. "Euler-Lagrange Differential Equation." From MathWorld--A Wolfram Web Resource. See Eq. (5).
↑ This derivation of the Beltrami identity corresponds to the one at — Weisstein, Eric W. "Beltrami Identity." From MathWorld--A Wolfram Web Resource.
↑ This solution of the Brachistochrone problem corresponds to the one in — Mathews, Jon; Walker, RL (1965). Mathematical Methods of Physics. New York, New York: W. A. Benjamin, Inc. pp. 307–9.

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