The Beltrami identity, named after Eugenio Beltrami, 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]
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,[2]
where C is a constant.[3]
Derivation
editThe following derivation of the Beltrami identity[4] starts with the Euler–Lagrange equation,
Multiplying both sides by u′,
According to the chain rule,
where u′′ = du′/dx = d2u / dx2.
Rearranging this yields
Thus, substituting this expression for u′ ∂L/∂u into the second equation of this derivation,
By the product rule, the last term is re-expressed as
and rearranging,
For the case of ∂L / ∂x = 0, this reduces to
so that taking the antiderivative results in the Beltrami identity,
where C is a constant.
Application
editAn example of an application of the Beltrami identity is the Brachistochrone problem, which involves finding the curve y = y(x) that minimizes the integral
The integrand
does not depend explicitly on the variable of integration x, so the Beltrami identity applies,
Substituting for L and simplifying,
which can be solved with the result put in the form of parametric equations
with A being half the above constant, 1/(2C ²), and φ being a variable. These are the parametric equations for a cycloid.[5]
References
edit- ^ Courant R, Hilbert D (1953). Methods of Mathematical Physics. Vol. Vol. I (First English ed.). New York: Interscience Publishers, Inc. p. 184. ISBN 978-0471504474.
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has extra text (help) - ^ Weisstein, Eric W. "Euler-Lagrange Differential Equation." From MathWorld--A Wolfram Web Resource. See Eq. (5).
- ^ Thus, the Legendre transform of the Lagrangian, the Hamiltonian, is constant on the dynamical path.
- ^ 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: W. A. Benjamin, Inc. pp. 307–9.