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.

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[edit]

The 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[edit]

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

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]

  1. ^ Courant R, Hilbert D (1953). Methods of Mathematical Physics. Vol. I (First English ed.). New York: Interscience Publishers, Inc. p. 184. ISBN 978-0471504474.
  2. ^ Weisstein, Eric W. "Euler-Lagrange Differential Equation." From MathWorld--A Wolfram Web Resource. See Eq. (5).
  3. ^ Thus, the Legendre transform of the Lagrangian, the Hamiltonian, is constant on the dynamical path.
  4. ^ This derivation of the Beltrami identity corresponds to the one at — Weisstein, Eric W. "Beltrami Identity." From MathWorld--A Wolfram Web Resource.
  5. ^ 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.