The Beltrami identity is a mathematical relation used in the calculus of variations to find a function u(x) that minimizes a functional, and is[1]
where
- C is a constant,
- u′ = du/dx,
- L = L[ u(x), u′(x) ].
It is derived from the Euler-Lagrange equation for the case of x not appearing explicitly in L. The Euler-Lagrange equation applies to functionals of the form[2]
![{\displaystyle I[u]=\int _{a}^{b}L[x,u(x),u'(x)]\,dx\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90e683a73b0dc65988f0967cf70d6f3a9d6be522)
where a, b are constants and u′(x) = du / dx. For the case of ∂L / ∂x = 0, the Euler-Lagrange equation reduces to the Beltrami identity,[3]
where C is a constant.
- ^ Weisstein, Eric W. "Euler-Lagrange Differential Equation." From MathWorld--A Wolfram Web Resource. See Eq. (5).
- ^ Methods of Mathematical Physics. Vol. Vol. I (First English ed.). New York, New York: Interscience Publishers, Inc. 1953. p. 184. ISBN 978-0471504474.
{{cite book}}:|access-date=requires|url=(help);|volume=has extra text (help); Unknown parameter|authors=ignored (help) - ^ Weisstein, Eric W. "Euler-Lagrange Differential Equation." From MathWorld--A Wolfram Web Resource. See Eq. (5).
The Beltrami identity is a simplified and less general version of the Euler-Lagrange equation in the calculus of variations. The Euler-Lagrange equation applies to functionals of the form[1]
where a, b are constants and u′(x) = du / dx. For the case of ∂L / ∂x = 0, the Euler-Lagrange equation reduces to the Beltrami identity,[2]
where C is a constant.
- ^ Methods of Mathematical Physics. Vol. Vol. I (First English ed.). New York, New York: Interscience Publishers, Inc. 1953. p. 184. ISBN 978-0471504474.
{{cite book}}:|access-date=requires|url=(help);|volume=has extra text (help); Unknown parameter|authors=ignored (help) - ^ Weisstein, Eric W. "Euler-Lagrange Differential Equation." From MathWorld--A Wolfram Web Resource. See Eq. (5).