The Beltrami identity, named after Eugenio Beltrami, is a special case of the Euler–Lagrange equation in the calculus of variations.
The Euler–Lagrange equation serves to extremize action functionals of the form
where and are constants and .[1]
If , then the Euler–Lagrange equation reduces to the Beltrami identity,
where C is a constant.[2][note 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.
^Weisstein, Eric W. "Euler-Lagrange Differential Equation." From MathWorld--A Wolfram Web Resource. See Eq. (5).
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The Beltramiidentity, named after Eugenio Beltrami, is a special case of the Euler–Lagrange equation in the calculus of variations. The Euler–Lagrange...
In that case, the Euler–Lagrange equation can be simplified to the Beltramiidentity L − f ′ ∂ L ∂ f ′ = C , {\displaystyle L-f'{\frac {\partial L}{\partial...
Eugenio Beltrami (16 November 1835 – 18 February 1900) was an Italian mathematician notable for his work concerning differential geometry and mathematical...
to obtain the expression of the path shape. (Simmons, Section 54). Beltramiidentity Brachistochrone curve Calculus of variations Catenary Cycloid Uniformly...
Lagrangian mechanics, and a ready-made solution exists in the form of the Beltramiidentity, which immediately supplies the first integral of this second-order...
{\displaystyle x} does not appear in the Lagrangian, we can use the Beltramiidentity L − y ′ ∂ L ∂ y ′ = C {\displaystyle {\mathcal {L}}-y'{\frac {\partial...
calculus of variations and the minimizing condition is given by the Beltramiidentity, L − φ ′ ∂ L ∂ φ ′ = const. {\displaystyle L-\varphi '{\frac {\partial...
In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation ∂ w ∂ z ¯ = μ ∂ w ∂ z . {\displaystyle {\partial...
Beltrami vector field, named after Eugenio Beltrami, is a vector field in three dimensions that is parallel to its own curl. That is, F is a Beltrami...
Laplace–de Rham operator, which is related to the Laplace–Beltrami operator by the Weitzenböck identity. The Laplacian can be generalized in certain ways to...
denote the derivative (pushforward) of u, ∇ the gradient, Δ the Laplace–Beltrami operator, RiemN the Riemann curvature tensor on N and RicM the Ricci curvature...
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tensors of rank 0), the connection Laplacian is often called the Laplace–Beltrami operator. It is defined as the trace of the second covariant derivative:...
models of hyperbolic space by Beltrami, not by Poincaré or Klein. All these models are extendable to more dimensions. The Beltrami–Klein model, also known as...
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which cover multiple neighborhoods. In 2023, the organizations serving Beltrami and Northeast Park merged, taking the number down from 70. The division...
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![{\displaystyle I[u]=\int _{a}^{b}L[x,u(x),u'(x)]\,dx\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90e683a73b0dc65988f0967cf70d6f3a9d6be522)
