Beltrami identity
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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
If
where C is a constant.[2][note 1]
Derivation
By the chain rule, the derivative of L is
Because
We have an expression for
that we can substitute in the above expression for
By the product rule, the right side is equivalent to
By integrating both sides and putting both terms on one side, we get the Beltrami identity,
Applications
Solution to the brachistochrone problem
An example of an application of the Beltrami identity is the brachistochrone problem, which involves finding the curve
The integrand
does not depend explicitly on the variable of integration
Substituting for
which can be solved with the result put in the form of parametric equations
with
Solution to the catenary problem
Consider a string with uniform density
The curve has to minimize its potential energy
Because the independent variable
It is possible to simplify the differential equation as such:
Solving this equation gives the hyperbolic cosine, where
The three unknowns
Notes
- ↑ Thus, the Legendre transform of the Lagrangian, the Hamiltonian, is constant along the dynamical path.
References
- ↑ Methods of Mathematical Physics. I (First English ed.). New York: Interscience Publishers, Inc.. 1953. p. 184. ISBN 978-0471504474.
- ↑ Weisstein, Eric W. "Euler-Lagrange Differential Equation." From MathWorld--A Wolfram Web Resource. See Eq. (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.
![]() | Original source: https://en.wikipedia.org/wiki/Beltrami identity.
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