Beltrami identity

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Short description: Special case of the Euler-Lagrange equations

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

I[u]=abL[x,u(x),u(x)]dx,

where a and b are constants and u(x)=dudx.[1]

If Lx=0, then the Euler–Lagrange equation reduces to the Beltrami identity,

LuLu=C,

where C is a constant.[2][note 1]

Derivation

By the chain rule, the derivative of L is

dLdx=Lxdxdx+Lududx+Lududx.

Because Lx=0, we write

dLdx=Luu+Luu.

We have an expression for Lu from the Euler–Lagrange equation,

Lu=ddxLu

that we can substitute in the above expression for dLdx to obtain

dLdx=uddxLu+uLu.

By the product rule, the right side is equivalent to

dLdx=ddx(uLu).

By integrating both sides and putting both terms on one side, we get the Beltrami identity,

LuLu=C.

Applications

Solution to the brachistochrone problem

The solution to the brachistochrone problem is the cycloid.

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

I[y]=0a1+y2ydx.

The integrand

L(y,y)=1+y2y

does not depend explicitly on the variable of integration x, so the Beltrami identity applies,

LyLy=C.

Substituting for L and simplifying,

y(1+y2)=1/C2  (constant),

which can be solved with the result put in the form of parametric equations

x=A(ϕsinϕ)
y=A(1cosϕ)

with A being half the above constant, 12C2, and ϕ being a variable. These are the parametric equations for a cycloid.[3]

Solution to the catenary problem

A chain hanging from points forms a catenary.

Consider a string with uniform density μ of length l suspended from two points of equal height and at distance D. By the formula for arc length, l=SdS=s1s21+y2dx, where S is the path of the string, and s1 and s2 are the boundary conditions.

The curve has to minimize its potential energy U=SgμydS=s1s2gμy1+y2dx, and is subject to the constraint s1s21+y2dx=l, where g is the force of gravity.

Because the independent variable x does not appear in the integrand, the Beltrami identity may be used to express the path of the string as a separable first order differential equation

LLy=μgy1+y2+λ1+y2[μgyy21+y2+λy21+y2]=C, where λ is the Lagrange multiplier.

It is possible to simplify the differential equation as such: gρyλ1+y2=C.

Solving this equation gives the hyperbolic cosine, where C0 is a second constant obtained from integration

y=Cμgcosh[μgC(x+C0)]λμg.

The three unknowns C, C0, and λ can be solved for using the constraints for the string's endpoints and arc length l, though a closed-form solution is often very difficult to obtain.

Notes

  1. Thus, the Legendre transform of the Lagrangian, the Hamiltonian, is constant along the dynamical path.

References

  1. Methods of Mathematical Physics. I (First English ed.). New York: Interscience Publishers, Inc.. 1953. p. 184. ISBN 978-0471504474. 
  2. Weisstein, Eric W. "Euler-Lagrange Differential Equation." From MathWorld--A Wolfram Web Resource. See Eq. (5).
  3. 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.