- Beltrami identity
The Beltrami identity is an identity in the
calculus of variations . It says that a function "u" which is an extremal of the integral:I(u)=int_a^b f(x,u,u') , dx
satisfies the differential equation
:frac{d}{dx}left(f-u'frac{partial f}{partial u'} ight)-frac{partial f}{partial x}=0.
Proof
The
Euler-Lagrange equation tells that:frac{partial f}{partial u}-frac{d}{dx}frac{partial f}{partial u'}=0.
Now consider the total differential of functional f(x,u,u'). Substituting the
Euler-Lagrange equation into it, we have:egin{align}frac{df}{dx} &= frac{partial f}{partial x} + frac{partial f}{partial u} u' + frac{partial f}{partial u'} u" \& = frac{partial f}{partial x} + left(frac{d}{dx}frac{partial f}{partial u'} ight) u' + frac{partial f}{partial u'} u".end{align}
Therefore,
:frac{d}{dx}left(f-u'frac{partial f}{partial u'} ight)-frac{partial f}{partial x}=0.
Application
In case the functional "f" is independent of "x", then the Beltrami identity can be simplified into
:egin{align}frac{d}{dx}left(f-u'frac{partial f}{partial u'} ight) &=0 \f-u'frac{partial f}{partial u'} &= ext{constant} \end{align}
Using the above form is an easier approach to solve for the optimal function "u" than directly applying the
Euler-Lagrange equation .
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