Question

Prove a transformation is a variational symmetry?

Answer 1

we require;

J(Y)=∫Y(b)X(a)XY˙2dX=J(y)=∫baxy′2dx.J(Y)=∫X(a)Y(b)XY˙2dX=J(y)=∫abxy′2dx.

where Y˙=dYdXY˙=dYdX

ie. The transformation does nothing to the value of the integral.

It has been proven that both XX and YY have inverse functions. Hence,

Y(X)Y˙(X)=(1+ϵ)y(x(X))=(1+ϵ)dydxdxdXY(X)=(1+ϵ)y(x(X))Y˙(X)=(1+ϵ)dydxdxdX

also,

dxdX=11+2ϵ(lnx+1)dxdX=11+2ϵ(ln⁡x+1)

but when I plug all this in I do not get the required equivalence.

Answer 2

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