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user15626
user15626
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Singularity at left endpoint for Variationalvariational calculus problem

Hi all, first, I'd like to apologize if the term "singularity" is being misused. I have the following integral:

$\int _{0}^{\pi/2} \sqrt{ r \left( x \right) ^{2} + \left( {\frac {d}{dx}} r \left( x \right) \right) ^{2}}\ln \left( { \frac {\sin \left( x \right) }{1-\cos \left( x \right) }} \right) {dx}$$\int _{0}^{\pi/2} \sqrt{ r \left( x \right) ^{2} + \left( {\frac {d}{dx}} r \left( x \right) \right) ^{2}} \left( \ln \left( { \frac {\sin \left( x \right) }{1-\cos \left( x \right) }} \right) + 1 \right) {dx}$

Note that the integrand shoots to infinity as x approaches 0. It's not too hard to establish that the integral will nonetheless always converge, regardless of the choice of r. My question is: will an extremizer satisfy the Euler-Lagrange Equation on the interval (0,Pi/2], and how can I tell? Sorry if this is trivial, I do not know much about the calculus of variations.

Singularity at left endpoint for Variational calculus problem

Hi all, I have the following integral:

$\int _{0}^{\pi/2} \sqrt{ r \left( x \right) ^{2} + \left( {\frac {d}{dx}} r \left( x \right) \right) ^{2}}\ln \left( { \frac {\sin \left( x \right) }{1-\cos \left( x \right) }} \right) {dx}$

Note that the integrand shoots to infinity as x approaches 0. It's not too hard to establish that the integral will nonetheless always converge, regardless of the choice of r. My question is: will an extremizer satisfy the Euler-Lagrange Equation on the interval (0,Pi/2], and how can I tell? Sorry if this is trivial, I do not know much about the calculus of variations.

Singularity at left endpoint for variational calculus problem

Hi all, first, I'd like to apologize if the term "singularity" is being misused. I have the following integral:

$\int _{0}^{\pi/2} \sqrt{ r \left( x \right) ^{2} + \left( {\frac {d}{dx}} r \left( x \right) \right) ^{2}} \left( \ln \left( { \frac {\sin \left( x \right) }{1-\cos \left( x \right) }} \right) + 1 \right) {dx}$

Note that the integrand shoots to infinity as x approaches 0. It's not too hard to establish that the integral will nonetheless always converge, regardless of the choice of r. My question is: will an extremizer satisfy the Euler-Lagrange Equation on the interval (0,Pi/2], and how can I tell? Sorry if this is trivial, I do not know much about the calculus of variations.

Source Link
user15626
user15626
  • 81
  • 5

Singularity at left endpoint for Variational calculus problem

Hi all, I have the following integral:

$\int _{0}^{\pi/2} \sqrt{ r \left( x \right) ^{2} + \left( {\frac {d}{dx}} r \left( x \right) \right) ^{2}}\ln \left( { \frac {\sin \left( x \right) }{1-\cos \left( x \right) }} \right) {dx}$

Note that the integrand shoots to infinity as x approaches 0. It's not too hard to establish that the integral will nonetheless always converge, regardless of the choice of r. My question is: will an extremizer satisfy the Euler-Lagrange Equation on the interval (0,Pi/2], and how can I tell? Sorry if this is trivial, I do not know much about the calculus of variations.