Extremum of functional via functional derivative
Multi tool use
$begingroup$
I have the following functional:
$$
H[lambda(T)]
= intfrac{1}{2}left | nabla m_{0}hleft(frac{x}{lambda(T)}right) right |^{2}+frac{1}{2}r_{0}m^{2}_{0}h^{2}left(frac{x}{lambda(T)}right)+frac{1}{4!}u_{0}m^{4}_{0}h^{4}left(frac{x}{lambda(T)}right)-B(x)m_{0}hleft(frac{x}{lambda(T)}right)dx,$$
with $r_{0},m_{0},u_{0}$ being real constants and x a real variable. B(x) is some real valued function. And I should find the $lambda(T)$ that minimizes that functional. To start I would compute the functional derivative with respect to $lambda(T)$ and set this to zero:
$$
int m_{0}nabla^{2} hleft(frac{x}{lambda(T)}right)frac{x}{lambda^{2}(T)}+r_{0}m_{0}hleft(frac{x}{lambda(T)}right)h^{'}left(frac{x}{lambda(T)}right)frac{x}{lambda^{2}(T)}-frac{1}{6}u_{0}m^{3}_{0}h^{3}left(frac{x}{lambda(T)}right)h^{'}left(frac{x}{lambda(T)}right)frac{x}{lambda^{2}(T)}+B(x)h^{'}left(frac{x}{lambda(T)}right)frac{x}{lambda^{2}(T)}=0.
$$
In this equation the $h^{'}left(frac{x}{lambda(T)}right)$ denotes the first derivative of $hleft(frac{x}{lambda(T)}right)$ with respect to its argument. Now I am not completely sure if I did the functional derivatives correctly.And otherwise I would not know how to solve this equation for $lambda (T)$ since all the explicit $lambda$ would cancel.
functional-analysis derivatives optimization
asked Nov 26 '18 at 12:47
zodiaczodiac
617
$endgroup$
add a comment |
$begingroup$
I have the following functional:
$$
H[lambda(T)]
= intfrac{1}{2}left | nabla m_{0}hleft(frac{x}{lambda(T)}right) right |^{2}+frac{1}{2}r_{0}m^{2}_{0}h^{2}left(frac{x}{lambda(T)}right)+frac{1}{4!}u_{0}m^{4}_{0}h^{4}left(frac{x}{lambda(T)}right)-B(x)m_{0}hleft(frac{x}{lambda(T)}right)dx,$$
with $r_{0},m_{0},u_{0}$ being real constants and x a real variable. B(x) is some real valued function. And I should find the $lambda(T)$ that minimizes that functional. To start I would compute the functional derivative with respect to $lambda(T)$ and set this to zero:
$$
int m_{0}nabla^{2} hleft(frac{x}{lambda(T)}right)frac{x}{lambda^{2}(T)}+r_{0}m_{0}hleft(frac{x}{lambda(T)}right)h^{'}left(frac{x}{lambda(T)}right)frac{x}{lambda^{2}(T)}-frac{1}{6}u_{0}m^{3}_{0}h^{3}left(frac{x}{lambda(T)}right)h^{'}left(frac{x}{lambda(T)}right)frac{x}{lambda^{2}(T)}+B(x)h^{'}left(frac{x}{lambda(T)}right)frac{x}{lambda^{2}(T)}=0.
$$
In this equation the $h^{'}left(frac{x}{lambda(T)}right)$ denotes the first derivative of $hleft(frac{x}{lambda(T)}right)$ with respect to its argument. Now I am not completely sure if I did the functional derivatives correctly.And otherwise I would not know how to solve this equation for $lambda (T)$ since all the explicit $lambda$ would cancel.
functional-analysis derivatives optimization
asked Nov 26 '18 at 12:47
zodiaczodiac
617
$endgroup$
add a comment |
$begingroup$
I have the following functional:
$$
H[lambda(T)]
= intfrac{1}{2}left | nabla m_{0}hleft(frac{x}{lambda(T)}right) right |^{2}+frac{1}{2}r_{0}m^{2}_{0}h^{2}left(frac{x}{lambda(T)}right)+frac{1}{4!}u_{0}m^{4}_{0}h^{4}left(frac{x}{lambda(T)}right)-B(x)m_{0}hleft(frac{x}{lambda(T)}right)dx,$$
with $r_{0},m_{0},u_{0}$ being real constants and x a real variable. B(x) is some real valued function. And I should find the $lambda(T)$ that minimizes that functional. To start I would compute the functional derivative with respect to $lambda(T)$ and set this to zero:
$$
int m_{0}nabla^{2} hleft(frac{x}{lambda(T)}right)frac{x}{lambda^{2}(T)}+r_{0}m_{0}hleft(frac{x}{lambda(T)}right)h^{'}left(frac{x}{lambda(T)}right)frac{x}{lambda^{2}(T)}-frac{1}{6}u_{0}m^{3}_{0}h^{3}left(frac{x}{lambda(T)}right)h^{'}left(frac{x}{lambda(T)}right)frac{x}{lambda^{2}(T)}+B(x)h^{'}left(frac{x}{lambda(T)}right)frac{x}{lambda^{2}(T)}=0.
$$
In this equation the $h^{'}left(frac{x}{lambda(T)}right)$ denotes the first derivative of $hleft(frac{x}{lambda(T)}right)$ with respect to its argument. Now I am not completely sure if I did the functional derivatives correctly.And otherwise I would not know how to solve this equation for $lambda (T)$ since all the explicit $lambda$ would cancel.
functional-analysis derivatives optimization
asked Nov 26 '18 at 12:47
zodiaczodiac
617
$endgroup$
I have the following functional:
$$
H[lambda(T)]
= intfrac{1}{2}left | nabla m_{0}hleft(frac{x}{lambda(T)}right) right |^{2}+frac{1}{2}r_{0}m^{2}_{0}h^{2}left(frac{x}{lambda(T)}right)+frac{1}{4!}u_{0}m^{4}_{0}h^{4}left(frac{x}{lambda(T)}right)-B(x)m_{0}hleft(frac{x}{lambda(T)}right)dx,$$
with $r_{0},m_{0},u_{0}$ being real constants and x a real variable. B(x) is some real valued function. And I should find the $lambda(T)$ that minimizes that functional. To start I would compute the functional derivative with respect to $lambda(T)$ and set this to zero:
$$
int m_{0}nabla^{2} hleft(frac{x}{lambda(T)}right)frac{x}{lambda^{2}(T)}+r_{0}m_{0}hleft(frac{x}{lambda(T)}right)h^{'}left(frac{x}{lambda(T)}right)frac{x}{lambda^{2}(T)}-frac{1}{6}u_{0}m^{3}_{0}h^{3}left(frac{x}{lambda(T)}right)h^{'}left(frac{x}{lambda(T)}right)frac{x}{lambda^{2}(T)}+B(x)h^{'}left(frac{x}{lambda(T)}right)frac{x}{lambda^{2}(T)}=0.
$$
In this equation the $h^{'}left(frac{x}{lambda(T)}right)$ denotes the first derivative of $hleft(frac{x}{lambda(T)}right)$ with respect to its argument. Now I am not completely sure if I did the functional derivatives correctly.And otherwise I would not know how to solve this equation for $lambda (T)$ since all the explicit $lambda$ would cancel.
functional-analysis derivatives optimization
functional-analysis derivatives optimization
asked Nov 26 '18 at 12:47
zodiaczodiac
617
asked Nov 26 '18 at 12:47
zodiaczodiac
617
asked Nov 26 '18 at 12:47
zodiaczodiac
617
asked Nov 26 '18 at 12:47
zodiaczodiac
617
asked Nov 26 '18 at 12:47
zodiaczodiac
617
617
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3014281%2fextremum-of-functional-via-functional-derivative%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3014281%2fextremum-of-functional-via-functional-derivative%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
dhCoMjmCj,hSt MSnaZgww,UNCo Rrw2Mi67DO2,J 0sWt,Eq2K,Q9s12L1jkxIfWO0Xv,8SVZfi2XcqyXZ8S89U,k,7FVENLi0,v4pe2,UbQ
