Getting Canonical Coordinates in differential equations from Lie Group
$begingroup$
I've been trying to understand how Lie Groups can help solve differential equations (this 12-page pdf is the most straightforward explanation I've seen).
My understanding is that when a first-order differential equation $frac{dy}{dx}=h(x,y)$ has a continuous "translation symmetry" $(x,y)mapsto(x,y+lambda)$, it becomes very easy to solve the equation using basic integration of one variable.
The problem is that most differential equations in standard $(x,y)$ coordinates don't have this nice translation symmetry, and so we need to change to a set of "canonical coordinates" $(X,Y)$ which do have the desired translation symmetry $(X,Y)mapsto(X,Y+lambda)$.
The main condition that these new canonical coordinates need to satisfy, is that the $Y$ coordinate should be aligned with the $lambda$ variable ($frac{dY}{dlambda}=1,frac{dX}{dlambda}=0$), so that we get the nice translation symmetry we want:
$$frac{dY}{dlambda}=frac{dY}{dx}frac{dx}{dlambda}+frac{dY}{dy}frac{dy}{dlambda}=Y_xxi+Y_yeta=1$$
$$frac{dX}{dlambda}=frac{dX}{dx}frac{dx}{dlambda}+frac{dX}{dy}frac{dy}{dlambda}=X_xxi+X_yeta=0$$
We also require that the symmetry condition be satisfied:
$$frac{dY}{dX}=frac{D_xY}{D_xX}=frac{Y_x+Y_y y'}{X_x+X_y y'}=h(X,Y)$$
Here's where I'm stuck. Basically all the terms in the above equations are unknowns, and it seems incredibly difficult to solve for the new coordinates $(X,Y)$. I've also seen the "linearized symmetry condition" below, but it still results in a difficult-to-solve equation:
$$eta_x-xi_y h^2 +(eta_y -xi_x)h - (xi h_x + eta h_y) = 0$$
Is there a nice, algorithmic approach for deducing a set of canonical coordinates? Or does it ultimately come down to "guess and check" with the above equation?
ordinary-differential-equations lie-groups
$endgroup$
add a comment |
$begingroup$
I've been trying to understand how Lie Groups can help solve differential equations (this 12-page pdf is the most straightforward explanation I've seen).
My understanding is that when a first-order differential equation $frac{dy}{dx}=h(x,y)$ has a continuous "translation symmetry" $(x,y)mapsto(x,y+lambda)$, it becomes very easy to solve the equation using basic integration of one variable.
The problem is that most differential equations in standard $(x,y)$ coordinates don't have this nice translation symmetry, and so we need to change to a set of "canonical coordinates" $(X,Y)$ which do have the desired translation symmetry $(X,Y)mapsto(X,Y+lambda)$.
The main condition that these new canonical coordinates need to satisfy, is that the $Y$ coordinate should be aligned with the $lambda$ variable ($frac{dY}{dlambda}=1,frac{dX}{dlambda}=0$), so that we get the nice translation symmetry we want:
$$frac{dY}{dlambda}=frac{dY}{dx}frac{dx}{dlambda}+frac{dY}{dy}frac{dy}{dlambda}=Y_xxi+Y_yeta=1$$
$$frac{dX}{dlambda}=frac{dX}{dx}frac{dx}{dlambda}+frac{dX}{dy}frac{dy}{dlambda}=X_xxi+X_yeta=0$$
We also require that the symmetry condition be satisfied:
$$frac{dY}{dX}=frac{D_xY}{D_xX}=frac{Y_x+Y_y y'}{X_x+X_y y'}=h(X,Y)$$
Here's where I'm stuck. Basically all the terms in the above equations are unknowns, and it seems incredibly difficult to solve for the new coordinates $(X,Y)$. I've also seen the "linearized symmetry condition" below, but it still results in a difficult-to-solve equation:
$$eta_x-xi_y h^2 +(eta_y -xi_x)h - (xi h_x + eta h_y) = 0$$
Is there a nice, algorithmic approach for deducing a set of canonical coordinates? Or does it ultimately come down to "guess and check" with the above equation?
ordinary-differential-equations lie-groups
$endgroup$
add a comment |
$begingroup$
I've been trying to understand how Lie Groups can help solve differential equations (this 12-page pdf is the most straightforward explanation I've seen).
My understanding is that when a first-order differential equation $frac{dy}{dx}=h(x,y)$ has a continuous "translation symmetry" $(x,y)mapsto(x,y+lambda)$, it becomes very easy to solve the equation using basic integration of one variable.
The problem is that most differential equations in standard $(x,y)$ coordinates don't have this nice translation symmetry, and so we need to change to a set of "canonical coordinates" $(X,Y)$ which do have the desired translation symmetry $(X,Y)mapsto(X,Y+lambda)$.
The main condition that these new canonical coordinates need to satisfy, is that the $Y$ coordinate should be aligned with the $lambda$ variable ($frac{dY}{dlambda}=1,frac{dX}{dlambda}=0$), so that we get the nice translation symmetry we want:
$$frac{dY}{dlambda}=frac{dY}{dx}frac{dx}{dlambda}+frac{dY}{dy}frac{dy}{dlambda}=Y_xxi+Y_yeta=1$$
$$frac{dX}{dlambda}=frac{dX}{dx}frac{dx}{dlambda}+frac{dX}{dy}frac{dy}{dlambda}=X_xxi+X_yeta=0$$
We also require that the symmetry condition be satisfied:
$$frac{dY}{dX}=frac{D_xY}{D_xX}=frac{Y_x+Y_y y'}{X_x+X_y y'}=h(X,Y)$$
Here's where I'm stuck. Basically all the terms in the above equations are unknowns, and it seems incredibly difficult to solve for the new coordinates $(X,Y)$. I've also seen the "linearized symmetry condition" below, but it still results in a difficult-to-solve equation:
$$eta_x-xi_y h^2 +(eta_y -xi_x)h - (xi h_x + eta h_y) = 0$$
Is there a nice, algorithmic approach for deducing a set of canonical coordinates? Or does it ultimately come down to "guess and check" with the above equation?
ordinary-differential-equations lie-groups
$endgroup$
I've been trying to understand how Lie Groups can help solve differential equations (this 12-page pdf is the most straightforward explanation I've seen).
My understanding is that when a first-order differential equation $frac{dy}{dx}=h(x,y)$ has a continuous "translation symmetry" $(x,y)mapsto(x,y+lambda)$, it becomes very easy to solve the equation using basic integration of one variable.
The problem is that most differential equations in standard $(x,y)$ coordinates don't have this nice translation symmetry, and so we need to change to a set of "canonical coordinates" $(X,Y)$ which do have the desired translation symmetry $(X,Y)mapsto(X,Y+lambda)$.
The main condition that these new canonical coordinates need to satisfy, is that the $Y$ coordinate should be aligned with the $lambda$ variable ($frac{dY}{dlambda}=1,frac{dX}{dlambda}=0$), so that we get the nice translation symmetry we want:
$$frac{dY}{dlambda}=frac{dY}{dx}frac{dx}{dlambda}+frac{dY}{dy}frac{dy}{dlambda}=Y_xxi+Y_yeta=1$$
$$frac{dX}{dlambda}=frac{dX}{dx}frac{dx}{dlambda}+frac{dX}{dy}frac{dy}{dlambda}=X_xxi+X_yeta=0$$
We also require that the symmetry condition be satisfied:
$$frac{dY}{dX}=frac{D_xY}{D_xX}=frac{Y_x+Y_y y'}{X_x+X_y y'}=h(X,Y)$$
Here's where I'm stuck. Basically all the terms in the above equations are unknowns, and it seems incredibly difficult to solve for the new coordinates $(X,Y)$. I've also seen the "linearized symmetry condition" below, but it still results in a difficult-to-solve equation:
$$eta_x-xi_y h^2 +(eta_y -xi_x)h - (xi h_x + eta h_y) = 0$$
Is there a nice, algorithmic approach for deducing a set of canonical coordinates? Or does it ultimately come down to "guess and check" with the above equation?
ordinary-differential-equations lie-groups
ordinary-differential-equations lie-groups
asked Nov 24 '18 at 22:52
eigenchriseigenchris
1,530616
1,530616
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%2f3012200%2fgetting-canonical-coordinates-in-differential-equations-from-lie-group%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%2f3012200%2fgetting-canonical-coordinates-in-differential-equations-from-lie-group%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