Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,853 answers , 20,624 comments
1,470 users with positive rep
501 active unimported users
More ...

Why can any pair of master coordinates be used to calculate a nonlinear mode of a nonlinear dynamical system?

+ 4 like - 0 dislike
1563 views

This is a question I have been asking myself for some time since the following technique is often used in the nonlinear dynamics community, but never managed to get an answer why it could be applied.

In symplectic mechanics, consider a dynamical system with no rigid body movements (all eigenvalues are non-zero, ie $\lambda_i\neq 0$). The centre manifold states that the dynamics can be written in the following form: $$ \begin{bmatrix} \dot{\mathbf{x}}_\mathrm{s} \\ \dot{\mathbf{x}}_\mathrm{c} \\ \dot{\mathbf{x}}_\mathrm{u} \end{bmatrix} =\begin{bmatrix} [ A_\mathrm{s}] & & \\ & [A_\mathrm{c}] & \\& &[A_\mathrm{u}] \end{bmatrix}\begin{bmatrix} \mathbf{x_\mathrm{s}} \\ \mathbf{x_\mathrm{c}} \\ \mathbf{x_\mathrm{u}} \end{bmatrix} +\text{nonlinear terms}$$ and that there exists a centre manifold (also stable and unstable manifolds, but they are not of interest here) of dimension the number of lines of $A_\mathrm{u}$.

Because all eigenvalues corresponding to the centre manifold ($\mathrm{Re}(\lambda_i)=0$) are purely imaginary and non-zero (no rigid body movements), $A_\mathrm{c}$ is composed of $2\times2$ blocks on the diagonal.

Now, we are searching for periodic orbits, so we focus and the centre manifold and we name the coordinates of $\mathbf{x}_\mathrm{u}$ such that $\mathbf{x}_\mathrm{u}=[u_1,v_1,\dots,u_n,v_n]^T$ (even dimension, as explained above).

In order to calculate the nonlinear normal mode $i$, many people write that any pair $(u_i,v_i)$ can be chosen as master coordinates and that all other pair $(u_j,v_j)$, $i\neq j$, can be written as: $$\left\lbrace \begin{aligned} u_j &=f_j(u_i,v_i) \\ v_j&=g_j(u_i,v_i)\end{aligned}\right.$$

Question: What ensures that choosing a pair of master coordinates is always possible? If not, could you give a counterexample?

This post imported from StackExchange Physics at 2014-08-17 11:31 (UCT), posted by SE-user anderstood
asked Aug 16, 2014 in Theoretical Physics by anderstood (20 points) [ no revision ]

I believe these are often only local coordinates, and there may be a singular set of problems (of measure zero) where such a choice is impossible. But I don't have a specific counterexample.

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsOverf$\varnothing$ow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...