Symmetry transformation

  • #1
698
18

Homework Statement


System of equations
[tex]\frac{du_j}{dt}=u_{j+1}+u_{j-1}-2u_j-\frac{K}{2 \pi}\sin(2\pi u_j)+\bar{F}+F_{ac}\cos(2\pi \nu_o t)[/tex]
where ##j=1,2,3,4##. So ##\{u_j\}## is set of coordinates. If we apply symmetry transformation
[tex]\sigma_{r,m,s}\{u_j(t)\}=\{u_{j+r}(t-\frac{s}{\nu_0})\}[/tex]
how to find condition for which
[tex]\sigma_{r,m,s}\{u_j(t)\}=\{u_{j}(t) \}[/tex]
We impose cyclic boundary condition.
[/B]


Homework Equations




The Attempt at a Solution


If I understand well
[tex]\frac{du_1}{dt}=u_{2}+u_{4}-2u_1-\frac{K}{2 \pi}\sin(2\pi u_1)+\bar{F}+F_{ac}\cos(2\pi \nu_o t)[/tex]
[tex]\frac{du_2}{dt}=u_{1}+u_{3}-2u_2-\frac{K}{2 \pi}\sin(2\pi u_2)+\bar{F}+F_{ac}\cos(2\pi \nu_o t)[/tex]
[tex]\frac{du_3}{dt}=u_{2}+u_{4}-2u_3-\frac{K}{2 \pi}\sin(2\pi u_3)+\bar{F}+F_{ac}\cos(2\pi \nu_o t)[/tex]
[tex]\frac{du_4}{dt}=u_{3}+u_{1}-2u_4-\frac{K}{2 \pi}\sin(2\pi u_4)+\bar{F}+F_{ac}\cos(2\pi \nu_o t)[/tex]
And to transformation be satisfied [tex]\sigma_{r,m,s}\{u_j(t)\}=\{u_{j}(t) \}[/tex]
it is important that for example
[tex]\sigma_{r,m,s}u_1=u_4[/tex]
[tex]\sigma_{r,m,s}u_2=u_3[/tex]
...

But I am not sure how to show explicite consequence from this
[tex]\sigma_{r,m,s}\{u_j(t)\}=\{u_{j}(t) \}[/tex]

[/B]
 

Answers and Replies

  • #2
18,668
8,634
Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 

Related Threads on Symmetry transformation

  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
10
Views
1K
  • Last Post
Replies
9
Views
2K
  • Last Post
Replies
3
Views
28K
  • Last Post
Replies
10
Views
1K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
1
Views
822
Top