Symplectic Notation: Confused by Subscripts i & j?

[aaaa202]

The attached is a section of the derivation of canonical transformation from the symplectic formulation. I tend to get very confused by the subscripts i and j. For me they both run from 1 to 2n and can be used interchangeably. But of course that is not the case since equation (9.53) on the attached picture specifically instructs you to transpose the matrix described by (9.51). Can someone explain what is wrong with just changing the indices in a pedagogic way - an example would be lovely too.

 

[gabbagabbahey]

The attached is a section of the derivation of canonical transformation from the symplectic formulation. I tend to get very confused by the subscripts i and j. For me they both run from 1 to 2n and can be used interchangeably. But of course that is not the case since equation (9.53) on the attached picture specifically instructs you to transpose the matrix described by (9.51). Can someone explain what is wrong with just changing the indices in a pedagogic way - an example would be lovely too.

I'm not sure exactly what you are asking. Can you give a specific example of the equation where you think you can't transpose i & j?

 

[Hypersphere]

Well, the indices do run from 1 to 2n, and there is no deep physics hidden in calling the indices i and j, specifically. Could be any letter or symbol, really. However, the order is important.

Changing the order of the indices for a matrix actually is the same as taking the transpose. Think of i as a row index and j as a column index. Just make up some old matrix and try it out.

 

[aaaa202]

perhaps I wasn't clear enough. My frustation is actually due to not being able to see why the order of i and j is important - it's probably trivial but I don't see it.

 

[gabbagabbahey]

My frustation is actually due to not being able to see why the order of i and j is important

In what term/equation?