What is Symplectic? Explaining the Group Theory Basics

[alexepascual]

What is "symplectic"?

I was interested in learning the meaning of this word as I saw it used in Goldstein. Looking in the web it appears this word has to do with a "symplectic group". I don't know much about group theory, but I was wondering if someone could give an explanation "for the un-initiated". Maybe what are the main characteristics of a symplectic group and how they apply to classical mechanics.
I'll appreciate it.

 

[alexepascual]

I am surprised there are no replies yet. If this is too complex to answer in this forum, I would appreciate your pointing me towards a good source, (book or web) where I might find information about this topic.
Thanks guys,
-Alex-

 

[brewnog]

Well since nobody else has said anything...

All I know is that a symplectic map is one where areas are preserved, so if you're making a projection of Earth onto a 2D map then the land areas of all the countries are kept in proportion. (As opposed to the, urm, other (anyone?) type of projection where shapes (?) are preserved.)

Guess your context is probably topological, but a quick dictionary search mentioned about the 'symplectic bone', obviously a medical term and probably not what you're after.

 

[masudr]

I would go through a long description, but I'll only be repeating stuff I learned off wikipedia. I suggest you look up symplectic spaces in Wikipedia. The full use of it is quite involved, and it is 3.20AM here so I'm not in a good condition at the moment, but it's main use is in Hamiltonain mechanics, and I suggest you learn that formalism before you try and see how symplectic spaces fit in.

 

[alexepascual]

Thanks Brewnog and Masudr. I'll look into the "symplectic map" but I doubt it that has any connection with the physics meaning. I'll try to find information on "symplectic spaces" in wikipedia, and I guess I'll have to make some time to continue reading Goldstein.
Thanks again,
Alex

 

[dextercioby]

Just curiosity:what edition of Goldstein??I've searched through the third and came up with nothing.
Advice:Symplectic geometry & its applications to both classical & quantum Hamiltonian (and Lagrangian,yet less used) are of great importance in theoretical physics.Check up on the 2 books:Marsden et al. and V.I.Arnold for complete descriptions...

Daniel.


P.S.These things are not really intuitive.They rarely can be expressed into layman terms,preferably without formulas...

 

[krab]

The reason answers are slow in coming is because your first question is purely a math question. You'll probably get more responses in the math forum. Anyway, the symplectic group is the set of matrices of given dimension 2n that can be used to describe linear symplectic maps. They have very special properties. For example, for n=1, it is all matrices of determinant 1. For n=2, there are 6 relations among the matrix elements, reducing the number of independent elements from 16 to 10. For n=3, there are 15 relations, and so on.

A symplectic map is an area-preserving map. This is the connection with Hamiltonian mechanics. It can be proved that the mapping that takes initial positions and momenta (phase space dimension = 2n, as above) at time t_1 to time t_2 is an area-preserving map if the motion is non-dissipative, i.e. governed by a Hamiltonian.

A Hamiltonian of linear motion is of the form
$$H=Ax^2+Bxp_x+Cxy+Dxp_y+...$$
it has all the combinations of quadratic terms containing the positions and momenta. In 3D, n=3, there are 21 terms, not 36. In other words, the 15 relations mentioned above come from the 15 possible relations
$${\partial^2H\over\partial x_i\partial x_j}={\partial^2H\over\partial x_j\partial x_i}$$, where x_i is the phase space vector (x,p_x,y,p_y,z,p_z).

 

[alexepascual]

Dextercioby:
The eddition of Goldstein I have is the second. Section 9-3 is titled "The Symplectic approach to canonical transformations. There is an equation there that is called "the symplectic condition" and a matrix that satisfies this condition is labeled as a "symplectic matrix". I have not read the chapter and I am not even sure I can dive into chapter 9 without first studying chapter 8. Honestly, I think that inorder to understand these things I would have to dedicate more time to these studies, but lately I have been preocupied with other things such as looking for work.
What Goldstein does not say is why a symplectic matrix is called that name or when the idea of a symplectic condicion or matrix originated. In some other place I had read that the word symplectic was invented by someone (can't remember the name) working on groups in order to distinguish these groups form a different category of groups.
When I have time I'll see if I can get from the library Marsden or Arnold.
For the moment, I have printed out some pages from Wikipedia and I'll be looking at those in my spare time. Thanks a lot for your advice.

Krab:
Thanks for your introduction to the subject, It helps getting me started.
I wonder how the concept of preservation of area is applied to mechanics, could you give me a short example?
Thanks again,
-Alex-