Antisymmetric but non-symplectic

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In summary, the conversation discusses the concept of a symplectic scalar product, which is defined as an antisymmetric, nondegenerate, and bilinear form. The speaker is unsure if this is the only type of scalar product that meets these criteria or if there could be other variations, similar to the different types of symmetric scalar products.
  • #1
Cantab Morgan
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I understand that a symplectic scalar product is bilinear and antisymmetric. But is that the only such scalar product? In other words, is it possible to have a bilinear and antisymmetric scalar product that is not symplectic?
 
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  • #2
A symplectic form is by definition an anti-symmetric, nondegenerate, bilinear form.
I am not sure what you mean by "scalar product" in this case, but if it should be nondegenerate then the answer to your question is no.
 
  • #3
Thanks for your reply, yyat! By scalar product I mean a map from pairs of points (vectors) onto a scalar.

For example, the traditional dot product is a Euclidean scalar product that's such a map. In two dimensions, we'd have

[tex]\left( \left( \begin{array}{c} x \\ y \end{array} \right) ,
\left( \begin{array}{c} x' \\ y' \end{array} \right) \right) = xx' + yy'
[/tex]

That's a concrete example of a scalar product that's symmetric and bilinear, but it's not the only one. For example, the Lorentz scalar product is also symmetric and bilinear.

[tex]\left< \left( \begin{array}{c} t \\ x \end{array} \right) ,
\left( \begin{array}{c} t' \\ x' \end{array} \right) \right> = tt' - xx'
[/tex]

Anyway, my understanding is that the symplectic scalar product [tex]\omega(,)[/tex] is given by

[tex]\omega\left( \left( \begin{array}{c} x \\ y \end{array} \right) ,
\left( \begin{array}{c} x' \\ y' \end{array} \right) \right) = xy' - x'y
[/tex]

I'm just curious as to whether that's the only possible bilinear antisymmetric scalar product, or whether there could be other flavors, just as there are different flavors of the symmetric scalar product.
 

1. What does it mean for a system to be "antisymmetric but non-symplectic"?

An antisymmetric but non-symplectic system is one in which the equations of motion exhibit antisymmetry, meaning that they are invariant under the interchange of two variables, but do not follow the laws of symplectic geometry. This means that the system does not conserve phase space volume and may exhibit chaotic behavior.

2. How is antisymmetry different from symmetry in a physical system?

Symmetry in a physical system refers to a property that remains unchanged under certain transformations, such as rotation or translation. Antisymmetry, on the other hand, refers to a property that is invariant under the interchange of two variables. While both symmetries are important in understanding physical systems, they have distinct mathematical properties and implications.

3. Can you provide an example of an antisymmetric but non-symplectic system?

One example of an antisymmetric but non-symplectic system is the double pendulum. While the equations of motion for this system exhibit antisymmetry, the system does not follow the laws of symplectic geometry due to its nonlinearity. This can lead to chaotic behavior, making the system difficult to predict and analyze.

4. How does antisymmetry affect the behavior of a physical system?

The presence of antisymmetry in a physical system can lead to complex and unpredictable behavior. This is because the system does not conserve phase space volume, meaning that trajectories can cross and overlap, leading to chaotic motion. Antisymmetry can also affect the stability of a system and make it more susceptible to small perturbations.

5. Can antisymmetric but non-symplectic systems be useful in scientific research?

Despite their complex behavior, antisymmetric but non-symplectic systems can be useful in scientific research. They can serve as models for real-world systems that exhibit chaotic behavior, such as weather patterns or biological systems. Studying these systems can also provide insights into the fundamental laws of physics and help us better understand the limitations of symplectic geometry.

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