Symplectic geometry of phase space

In summary, a symplectic manifold is a manifold with a symplectic form, which is a non-degenerate 2-form. Hamilton's formalism reduces the equations of motion to the flow of a Hamiltonian vector field, making symmetries more manifest. State space and phase space are different manifolds, with the latter being the cotangent bundle of the former. Using generalized momenta as coordinates simplifies the symplectic form compared to using generalized velocities. A good resource for further understanding is V. I. Arnold's book, "Mathematical Methods of Classical Mechanics."
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CrazyNeutrino
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What is a symplectic manifold or symplectic geometry? (In intuitive terms please)
I have a vague understanding that it involves some metric that assigns an area to a position and conjugate momentum that happens to be preserved. What is 'special' about Hamilton's formulation that makes it more useful than Lagrange's? Why is it that phase space has this special geometry but state space does not? What makes the generalized momentum a more useful coordinate than the generalized velocity?
 
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A symplectic manifold is a manifold equipped with a symplectic form. The symplectic form is a non-degenerate 2-form, which makes it an anti-symmetric rather than symmetric tensor. It is not a metric.

Hamilton’s formalism reduces the equations of motion to the flow of a Hamiltonian vector field, which is a set of first order differential equations and the symmetries of the time evolution phase space are often more manifest and boil down to the Poisson bracket.

State space and phase space are different manifolds (the latter being the cotangent bundle of the former).

Using the generalised momenta as coordinates, the symplectic form takes a much easier form than if you use generalised velocities. Much like using Cartesian coordinates instead of curvilinear ones makes the metric take an easier form in Euclidean space.
 
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  • #3
Thank you, this is the most concise explanation I've seen yet!
 
  • #4
A very good (but of course also very mathematical) explaning these things in detail is

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer Verlag (1989)
 

1. What is symplectic geometry of phase space?

The symplectic geometry of phase space is a mathematical framework used to study the dynamics of systems in classical mechanics. It involves the study of symplectic manifolds, which are geometric spaces that preserve certain mathematical structures known as symplectic forms. These structures are used to describe the motion of particles in phase space, where each point represents the state of a physical system at a given time.

2. How is symplectic geometry used in physics?

Symplectic geometry is used in physics to describe the symmetries and conservation laws of physical systems. It provides a powerful mathematical language for analyzing the dynamics of particles and fields, and has applications in various fields such as mechanics, electromagnetism, and quantum mechanics. It is also used in the study of Hamiltonian systems, which are systems described by a set of differential equations known as Hamilton's equations.

3. What is the significance of symplectic forms in symplectic geometry?

Symplectic forms are fundamental mathematical objects in symplectic geometry that are used to describe the dynamics of systems in phase space. They are non-degenerate and closed differential 2-forms, which means that they do not vanish and satisfy certain mathematical equations. These forms play a crucial role in preserving the symplectic structure of phase space, which is essential for understanding the behavior of physical systems.

4. How does symplectic geometry differ from other areas of geometry?

Symplectic geometry differs from other areas of geometry in that it focuses on the study of symplectic manifolds, which have a different geometric structure than other manifolds. Unlike Riemannian manifolds, which are used in the study of general relativity, symplectic manifolds are not equipped with a metric structure. Instead, they have a symplectic structure that is used to describe the dynamics of systems in phase space. Additionally, symplectic geometry is closely related to the study of group actions and symmetries, making it a powerful tool in physics.

5. What are some practical applications of symplectic geometry of phase space?

The symplectic geometry of phase space has practical applications in various fields, including classical mechanics, astrophysics, and quantum mechanics. In classical mechanics, it is used to describe the motion of particles and the conservation laws that arise from symmetries. In astrophysics, it is used to study the dynamics of celestial bodies, such as planets and galaxies. In quantum mechanics, it is used to understand the behavior of quantum systems and the symmetries that govern their dynamics. Additionally, symplectic geometry has applications in control theory, robotics, and other areas of engineering.

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