Introduction E9F5FC

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Consider the simplest illustration

Consider a simple harmonic oscillator, such as a pendulum. It may be thought of a single particle with a position and a momentum. We will consider the phase space given by the possibilities for the position and the momentum.

The symplectic area is a region of this phase space. We can consider a variation of the possibilities for position and likewise for momentum.

Consider, in particular, two extreme cases. In one extreme, kinectic energy is near its minimum (near zero) but potential energy is near its maximum. This is when the pendulum has swung completely to one side. In the other extreme, kinectic energy is near its maximum but potential energy is near its minimum. This is when the pendulum is hanging down vertically and swinging with greatest velocity.

Define a region in phase space which describes the first extreme, where the velocity is close to zero but the position can vary substantially, corresponding to different amplitudes. Then the area - the variety of possibilies - will be preserved as time evolves. In particular, when the pendulum hangs down vertically - let us call this position zero - then the momentum varies substantially. Note that in this case of a pendulum, the frequencies are the same, the possibilities happen to be in sync, and so the area exhibits periodic behavior and traces a closed loop.

Interpreting the area

The symplectic form {$dp \wedge dq$} measures the area. By Stokes' theorem, the integral of this area form equals the integral of the action {$p \, dq$} around the boundary {$\partial S$}. If we have a rectangle given by {$p_{max}$} and {$p_{min}$} and {$q_{max}$} and {$q_{min}$} then the integral of the action - and the area - is given by {$(p_{max} - p_{min})(q_{max} - q_{min})=\Delta p \Delta q$} if we integrate clockwise.

Orientation of the area

If we integrate clockwise, then position and momentum increase together. If we integrate counterclockwise, then one increases as the other decreases.

Further investigations

• Describe the change in the shape of the region of possibilities as the pendulum moves. Show that the area is preserved.
• Read further about Heisenberg's uncertainty principle.
• Understand action in physics. Action is energy x time = momentum x distance. In particular, physics is based on the minimization of action.
• How is symplectic area related to the constraints on symplectic matrices? and to the Lie algebra/group {$C_n$}?

Readings

• What is symplectic geometry? Lisa McDuff
• Heisenberg's uncertainty principle
• Bohr–Sommerfeld quantization condition
• Stokes' theorem
• Action
• Calculus of variations