Weyl distance function
The Weyl distance function is a way to measure how far apart two points are in a special kind of space called a symplectic manifold. A symplectic manifold is like a fancy mathematical surface where you can do special math stuff called symplectic geometry.
To understand the Weyl distance function, let's imagine we are playing a game on a symplectic manifold. The game is called "Guess the Path". In this game, we have a starting point (let's call it Point A) and an ending point (let's call it Point B), and we want to find the shortest path between them.
But in symplectic geometry, paths can be a bit tricky. We can't just walk in a straight line like we're used to in normal geometry. Instead, we have to follow special rules called Hamilton's equations. These rules tell us how the coordinates of our path can change as we move from Point A to Point B.
Now, the Weyl distance function comes into play. It helps us figure out which path is the shortest. Think of it like a special ruler that measures the distance between two points in symplectic space.
To use the Weyl distance function, we first need to know the symplectic structure of our manifold. This structure tells us how the space behaves and how we can do our math.
Once we know the symplectic structure, we can start finding the shortest path. We start at Point A and use Hamilton's equations to move a little bit in the right direction. Then we check how far away we are from Point B using the Weyl distance function.
If we're still far away, we adjust our direction a little bit and move again. We keep doing this until we eventually reach Point B. At each step, we use the Weyl distance function to check if we're getting closer or farther away.
By following these steps, we can find the shortest path between Point A and Point B on our symplectic manifold. And the Weyl distance function helps us make sure we're going in the right direction and not getting lost!
So, in summary, the Weyl distance function is like a special ruler that helps us find the shortest path between two points on a symplectic manifold. It uses Hamilton's equations and the symplectic structure of the space to guide us along the right path.
To understand the Weyl distance function, let's imagine we are playing a game on a symplectic manifold. The game is called "Guess the Path". In this game, we have a starting point (let's call it Point A) and an ending point (let's call it Point B), and we want to find the shortest path between them.
But in symplectic geometry, paths can be a bit tricky. We can't just walk in a straight line like we're used to in normal geometry. Instead, we have to follow special rules called Hamilton's equations. These rules tell us how the coordinates of our path can change as we move from Point A to Point B.
Now, the Weyl distance function comes into play. It helps us figure out which path is the shortest. Think of it like a special ruler that measures the distance between two points in symplectic space.
To use the Weyl distance function, we first need to know the symplectic structure of our manifold. This structure tells us how the space behaves and how we can do our math.
Once we know the symplectic structure, we can start finding the shortest path. We start at Point A and use Hamilton's equations to move a little bit in the right direction. Then we check how far away we are from Point B using the Weyl distance function.
If we're still far away, we adjust our direction a little bit and move again. We keep doing this until we eventually reach Point B. At each step, we use the Weyl distance function to check if we're getting closer or farther away.
By following these steps, we can find the shortest path between Point A and Point B on our symplectic manifold. And the Weyl distance function helps us make sure we're going in the right direction and not getting lost!
So, in summary, the Weyl distance function is like a special ruler that helps us find the shortest path between two points on a symplectic manifold. It uses Hamilton's equations and the symplectic structure of the space to guide us along the right path.