Statics and Dynamics
Wick rotation relates statics problems in dimensions to dynamics problems in dimensions, trading one dimension of space for one dimension of time. A simple example where is a hanging spring with fixed endpoints in a gravitational field. The shape of the spring is a curve . The spring is in equilibrium when the energy associated with this curve is at a critical point; this critical point is typically a minimum, so this idea is usually called "the principle of least energy". To compute the energy, we integrate over the energy density at each point:
where is the spring constant and is the gravitational potential.
The corresponding dynamics problem is that of a rock thrown upwards; the path the rock follows is a critical point of the action. Action is the integral of the Lagrangian; as before, this critical point is typically a minimum, so this is called the "principle of least action":
We get the solution to the dynamics problem (up to a factor of ) from the statics problem by Wick rotation, replacing by and the spring constant by the mass of the rock :
Read more about this topic: Wick Rotation
Famous quotes containing the word dynamics:
“Anytime we react to behavior in our children that we dislike in ourselves, we need to proceed with extreme caution. The dynamics of everyday family life also have a way of repeating themselves.”
—Cathy Rindner Tempelsman (20th century)