Editing Wick rotation (section)

== Statics and dynamics ==
Wick rotation relates statics problems in {{mvar|n}} dimensions to dynamics problems in {{math|''n'' − 1}} dimensions, trading one dimension of space for one dimension of time. A simple example where {{math|1=''n'' = 2}} is a hanging spring with fixed endpoints in a gravitational field. The shape of the spring is a curve {{math|''y''(''x'')}}. The spring is in equilibrium when the energy associated with this curve is at a critical point (an extremum); this critical point is typically a minimum, so this idea is usually called "the principle of least energy". To compute the energy, we integrate the energy spatial density over space:
: <math>E = \int_x \left[ k \left(\frac{dy(x)}{dx}\right)^2 + V\big(y(x)\big) \right] dx,</math>
where {{math|''k''}} is the spring constant, and {{math|''V''(''y''(''x''))}} is the gravitational potential.

The corresponding dynamics problem is that of a rock thrown upwards. The path the rock follows is that which extremalizes the [[Action (physics)|action]]; as before, this extremum is typically a minimum, so this is called the "[[principle of least action]]". Action is the time integral of the [[Lagrangian mechanics|Lagrangian]]:
: <math>S = \int_t \left[ m \left(\frac{dy(t)}{dt}\right)^2 - V\big(y(t)\big) \right] dt.</math>

We get the solution to the dynamics problem (up to a factor of {{mvar|i}}) from the statics problem by Wick rotation, replacing {{math|''y''(''x'')}} by {{math|''y''(''it'')}} and the spring constant {{mvar|k}} by the mass of the rock {{mvar|m}}:
: <math>iS = \int_t \left[ m \left(\frac{dy(it)}{dt}\right)^2 + V\big(y(it)\big) \right] dt = i \int_t \left[ m \left(\frac{dy(it)}{dit}\right)^2 - V\big(y(it)\big) \right] d(it).</math>