Foucault Pendulum，Parallel Transport and Classical Geometrical Phase

A Rotating frame.

For a general treatment, we consider a frame that is rotating by angular velocity Omega around an arbitrary axis where we set its coordinate origin.

In a frame that is inert (lab), absolute position can be described by the rotating frame, given its changing (rotating) bases at any time and coordinates associated with them.

The second equation converting derivatives to its Cross product comes from a more general rotation analysis.

We’ve reached:

This is understood as the absolute velocity in Lab is the sum of relative motion in that frame and a tangential velocity due to the frame’s rotation.

Take derivatives further to get Acceleration (Newtonian) law of motion:

Multiplying mass to get (inert) Force as:

For the simplest case, where rotation is perpendicular to displacement vector as well as its velocity (as respective with that frame):

Foucault Pendulum: A Pendulum on Earth.

If earth were not to rotate, a pendulum will keep swinging in the same plane defined by g and l for ever.

But our earth is indeed rotating along its polar axis. With pendulum’s non-zero velocity, it’s to expect a Coriolis effect on its plane of oscillation.

A full treatment is given as follows well as elsewhere[2].

Newtonian equation is obtained using inert forces:

or by components (* also after some careful visualization):

(2)

due to Cross Product of rotation axis of Earth and the x and y pendulum axis at a given altitude.  

Adiabatic Process

Adiabatic Process is known when a system is modified so slow that in each of the infinitesimal step it maintains energy. 

Now let’s consider what is meant for this equation. For rotation of earth being far slower 

, which is valid for all practical purposes

Pendular Motion due to gravity can actually be decoupled from that of Coriolis force. 

In other sense, we consider Coriolis term as an adiabatic perturbation. Thus, in addition to usual swing of our pendulum, its x and y positions ( or equivalently, the x-y coordinate axes in a counter direction) are taking a peculiar motion described by:

It’s to be found this motion is actually a rotation of [x,y] with a general solution:

It corresponds to a clockwise (in North Semi-sphere of earth, counter-clockwise in South) rotation of the line traced by our swinging pendulum. The actual rotary frequency of that pendular plane is half of what we’ve got here[3]

Geometrical origin of phase:

Consider you are sitting at northern pole of earth. What’s more, there is a straight rod on you hands. Now, hold your stick, move slowly so as each time you step forward, the stick is parallel transported. The process is called “Parallel Transport” and is in equivalence adiabatic (the stick’s internal state is conserved each step). Imagine you set out from the pole, move down along one longitude to equator, then move to another latitude while remaining on the equator, climb up the respective longitude and finally come back to the same pole, making a closed LOOP. Guess what, you will end up having a global phase difference between the angles of this stick when you start out and when you come back.

This phase difference has nothing to do with how fast you move (so long as it is slow enough) and what the initial angle is. It is purely geometrical, honored by its name “Geometrical Phase”.

The phase is geometrical for its value is totally determined by the area covered with the loop. Not quite. It is the solid angle spanning that area that sets the difference[4]. Usually it will multiply with another constant representing the spin of the traversing system.

Geometrical Phase and Foucault’s Pendulum.

From the local non-inert frame of a pendulum situated at a given latitude Psi , we concluded after one day, pendular plane will make a phase difference (angle) of 2PiCOS(Psi) . Now it can be explained by parallel transport in differential geometry as in somewhere else[5], or by the simpler parallel transport in Geometrical Phase.

Consider a pendulum at latitude Psi. This adiabatic process (or parallel transport) traverse along one longitude to complete a loop when it comes back to where it sets off (we call it a day) with area covered by this loop (enclosed by great circles)

Solid angle spanning this area is known by:

This sign convention is a matter of order of comparison. In exact agreement with one concluded from non-inert frame dynamics.[6]

［1］注意相对速度本身对时间求导时，作为一个依赖参考系基的向量，本身也会引入角速度叉乘项。

［2］参见《经典力学》教程

［3］从几何上而言，是因为在摆动同时存在时，这个系统的旋转角度为2重简并。旋转45度和旋转135度是不可区分的。如果严格求解那个微分方程，可以得到数学上相同的结果，也既是1/2 倍所得旋转角速度。

［4］参考关于berry phase的著作。

［5］关于微分几何的纯数学讨论，（其实也是平行移动和参数球面绝热过程的分析），见http://blog.renren.com/share/223331581/10898339880 《从傅科摆到矢量平行移动》

［6］对于自旋为1/2的电子，或者锁定动量方向的光子（2 level system），其在参数球面上环形移动局部变量不变导致的全局相位差需要再乘以1/2的因子。事实上量子几何相位中，相位变化为（自旋＊立体角）。这和量子力学中的旋转定义一致。参考berry's phase， Pancharatnam 给出的偏振光在poincare 参数球面移动而形成的Pancharatnam‘s phase。