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## Main Question or Discussion Point

I know it's just a mental block - for some reason I can't get the phrase "restoring force" out of the ongoing debate in my noggin. But this is a question about symmetry specifically, and the notion of a force appearing because of an asymmetry, or "symmetry-breaking".

The simplest notion I have of the principle of broken symmetry is a coupled pendulum. A single string pendulum with one point of connection is more symmetrical than a pendulum hanging from a spring coupled at either end (which is another pendulum).

With the constrained motion introduced to the "system", the pendulum will "relax' into a mode that's a linear, perpendicular one - to the line made by the spring length. The spring breaks the symmetry, and "absorbs" any motion of the pendulum except transverse to the plane formed by it + the hanging weight. Any motion of the weight parallel to the plane of the spring does work on the spring, making it "rotate' in the plane, and energy is lost to frictional coupling etc. This is the "resulting force" from the asymmetry.

How does the symmetry argument apply to precessional rotation? A gyroscope that isn't spinning is more symmetrical in the inertial field of planet earth. The symmetry is broken when it's given some angular momentum - but the precession rate is independent of the rotation of the inertial field - i.e. planet earth?

I'm looking for a succinct explanation of the symmetry (of precessional rotation) and what breaks it - without referring to the torque as a time derivative except as a result of the asymmetry produced by the angular momentum of the precessing body...?

The simplest notion I have of the principle of broken symmetry is a coupled pendulum. A single string pendulum with one point of connection is more symmetrical than a pendulum hanging from a spring coupled at either end (which is another pendulum).

With the constrained motion introduced to the "system", the pendulum will "relax' into a mode that's a linear, perpendicular one - to the line made by the spring length. The spring breaks the symmetry, and "absorbs" any motion of the pendulum except transverse to the plane formed by it + the hanging weight. Any motion of the weight parallel to the plane of the spring does work on the spring, making it "rotate' in the plane, and energy is lost to frictional coupling etc. This is the "resulting force" from the asymmetry.

How does the symmetry argument apply to precessional rotation? A gyroscope that isn't spinning is more symmetrical in the inertial field of planet earth. The symmetry is broken when it's given some angular momentum - but the precession rate is independent of the rotation of the inertial field - i.e. planet earth?

I'm looking for a succinct explanation of the symmetry (of precessional rotation) and what breaks it - without referring to the torque as a time derivative except as a result of the asymmetry produced by the angular momentum of the precessing body...?