Physics - Energy and Momentum of Rotation

Prerequisites

Kinetic Energy of Solid Body

For purely linear motion the kinetic energy of a solid body is:

T = 0.5 m v2

For purely rotational motion the kinetic energy, about its centre of mass,
of a solid body is:

T = 0.5 wt [I] w

So if we have combined linear and rotational motion can we just add the kinetic
energy due to these parts separately? I think the answer is no because the energy
of a particle is proportional to the square of its velocity. We need to add
on another factor:

T = 0.5 m v2 + 0.5 wt [I] w + m(v(w x r))

This can be derived as follows:

T for particle on solid body = 0.5 sum(mi vi2 )

but the velocity of a particle on a solid body is (v + w x r) so the kinetic
energy is:

T = 0.5 sum(mi (v + w x r)(v + w x r))

this can be expanded out to give:

T = 0.5 sum(mi (v2 + (w x r)(w x r) +2v(w x r))

So, there is a part that is due purely to linear motion, a part that is due
purely to rotational motion, and a part that is due to the product of linear
motion and rotational motion.

Next

Symmetry

Symmetry is an important topic for maths and physics.

Symmetry is important for many branches of mathematics including geometry (see this page) and group theory (see this page). Its importance can become apparent in unexpected places, for example, solving quintic equations.

We say that an object is symmetric, with respect to a given mathematical operation, if this operation does not change the object.

Nothers Theorem (discussed further on this page) says that, for every symmetry exhibited by a physical law,
there is a corresponding observable quantity that is conserved. Virtually every theory, including relativity and quantum physics is based on symmetry principles.

Successive Rotations

When we apply a sequence of rotations in three dimensions and then calculate the resultant total rotation we find it follows laws which may not be intuitive.

For instance, the order of rotations is important, that is: if we apply a sequence of rotations, but just change the order, the result is different.

Where I can, I have put links to Amazon for books that are relevant to
the subject, click on the appropriate country flag to get more details
of the book or to buy it from them.

New Foundations for Classical Mechanics (Fundamental Theories of Physics). This
is very good on the geometric interpretation of this algebra. It has lots of insights
into the mechanics of solid bodies. I still cant work out if the position, velocity,
etc. of solid bodies can be represented by a 3D multivector or if 4 or 5D multivectors
are required to represent translation and rotation.

Commercial Software Shop

Where I can, I have put links to Amazon for commercial software, not
directly related to the software project, but related to the subject being
discussed, click on the appropriate country flag to get more details of
the software or to buy it from them.