Dear Marek,

    What would be the advantage that you could find to work with 9N degrees of freedom instead of 9N? Are you thinking in N rigid or free classical, quantum or spinors (Dirac, Weyl, Majorana, ...) bodies?

It then amounts to 12 N.  The Hamilton and Lagrange formalism works with any number of degrees of freedom.  But the phase space is always even dimensional since there is a (generalised) momentum associated to each (generalised) coordinate. In this case, there are the three Euler angles and the three components of the angular momentum.

Rigid bodies is treated in introductory university courses on mechanics. And you can f.i. find a chapter devoted to it in the text by V.I. Arnold, Mathematical Methods of Classical Mechanics.

The contributions from rotational (and vibrational) degrees of freedom of polyatomic molecules is a standard treatment in Statistical Mechanics courses. But, in f.i. a many-body modelling of the solar system dynamics, the coupling between positional and orientational degrees of freedom is so weak that it can be ignored in the first approximation. But it not entirely absent, as demonstrated by the bound rotation of the moon.

The Lagrangian (kinetic energy part) for the orientational part of a general rigid body can be found in eq. (59) of the linked file. The orientation is described by the three Euler angles, the kinetic energy is quadratic in the time derivatives of these angles, but also depends on the angles themselves, and the inertia tensor of the body. Rather messy altogether; hence the general case is not written out in most treatments I found on the web.

http://www.lehman.edu/faculty/dgaranin/Mechanics/Mechanis_of_rigid_bodies.pdf

There are applications--for instance in magnets-where the elementary variables are the magnetic moments, that have three components. Then N such moments would have require a 3N-dimensional phase space. The appropriate mechanics for odd-dimensional phase spaces  was introduced by Nambu in 1973, cf. http://inspirehep.net/record/81196

I think 6N phase space is the basic description of object motion. The motion of every point of the object have been described by every 'ponit' of phase space. Even if the object is rigid body having anglur momentum, the translation and rotation of the object can be completely described by 6N phase space. 

Jun> Even if the object is rigid body having anglur momentum, the translation and rotation of the object can be completely described by 6N phase space. 

How? Consider a single rigid body, N=1, and tell me how you will construct generators which rotate its intrinsic spin. One may of course construct a rigid body from many point-like objects, and impose (fixed-distance) constraints between them, but this was perhaps not the point of Marek's question(?).

Excuse me, if generators means canonical hamiltonian generators? For a single rigid body, the time change rate of its intrinsic spin should be equal to the torque applied to it. If the torque is the generator?

Angular momentum is a convenient way to describe momentum of every point. If to change positon to angle and momentum to angular momentum, kinetic theory of rotation can have the same formulas as that of translation. So there may be 6N phase space for rotation. I understand Marek's point is to find the general formulation of phase space for translation and rotation. But I think to separately treat translation of every point and rotation of intrinsic  spin of every point of a object or multiple objects is better. 

   It is clear that we can find a phase space of 9N instead of the 6N, assuming that the angular momentum is represented by three "coordinates" (which in general is not true because the angular momentum is 2-tensor and not a vector)? But for what? What would be the advantage to have the information coded in this form when the angular momentum L=r x p might be obtained directly from the 6N coordinates?

   Another thing very different of this classical point of view is we thought in spinors written within a Clifford algebra and we want to represent the operators position and momentum.

Spin does not exist in classical particles. Orbital angular momentum is written in terms of position and linear momentum, so it is not an extra independent variable.

Let me clarify my question and explain how it come to my mind. We usually solve mechanical problems for either rotating bodies (but not translating at the same time) or for translating ones - disregarding their rotation.  This is fine in many situations, but certainly not always.  Think about the ping-pong or football ball - its rotation severely influences the shape of trajectory.  Well, if you have objections to this example because of the air around and its involvement then think about billiard bills instead.  Or about two toy cars (without engines of any kind) on the inclined plane, both having the same mass and being initially at rest, but differing in internal structure: namely only one is equipped with flying wheel, hidden inside and thus not visible for observer.  Not knowing about the presence of this flying wheel we will be surprised with differences in their motion.  This last example takes us close to the problem of electron spin, not expressible as a vector product of two well known vector quantities, and, perhaps, to spin-orbit coupling (I would rather prefer the name "spin-trajectory", since orbit suggests a closed path).  Ok, maybe we should consider even 12N-dimensional description (3N position, 3N momenta, 3N angular momenta and 3N angular accelerations) -- but 9N seems a minimal choice.  I don't see how this approach could produce "quantized" (i.e. discrete) values of angular momenta, but - who knows?  The real question is what extra equations, besides those given by Hamilton, would be needed?  Let's assume that angular momentum is a primary property of a chunk of matter (like its position), thus not necessarily expressible in the way we know today.  A marriage of classical and quantum mechanics is in sight then!

Hello,

Good question, I liked your insight..

You'll find that this was attempted long ago 

https://arxiv.org/abs/quant-ph/9601011

 However, I would love to see you publishing an extended study in this regard ! 

Best

Dear Salwa,

Indeed, very interesting article.  Thank you!  The authors seem to share my feelings, but they concentrate first of all on Zitterbewegung, then "discover" antimatter particles (as a byproduct) and quietly ignore the case of ordinary massless photon.  However, at the end authors write: The lesson is clear: even if the kinematics of the spinning particle traces that of the spinless case, there are additional degrees of freedom present, and these have to be taken into account to implement the canonical formalism.  In this sense, the study of spinning particles in ordinary (x,p) phase space is hopeless.

Sounds like rather sad conclusion, but in reality this opinion is in full accordance with my question.

Usually to model spin one uses spinors - `things that Clifford algebras act on'.  More than this, the spin statistics theorem tells us that the quasi-classical limit of 1/2 spin degrees of freedom are anticommuting objects.

Thus, the quasi-classical mechanics of spinning particles is based on the theory of supermanifolds.  To my mind, these 'generalised manifolds' represent a half-way house between the true classical (no spin) world and the true quantum or noncommutative world.

Very loosely, a supermanifold is a `manifold' with coordinates (x^\mu  \theta^i) where the important thing is that \theta^i \theta^j  =  -  \theta^j  \theta^i.  

One can develop almost all the theory of Hamiltonian and Lagrangian mechanics on supermanifolds,  as a direct generalisation of the classical case, or one can just work formally and generalise things. The subtlety is with the notion of (phase or configuration) trajectories, this requires a more categorical approach. I wrote a paper on this not so long ago...

https://arxiv.org/abs/1606.02604

(This has been accepted for publication and I will update the arXiv version soon.)

But I am not sure this is really what you had in mind.

Here's a recent paper describing mechanics of Vlasov-Maxwell system with spin (wave packets interpreted as particles with internal degrees of freedom) 

https://arxiv.org/abs/1611.03064

Dear Dr Hirvijoki,

I read the above reference, could anyone describe the essence?. Are there such also for mechanical systems, easy to read? I think I will disagree with the discretization, and also the extraction of eigenfrequency, the latter is detetmined by the 'linear'(harm osc') part of Hamiltonian, and the particle location should be at one side of the boundary, but the equivalent mass or other measure for impact on boundaries will be indeterminate.

How many more dof than at beginning is such approach? And does it imply a discretization of space to skew cells, where the inertia is at one side?

Concerning the question, I have written about aqua plane, in some articles.

For other cases, classical, the material points could be considered homogenous, but I agree somewhat with the formulation of the first question,  in the sence that one maybe assume that a priori, too often, and also design things symmetric from a habit, and since easy to transport.