Re: [ESPResSo-users] mathematical description of the argument rinertia by using rigid arrangement of particles

[Rudolf Weeber]

Hi Lena,
On Fri, Jun 19, 2015 at 07:52:55AM +0000, Holst, Lena wrote:
> I have an question about the argument "rinertia", that can be set by using 
> rigid arrangement of particles. "rinertia" has the elements "x y z".
> Does means x: J_x = sum_i(m_i(x_i^2 + y_i^2))
>                      Y: J_y = sum_i(m_i(x_i^2 + z_i^2))
>                      Z: J_z = sum_i(m_i(x_i^2 + y_i^2))
> Where i reaches from 1 to N (N is the number of virtual sites), m is the mass 
> of one virtual site, and x,y,z the coordinates of the position of the virtual 
> sites, if the point of origin is the position of the non-virtual site. J_x, 
> J_y, J_z are the moments of inertia ordered to the coordinate axis.

So, to my understanding, the three arguments of rinertia are the three main 
inertial moments of the body, i.e., the inertial moments along the main axes of 
inertia, or more formally, the Eigenvalues of the inertial tensor.

The fact that Espresso uses only the main inertial moments implies that the 
main interital axes of the rigid body have to be co-aligned with the 
particle-fixed coordinate system of the central particle.
If the quaternions of the central particle are (1 0 0 0), the particle-fixed 
and the lab-fixed frames coincide.

Thus:
1. Calculate the full inertial tensor of the rigid body
2. Find the main axes of inertia and the respective moments
3. Orient the rigid body such that the main inertia axes co-align with the 
Cartesian axes in the lab frame.
4. Place the center of mass particle with quaternions (1 0 0 0) and attache the 
rigid body from step 3. Set the rinerita of the central particle to the three 
moments along the respective axes.

Please note the current discussion regarding the Langevin thermostat on the 
Espresso developer mailing list.
The discussions regarding the Langevin gamma for particles with different mass 
are also valid for the rotational thermostating of particles with different 
inertial moments.

Regards, Rudolf