Re: Rigid Body dynamics and solution methods

Just to sum up what I really have done... It seems like this could be an
advantage for me and you.

Now I solve the systems of equations:

M * mu = - F - T

to find the constraint force multipliers and next calculate the
accelerations :

ublas::vector<double> accel = -dV + prod(trans(dG), mu);
for (unsigned int i = 0; i < m.size(); ++i) {
accel(i) = accel(i)/m(i);
}

now it should be possible to use a method for solving the equations of
motion!


Is this a bad method?

It is not completely wrong, but it has enabled you to ignore a very pertinent
fact, namely that the accelerations depend on the velocities. In a pure
rigid-body case these terms give rise to centripetal and Coriolis-like
forces.

The accelerations are calculated according to the velocities.

If you attempt to integrate such a Hamiltonian in generalised
co-ordinates using standard velocity-Verlet or any other straightforward
integration algorithm you will find that energy is not conserved. This
is because you are implicitly using the velocities from step n but
co-ordinates from n+1 to evaluate the accelerations. Please re-read
my earlier posting where you will find a better discussion of this issue.


Just to make it clear:

Using vel-verlet I first calculate the acceleration using the same time-step of both velocities and accelerations.

Afterwards for calculating v(t+1/2), I only calculate the acceleration like:
a_i = -dV_i/m_i

and therefore leave out the parts of the constraint forces. This is due to the fact that I doesn't know the velocity yet.


And by re-reading your earlier posting you refer me to SHAKE - but then again and a lot of reading's.

You refer to a couple of articles:
A. K. Mazur, Journal of Computational Chemistry 11, 1354-1364 (1998)
K. Refson, physica 131B, 356-266 (1985)

I can't seem to find any of those at
http://www3.interscience.wiley.com/cgi-bin/bookhome/93517240
http://www.sciencedirect.com/science/journal/09214526


I should have access to both journals from the university network, but I cannot really find any of those articles. Could you check the spelling, or maybe direct me to the correct journals?


Thanks for helping.


/ Preben
.

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