# Rotational degrees of freedom¶

## Overview¶

HOOMD-blue natively supports the integration of rotational degrees of freedom. Every particle in a hoomd simulation may have rotational degrees of freedom. When any torque-producing potential or constraint is defined in the system, integrators automatically integrate both the rotational and translational degrees of freedom of the system. Anisotropic integration can also be explicitly enabled or disabled through the aniso argument of hoomd.md.integrate.mode_standard. hoomd.md.pair.gb, hoomd.dem, hoomd.md.constrain.rigid are examples of potentials and constraints that produce torques on particles.

The integrators detect what rotational degrees of freedom exist per particle. Each particle has a diagonal moment of inertia tensor that specifies the moment of inertia about the 3 principle axes in the particle’s local reference frame. Integrators only operate on rotational degrees of freedom about axes where the moment of inertia is non-zero. Ensure that you set appropriate moments of inertia for all particles that have them in the system.

Particles have a number of properties related to rotation accessible using the particle data API (hoomd.data):

• orientation - Quaternion to rotate the particle from its base orientation to its current orientation, in the order $$(real, imag_x, imag_y, imag_z)$$
• angular_momentum - Conjugate quaternion representing the particle’s angular momentum
• moment_inertia - principal moments of inertia $$(I_{xx}, I_{yy}, I_{zz})$$
• net_torque - net torque on the particle in the global reference frame

GSD files store the orientation, moment of inertia, and angular momentum of particles.

## Quaternions for angular momentum¶

Particle angular momenta are stored in quaternion form as defined in Kamberaj 2005 : the angular momentum quaternion $$\mathbf{P}$$ is defined with respect to the orientation quaternion of the particle $$\mathbf{q}$$ and the angular momentum of the particle, lifted into pure imaginary quaternion form $$\mathbf{S}^{(4)}$$ as:

$\mathbf{P} = 2 \mathbf{q} \times \mathbf{S}^{(4)}$

in other words, the angular momentum vector $$\vec{S}$$ with respect to the principal axis of the particle is

$\vec{S} = \frac{1}{2}im(\mathbf{q}^* \times \mathbf{P})$

where $$\mathbf{q}^*$$ is the conjugate of the particle’s orientation quaternion and $$\times$$ is quaternion multiplication.