5.1.3. Rigid Body#
Newton dynamical equations - strong form. Dynamical equations governing the motion of a rigid body, referred to its center of mass
with momentum
Weak form. Weak form of dynamical equations is derived with scalar multiplication of the strong form by an arbitrary test functions
Lagrange equations. Lagrange equations are derived from the weak form, with a proper choice of the weak test functions. The “translational part” is recasted after choosing
Following the same steps show to derive Lagrange equations for a point system, the translational part becomes
being
The “rotational part” is recasted after choosing
Angular velocity
and the inertia tensor as
being the components
The first term becomes
The second term becomes
The third term can be written as the sum of the derivative of a potential function and a generalized force,
The rotational part of the wak form becomes
being
Adding together the contributions of the momentum and the angular momentum equations, the Lagrange equation can be formally written with the same expression found for the system of points,
being