The previous two sections presented D-NURBS curve and surface geometry
in a unified way. D-NURBS physics are based on the work-energy version
of Lagrangian dynamics [14]. In an abstract physical
system, let 
be a set of generalized coordinates. These N
functions of time are assembled into the vector 
. Let
be the generalized applied force that acts on 
. We
assemble the 
into the vector 
. We also assume that 
is the concatenation of N vectors 
.
To proceed with the Lagrangian formulation, we will define kinetic energy T, potential energy U, and Raleigh dissipation energy F which are functions of the generalized coordinates and their derivatives. The Lagrangian equations of motion are then expressed as
Variants of this equation have served as the basis for deformable model formulations [36]. Using (10), we can take an arbitrary geometric model, such as a NURBS, introduce appropriate kinetic, potential, and dissipation energies, and systematically formulate a physics-based, dynamic generalization of the model [17].
In the sequel, we will discuss only D-NURBS surfaces (we can consider
D-NURBS curves as a special case with a simpler expression in fewer
variables). To define energies and derive the D-NURBS equations of
motion, let 
be the mass density function defined over the
parametric domain of the surface. The kinetic energy of the surface
is
where (using (9))
is an 
mass matrix. Similarly, let 
be the
damping density function. The dissipation energy is
where
is the damping matrix.
For the elastic potential energy of D-NURBS, we can adopt the thin-plate under tension energy model [35], which was also used in [5, 42] (other energies are possible, including the nonquadratic, curvature-based energies in [36, 19]):
The 
and 
are elasticity
functions which control local tension and rigidity, respectively, in
the two parametric coordinate directions.
In view of (8), the stiffness matrix is
where the subscripts on denote parametric partial
derivatives.
In Appendix B, we show that by applying (10), the D-NURBS equations of motion are given by
where the generalized force vector, obtained through the principle of
virtual work [14] done by the applied force distribution
, is
and where
