Force on individual membrane points I am working on the dynamics of a membrane under various forces. My membrane is made up of discrete points $(x_i,y_i)$ with $i = 1,2,\cdots n$. I can model the total energy of the membrane as made up of different components, corresponding to bending, pressure and tension. Say these individual components of membrane energy are $Em_1,\; Em_2,\; Em_3$. The total energy $Em$ is the sum of the three components. I have been taught that I can get the membrane points by solving $\gamma\frac{dx_i}{dt} = -\frac{\partial Em}{\partial x_i}$ and $\gamma\frac{dy_i}{dt} = -\frac{\partial Em}{\partial y_i}$. However, I don't understand the physics behind this equation, since I haven't seen it anywhere before. Is $-\frac{\partial Em}{\partial x_i}$ the force on point $x_i$? The expressions for $Em_i$ are purely algebraic, they don't have any time derivatives. Won't that mean the force is constant, which shouldn't be the case since the membrane is evolving in time?
 A: This looks like a relaxation equation, with an empirically pre-determined friction coefficient $\gamma$, intended to make the nodes of the membrane move towards positions at which the net force on each node $(x_i,y_i)$ is zero. The physics is essentially that of particles moving through a viscous medium, so that the frictional forces (proportional to velocity) exactly balance the forces derived from the membrane energies. This is not Newtonian mechanics: there is no consideration of  particle mass or acceleration. Or, if you prefer, you can imagine that the masses are so small, and hence the accelerations so large, that the velocities rapidly evolve until the frictional forces exactly balance the ones derived from the $Em$ terms.
Your expressions $Em_1$ etc will typically depend on several of the node coordinates. You analytically differentiate each $Em$ term with respect to each of the coordinates, and apply the negative sign, to give forces acting on each of them, as you suspected. There is no time differentiation of the $Em$ terms, as you say. Sum up all the forces acting on each node: call these $(X_i,Y_i)$ (the right hand sides of your differential equations). I guess the aim is to numerically solve the differential equations: crudely, something like $\Delta x_i = \gamma^{-1} X_i \Delta t$ and similarly for the $y_i$. This process then needs iterating: recalculate the forces and once more advance the positions. As the positions change, so will the forces. It's basically a minimization problem (for the energy, relative to the positions of the nodes) tackled by inventing a differential equation that leads "downhill": effectively a steepest descent method.
The details depend on the exact advice you have been given regarding the method. For example, the values of $\gamma$ and $\Delta t$ need specifying, and you might even adjust $\Delta t$ during the process. The coefficient $\gamma$ might have some physical significance, but equally this might just be a convenient numerical method to achieve the desired result. The outcome should be a set of nodal positions $(x_i,y_i)$ on which the forces are zero, and hence they constitute a local minimum with respect to the total energy.
