Hi, I'm wondering how to formulate Newton's second law in terms of Dirac delta functions and standard mass density distributions. We define the Dirac delta function of a point mass with mass m located at point x0 as follows: [tex]\rho(x)=m\delta (x-x_0)[/tex] If our point mass is represented by a Dirac delta function, our acceleration will need to be too. So we need an acceleration distribution for our point mass that assigns an instantaneous acceleration a to x0: [tex]a(x)=a\delta (x-x_0)[/tex] If that is correct, then is the correct Dirac delta function formulation of f=ma: [tex] F_i = [\int m\delta (x-x_i)dx] [\int a\delta (x-x_i)dx][/tex] ...for the force of particle i? And, for the total force FT for N particles, is it: [tex] F_T = \sum_i [[\int m\delta (x-x_i)dx] [\int a\delta (x-x_i)dx]][/tex] Okay, so if that's all good, is this the equation for standard mass density distributions (not for point particles, but for mass densities distributed over tiny regions): [tex] F_T = \sum_i [\int [\rho(x) a(x)]dx][/tex] Where ρ(x) is now defined at dM/dV and a(x) as da/dV. Notice that the Dirac delta formulation has TWO integrals while the mass density formualtion has ONE integral. I think that's right because you cannot multiply delta functions because, pre-integration, that would be multiplying infinities? Whereas with the densities, you are instead multiplying finite values and so you CAN multiply BEFORE the integration? Or am I just completely confused here? Any advice would be most welcome!