Hessian matrix of the Newtonian potential is zero?

[Chain]

So I'm looking at the hessian of the Newtonian potential:

\partial^2\phi / \partial x_i \partial x_j

Using the fact that (assuming the mass is constant):

F = m \cdot d^2 x / d t^2 = - \nabla \phi

This implies:

\partial^2\phi / \partial x_i \partial x_j = -m \cdot \frac{\partial}{\partial x_j} (d^2 x_i / d t^2) = -m \cdot \frac{\partial}{\partial x_j} (\partial^2 x_i / \partial t^2)

As we can swap the total derivatives for partial derivatives since for Cartesian coordinates:

\partial x_i / \partial x_j = \delta_{ij}

Using the fact that we can swap the order of differentiation for mixed partials (assuming continuity of the partial derivatives) we obtain:

\partial^2\phi / \partial x_i \partial x_j = -m \cdot \partial^3 x_i / \partial x_j \partial t^2 = -m \cdot \frac{\partial}{\partial t^2} \partial x_i / \partial x_j = -m \cdot 0 = 0

Hence I obtain the result that the hessian of the Newtonian potential is zero which can't possibly be correct but I can't find the error in my calculation.

Any help would be much appreciated :)

 

[dextercioby]

What you wrote doesn't make too much sense and the mathematical manipulations are illegal. Acceleration depends on time, coordinate depends on time: a(x) = a(t(x)). Good luck reverting x(t) into t(x).

 

[Chain]

So the problem is in the last step where I swap the order of differentiation because it is not possible to find time as a function of position?

I guess the proper expression for the differential of acceleration with respect to a spatial coordinate is:

\partial a(t(x)) / \partial x = \frac{\partial a(t)}{\partial t} \cdot \frac{\partial t}{\partial x} = \frac{\partial a(t)}{\partial t} \cdot (\frac{\partial x}{\partial t})^{-1}

Which is clearly non-zero.