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Let $E(x,y,z)$ denote the gravitational field,

then in an uniform gravitational field( $E(x,y,z) = k$), the equation

$\mathbf{s(t)} = \mathbf{u}t + \frac{1}{2}kt^2 $ holds (where $\mathbf{s}$ is the displacement as a function of time and $\mathbf{u}$ is the initial velocity).

So, my question is:

Is it possible to obtain an expression which relates $s(t)$ with $E(x,y,z)$ in a non-uniform gravitational field.

The answer, I believe is no. According to me this is an iterative process.

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  • $\begingroup$ Of course it is not. The equation of motion you mentioned is valid only if the acceleration is constant which in this case is not because the gravitational field is not constant $\endgroup$ Sep 25 '16 at 14:19
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The answer depends on the form of $E$: for some forms you can solve the differential equations exactly, for some, obviously, you can't.

A rather famous example where you can is for a particle moving in the field of a point mass (or of a spherically-symmetric distribution of mass). Some guy called Newton discovered this.

And it turns out that you can solve the problem of two point-masses (spherically-symmetric distributions of mass) moving in each others' fields, but not in general that of three (or more). This again is a very famous result.

All this is very well known: it would have been a good idea to do some research (on, say, Wikipedia) before asking.

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