Field equations of Newton.

1. Sep 23, 2014

avito009

Am I right when I say during Newton's time there was no idea of fields?

Now I have been looking for books and courses which are meant for amateurs. So I came across this video of one of my favourite professors Prof Leonard Susskind. http://theoreticalminimum.com/courses/general-relativity/2012/fall/lecture-9.

In this lecture he has mentioned about Newtons field equations. How can there be newtons field equations? Can somebody explain me what it means and what the variables stand for?

F= ma= -m∇Φ(x)

a= -∇Φ(x)

2. Sep 23, 2014

ShayanJ

Newton didn't know about fields when he proposed his gravity law. But that doesn't mean his law can't be formulated in terms of fields.
In this formulation, there is a scalar field called $\Phi$ called the gravitational potential and a vector field $\vec g=-\vec \nabla \Phi$ called gravitational acceleration such that a particle at position $\vec r$ has acceleration $\vec g (\vec r)$.

3. Sep 23, 2014

voko

The term "field" did not exist in Newton's time. However, the concept is implicit in Newton's gravitational law, because it assigns a particular value and direction of the force of gravity to every spatial location.

4. Sep 26, 2014

avito009

Do you mean that the spacial location is the r (Distance from centre of object of mass M)? Also how does the vector "a" (Mentioned as "g" by Shyan) have a direction?

5. Sep 27, 2014

Orodruin

Staff Emeritus
The magnitude of the force depends on the distance r between the objects and therefore on where in space the objects are located. Having a direction is what sets vectors apart from normal numbers. In the case of gravity, the force (and hence acceleration) has the direction "towards the gravitating body".

6. Sep 27, 2014

lpetrich

Poisson's equation, $\nabla^2 \Phi = 4 \pi G \rho$, is the appropriate field equation for Newtonian gravity. The potential Φ is a scalar, and g is a vector because it has for each space dimension the gradient of Φ along that dimension.

7. Sep 27, 2014

voko

You cannot say "a spatial location is the distance from something", because there are infinitely many spatial locations at a distance from something, all in different directions. In addition to the distance, you must specify a direction.