The Field Equations of Newton: Understanding the Basics

[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)

 

[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).

 

[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.

 

[avito009]

it assigns a particular value and direction of the force of gravity to every spatial location.

Do you mean that the spatial 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?

 

[Orodruin]

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".

 

[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.

 

[voko]

Do you mean that the spatial location is the r (Distance from centre of object of mass M)?

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.