Physics+ Newton Laws, Central Field: UNIZOR.COM - Physics+ - Laws of Newton

Notes to a video lecture on UNIZOR.COM

Central Force Field

The subject of this lecture is a theory behind practical problems of space travel around some gravitating mass or a movement of an electron around an electrically charged nucleus, or other types of movement in a central force field, where the force at any point A is always directed towards or away from some central point O and its magnitude is the same at all points that lie on the same distance from the center O.

Consider an inertial reference frame in our three-dimensional space with origin at point O and a force field F defined in all points with the following conditions satisfied:
(a) for any point A in our space the vector of force F(A) at this point is always colinear with position vector r=OA;
(b) for any two points A and B lying on the same distance from the center O vectors F(A) and F(B) have the same magnitude and both directed either towards or away from the center O.

Obviously, from condition (b) follows that all force vectors at all points lying on the same distance around center O have the same magnitude and all of them are directed the same way relatively to center O - either towards it or away from it.

Defined in such way, the force field is called central force field.

The vector of central force at any point A is defined by position vector r=OA and can be expressed as a scalar f(r) that depends only on the distance r=|r| of a point A from the origin of coordinates (positive or negative to differentiate between the directions towards or away from center O) multiplied by the vector of position r.
F(A) = F(r) = f(r)·r.

Imagine that at time t=0 some object of point mass m is at position defined by a vector r₀ and its initial velocity is v₀.

Velocity vector, by definition, is the first derivative of position
v(t) = r'(t)

Vector of acceleration is the first derivative of velocity by time or the second derivative of position
a(t) = v'(t) = r"(t).

Now we can express the Newton's Second Law as
F = f(r)·r = m·a = m·r"
and express an acceleration in terms of a central force and mass
a = f(r)·r/m

Consider a vector of angular (rotational) momentum of motion of our object L defined as the vector (cross) product of the position vector r(t) by the momentum of motion p=m·v(t):
L = r⨯(m·v) = m·r⨯v
and analyze how this angular momentum changes with the time.
More precisely, let's prove that the angular momentum of an object in a central field is a constant (the Law of Conservation of Angular Momentum in Central Field).

The first derivative of an angular momentum L is
L'(t) =
= m·[r(t)⨯v'(t) + r'(t)⨯v(t)]
Take into account that the derivative of a position is a velocity and the derivative of a velocity is an acceleration.
Then L'(t) =
= m·[r(t)⨯a(t) + v(t)⨯v(t)]

The second component in the above expression is a zero-vector because the vector product of collinear vectors is always a zero-vector.
Substituting acceleration for its expression in terms of force, position and mass a=f(r)·r/m, we obtain
L'(t) = f(r)·r(t)⨯r(t) = 0
because, again, a vector (cross) product of two collinear vectors is a zero-vector.

As we see, the derivative by time of the vector of angular momentum of an object in a central force field equals to zero-vector, which means that the vector of angular momentum in a central force field is a constant.

Notice that in our proof of the Angular Momentum Conservation Law we relied only on the central character of the force field, not its specific form for gravitational or electrostatic fields.