Gravitational Field - Classical Mechanics

Classical Mechanics

In classical mechanics as in physics, the field is not real, but merely a model describing the effects of gravity. The field can be determined using Newton's law of universal gravitation. Determined in this way, the gravitational field g around a single particle of mass M is a vector field consisting at every point of a vector pointing directly towards the particle. The magnitude of the field at every point is calculated applying the universal law, and represents the force per unit mass on any object at that point in space. Because the force field is conservative, there is a scalar potential energy per unit mass, Φ, at each point in space associated with the force fields; this is called gravitational potential. The gravitational field equation is

where F is the gravitational force, m is the mass of the test particle, R is the position of the test particle, is a unit vector in the direction of R, t is time, G is the gravitational constant, and ∇ is the del operator

This includes Newton's law of gravitation, and the relation between gravitational potential and field acceleration. Note that d2R/dt2 and F/m are both equal to the gravitational acceleration g (equivalent to the inertial acceleration, so same mathematical form, but also defined as gravitational force per unit mass). The negative signs are inserted since the force acts antiparallel to the displacement. The equivalent field equation in terms of mass density ρ of the attracting mass are:

which contains Gauss' law for gravity, and Poisson's equation for gravity. Newton's and Gauss' law are mathematically equivalent, and are related by the divergence theorem. Poisson's equation is obtained by taking the divergence of both sides of the previous equation. These classical equations are differential equations of motion for a test particle in the presence of a gravitational field, i.e. setting up and solving these equations allows the motion of a test mass to be determined and described.

The field around multiple particles is simply the vector sum of the fields around each individual particle. An object in such a field will experience a force that equals the vector sum of the forces it would feel in these individual fields. This is mathematically:

i.e. the gravitational field on mass m_j is the sum of all gravitational fields due to all other masses m_i, except the mass m_j itself. The unit vector is in the direction of R_i − R_j.