Force was first described by Archimedes, a Greek in the Hellenic civilization. The total (Newtonian?) force on a point particle at a certain instant in a specified situation is defined as the rate of change of its momentum:
F = Limit as T goes to zero of (mv - mvo)/T
Where m is the inertial mass of the particle, vo is its initial velocity, v is its final velocity, and T is the time from the initial state to the final state.
Force was so defined in order that its reification? would explain the effects of superimposing situations: If in one situation, a force is experienced by a particle, and if in another situation another force is experience by that particle, then in a third situation, which (according to standard physical practice) is taken to be a combination of the two individual situations, the force experienced by the particle will be the vector sum of the individual forces experienced in the first two situations. This superposition of forces, along with the definition of [inertial frame]?s and inertial mass, are the emperical content of [Newton's Laws]?.
Since force is a vector it can be resolved into components. For example, a 2D force acting in the direction North-East can be split in to two forces along the North and East directions respectively. The vector-sum of these component forces is equal to the original force.
The content of above definition of force can be further explicated. First, the mass of a body times its velocity is designated its momentum (labeled p). So the above definition can be written:
F = Δp/Δt
If F is not constant over Δt, then this is the definition of average force over the time interval. To apply it at an instant we apply an idea from Calculus. Graphing p as a function of time, the average force will be the slope of the line connecting the momentum at two times. Taking the limit as the two times get closer together gives the slope at an instant, which is called the derivative:
F = dp/dt
Many forces are thought of as being produced by [potential field]?s. For instance, the gravitational force acting upon a body can be seen as the action of the [gravitational field]? that is present at the body's location. The potential field is defined as that field whose gradient is the force produced at every point:
F = -∇U
In most expositions of mechanics, force is usually taken as a primitive, without an explicit definition. Rather it is taken to be defined implicitly by the (often vague) presentation of the theory within which it is contained. Various physicists, philosophers and mathematicians, such as [Ernst Mach]?, [Clifford Truesdell]? and [Walter Noll]? have contributed to the intellectual effort of obtaining a more rational, non-circular, and explicit definition of force.
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