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 = (mv - mvo)/T
Where m is the mass of the body, vo is the initial velocity, and v is the final velocity, and T is how long the force is applied.
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.
Similarly one force can be resolved in to its multiple component forces, each of which runs along the axis of a dimension. For example, a 2D force acting in the direction North-East can be split in to two forces along respectively the North and East directions. When these forces are vector-summed this is equal to the original force.
The basic laws governing the behaviour of forces is described in Newton's Laws of Motion.
The above definition works fine, as long as the force acting on the body is constant. Since that is not always the case, a more robust definition is required. First, the mass of a body times its velocity is designated its momentum (labeled p), a distinction that is trivial now, but that becomes important when dealing with relatvity. So the above definition becomes:
F = Δp/Δt
If F is not constant over Δt, then this definition is only an estimate (often a bad one at that). To improve the estimate we borrow an idea from Calculus. Graphing p as a function of time, as long as p is a straight line, the force will be the slope of that line. So, if p doesn't form a straight line, force is still defined as a slope. To find the slope at a point, just take 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. Force due to a potential is defined as the gradient of the field:
F = -∇U
In mechanics, force is usually taken as a mathemetical primitive, something that is defined with certain properties. Various mathematicians such as [Clifford Truesdell]? and [Walter Noll]? have put a considerable amount of intellectual effort to provide rational definitions of force.
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