In our version of modeling physics, a model is a specific Law of Change that is applied to a system.
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Many Definitions of Model
The word "model" has many meanings in everyday language, and it has many meanings in physics as well. The Models in Physics page gives a summary of what is usually called the modeling approach to mechanics. In this WIKI, however, we will use a more narrow definition of model. We will use it to refer to a specific Law of Change (which may have more than one equivalent form) that can be applied to systems which meet certain specified restrictions.
Law of Change
Definition
A Law of Change is an equation which expresses the time evolution of some property of a system.
Example – Momentum
For example, the equation:
\begin
[ \vec
_
= \vec
_
+ \int_{t_{i}}^{t_{f}} \vec
^
\;dt]\end
expresses the time evolution of the momentum of a system in terms of the external forces acting on the system. It is therefore a Law of Change (in this case, belonging to the Momentum and External Force model).
Integral vs. Differential
Many Laws of Change can be equivalently expressed using derivatives or using integrals (or using explicitly integrated quantities).
Example – Momentum
For example, the Law of Change from the momentum model that was discussed above is an integral form. This Law could also be expressed as:
\begin
[ \frac{d\vec{p}}
= \vec
^
]\end
Hierarchy of Models
Restrictions to the Law of Change – Sub-models
The [Model Hierarchy] presented in this WIKI classifies some models as sub-models or special cases of other models. These sub-models have a Law of Change which is a special case of the model of which it is a sub-model.
Example – Point Particle Dynamics
For example, the Point Particle Dynamics model is a sub-model of the Momentum and External Force model. The differential form of the Law of Change for the Momentum and External Force model is:
\begin
[ \frac{d\vec{p}}
= \vec
^
]\end
For a point particle system, the momentum can be written as:
\begin
[ \vec
= m\vec
]\end
where the mass is constant. Thus, we can write:
\begin
[ \frac{d\vec{p}}
= m\frac{d\vec
}
= m\vec
= \vec
^
]\end
which is the Law of Change for Point Particle Dynamics. In this way, the Law of Change for Point Particle Dynamics is a special case of the Law of Change for Momentum and Force, and so Point Particle Dynamics is a sub-model of Momentum and Force in the hierarchy.