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{excerpt}In our version of [modeling physics|Models in Physics], a physical model consists of one or more describes the [system|system], the state of its constituents (including perhaps geometric and temporal structure), their internal and external interactions, and has [Laws of Change|law of change] along withthat determine the changes of state (i.e. behavior).  Models combine the conditionsdefinitions, forconcepts, theirprocedures, applicationinteractions, including restrictions on the [systems|system] which are compatible with the [Law(s) of Change|law of change] and the [interactions|interaction]laws of nature and other relationships that model some aspect of the physical world.  Models intermediate between laws of nature, which are relevantrelationships toamong theabstract [Lawquantities, of Change|law of change].and experimental/experiential reality.  {excerpt}

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h2. ManyProperties of Definitionsa ofPhysical Model

TheA wordphysical "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) involving some class of relevant [interaction|interaction] that can be applied to [systems|system] which meet certain specified restrictions.  model is a mentally linked collection of physical laws, concepts, equations, and associated descriptions that relate to a particular common pattern found in nature.  Examples are motion with constant acceleration, harmonic motion, mechanical energy conservation, and applying ΣF = ma to a point particle.  A model consists of the following pieces:

1.    the physical systems/situations where the model applies and vocabulary of involved objects, state variables, and agents (interactions) involved.
2.    specification of the independent and dependent (measurable) state variables that characterize the system and which the model interrelates
3.    what physical theories underlie the model and the resulting Laws of Change
4.    the behavior/change in state (geometric and temporal) and interaction structure
5.    descriptions of the model and interpretation of its predictions as expressed in all various useful representations

h2. Law of Change


h4. Definition

A Law of Change is an equation which represents the _time evolution_ of some property of a system.  

h5. Example -- Momentum

For example, the equation:

{latex}\begin{large}\[ \vec{p}_{f} = \vec{p}_{i} + \int_{t_{i}}^{t_{f}} \vec{F}^{\;\rm ext}\;dt\]\end{large}{latex}

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).

h4. Integral vs. Differential

Many Laws of Change can be equivalently expressed using derivatives or using integrals (or using explicitly integrated quantities).  

h5. 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:

{latex}\begin{large}\[ \frac{d\vec{p}}{dt} = \vec{F}^{\;\rm ext}\]\end{large}{latex}

h2. Hierarchy of Models


h4. 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.  

h4. 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:

{latex}\begin{large}\[ \frac{d\vec{p}}{dt} = \vec{F}^{\;\rm ext} \]\end{large}{latex}

For a [point particle] system, the momentum can be written as:

{latex}\begin{large}\[ \vec{p} = m\vec{v}\]\end{large}{latex}

where the *mass is constant*.  Thus, we can write:

{latex}\begin{large}\[ \frac{d\vec{p}}{dt} = m\frac{d\vec{v}}{dt} = m\vec{a} = \vec{F}^{\;\rm ext}\]\end{large}{latex}

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.