Model
| Excerpt |
|---|
In modeling physics a physical model describes the system, the state of its constituents (including perhaps geometric and temporal structure), their internal and external interactions, and has Laws of Change that determine the changes of state (i.e. behavior). Models combine the definitions, concepts, procedures, interactions, laws of nature and other relationships that model some aspect of the physical world. Models intermediate between laws of nature, which are relationships among abstract quantities, and experimental/experiential reality. |
Properties of a Physical Model
A physical 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
Law of Change
Definition
A Law of Change is an equation which represents the time evolution of some property of a system.
Example - Momentum
For example, the equation:
| Latex |
|---|
| Wiki Markup |
{table:border=1|frame=void|rules=cols|cellpadding=8|cellspacing=0} {tr:valign=top} {td:width=350|bgcolor=#F2F2F2} {live-template:Left Column} {td} {td} {excerpt}In [modeling physics|Models in Physics] a physical model 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] that determine the changes of state (i.e. behavior). Models combine the definitions, concepts, procedures, interactions, laws of nature and other relationships that model some aspect of the physical world. Models intermediate between laws of nature, which are relationships among abstract quantities, and experimental/experiential reality. {excerpt} || Page Contents || | {toc:indent=10px} | h2. Properties of a Physical Model A physical 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
...
...
...
...
...
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:
| Latex |
|---|
}\begin{large}\[ \frac{d\vec{p}}{dt} = \vec{F}^{\;\rm ext}\]\end{large}{latex} h2. Hierarchy of Models h4. Restrictions to |
Hierarchy of Models
Restrictions to the Law of Change - Sub-models
The hierarchy of Models 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:
| Latex |
|---|
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
...
...
...
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
...
...
...
...
.
...
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.
...