Mathematical Modelling

Mathematical modeling is just describing physical systems using equations (including differential equations) and where applicable solving those equations.

Solutions to these equations can be obtained analytically or numerically. Analytic solutions are typically in the form of functions, for example e^x is an analytic solution of the differential equation dx/dt = x. Numeric solutions are obtained via computation by turning a differential equation into a difference equation (which is essentially a discrete form of differential equation) and solving the difference equation in an iterative fashion to build up a numeric solution. For example, the differential equation,

dx/dt = x can be turned into the difference equation

x_(i-1) - x_(i+1) = x_i (I can't guarantee this particular equation is exactly correct, but you get the idea)

We can specify x_0, x_1, x_2 etc. as boundary conditions and solve for the rest of x. Clearly, the more points we use, the more accurate our calculation will be, which is why computers are used in this type of modeling.

Coordinate geometry comes in when it comes to defining the variables of the system of interest. Typically you would like to choose a coordinate system that minimises the number of variables. The most common coordinate systems are Cartesian (x, y and z), Cylindrical (x, theta and z) and Spherical (x, theta and phi), the most appropriate system will depend on what symmetry (if any) exists in the physical system. We can even get clever and use coordinate systems that move or rotate with time to further simplify our problem.

Claude.

 

Here is an example, if you know calculus and introductory physics.

1) Taken Newton's law F = ma and write it as F = m x''(t), x(t) is a particles position and so x'(t) is its velocity and x''(t) is acceleration.

2) Consider the following assumption (Hooke): the force exerted by a spring is proportional to its extension, and opposite in direction to that extension. In equation form this says F = - k x, where k is a constant of proportionality, x is the extension from equilibrium, and the minus sign insures that the force is opposite the extension.

3) Newtons second law now becomes m x''(t) = -k x(t). Take the case k = m = 1 and this reduces to x'' = -x, or "the second derivative is the negative of the original function". Functions that have this property are the sine and cosine, which upon reflection seem to describe the oscillation of a spring quite well.

 

You should start with coordinate geometry. To describe collisions you need to be able to say where things are in a mathematical way, like an address on a piece of graph paper. The coordinates are often called x, y, z for the 3 directions we have ( up, down, sideways). So you can describe a position with 3 numbers ( called a vector). Velocity in each direction is just the change of position per unit of time. Newton's laws tell us how things behave when they are pushed around or collide.

 

You need basic coord geometry and real analysis then

http://theory.uwinnipeg.ca/physics/
www.glenbrook.k12.il.us/gbssci/Phys/Class/BBoard.html[/URL]

or a book for less than $15

[URL]https://www.amazon.com/Basic-Physics-Self-Teaching-Guide-Guides/dp/0471134473[/URL]

 

Unfortunately it's the opposite case:
For most of the problems in mechanics u can write down diff. eq.,but for very few of them we can find their solutuions exactly.

 

Newton's laws ARE the differential equations.

 

well if you think of that there are only two rules of newton.
i wonder if you can derive the third rule (equal and opposite reaction) from F=dp/dt, it will be neat if we had only one rule from which to deduce the others.

p.s
im replying to integral's last comment.

 

How does this relate?

 

Numeric methods give approximate solutions - you can increase the accuracy by increasing the number of discrete points over a given interval.

Numeric solutions are not in the form of functions however as analytic solutions are. They will be in the form of a finite set of discrete values. These values can specify coordinates on a graph, or coefficients in a series, such as a Taylor or Fourier series.

Claude.