Math Modelling with Differential Equations for Physics

In summary, mathematical modeling is the process of describing physical systems using equations, including differential equations, and solving them analytically or numerically. Coordinate geometry is used to define variables and simplify problems, and is essential in describing collisions in physics. While there may be only two rules in Newton's laws, the third rule of equal and opposite reactions can be derived from F=dp/dt. There are books available that link physics to coordinate geometry and apply mathematical modeling to physics, including ebooks and affordable options.
  • #1
hi guys,
i am an 11th grade student who loves physics and i have been doing lots of advanced problems from books like irodov,etc.

I recently came across something called mathmatical modelling via differential equations.
And my teacher once mentioned that u can solve most of problems of mechanics via placing the system on a coordinate system and then writing differential equations and stuff like that.
Is it true?
Can u solve problems of collisions?
Is this a part of mathematical modelling?

I want to learn more about the use of coordiate geometry in physics especially mechanics.

Please help.
 
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  • #2
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.
 
  • #3
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.
 
  • #4
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.
 
  • #5
Thanx a lot guys.That made lots of stuff clear.
So can u all suggest me some books that link physics to coordinate geometry and books on applying mathemnatical modelling to physics.
Even ebooks would do.
 
  • #6
Thanx a lot guys.That made lots of stuff clear.
So can u all suggest me some books that link physics to coordinate geometry and books on applying mathemnatical modelling to physics.
Even ebooks would do.
 
  • #7
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/dp/0471134473/?tag=pfamazon01-20[/URL]
 
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  • #8
einstein_genx said:
And my teacher once mentioned that u can solve most of problems of mechanics via placing the system on a coordinate system and then writing differential equations and stuff like that.
Is it true?
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.
 
  • #9
so can we get approximate solutions?
i am studying for a competitive exam where options are given.
if i can get approximate solutions that would be gr8.

I wanted to ask something.
Suppose we have a problem of collisions.
is it better to write down 5 to 6 equations using Newtons laws of motion or is there a better shorter method through diff. equations.
 
  • #10
einstein_genx said:
so can we get approximate solutions?
i am studying for a competitive exam where options are given.
if i can get approximate solutions that would be gr8.

I wanted to ask something.
Suppose we have a problem of collisions.
is it better to write down 5 to 6 equations using Newtons laws of motion or is there a better shorter method through diff. equations.


Newton's laws ARE the differential equations.
 
  • #11
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.
 
  • #12
loop quantum gravity said:
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?
 
  • #13
einstein_genx said:
so can we get approximate solutions?
i am studying for a competitive exam where options are given.
if i can get approximate solutions that would be gr8.
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.
 

1. What is the purpose of using differential equations in math modelling for physics?

Differential equations are used to describe the relationship between the variables in a system and how they change over time. In physics, this is important for understanding the behavior of various physical phenomena and predicting future outcomes.

2. What types of problems can be solved using differential equations in math modelling for physics?

Differential equations can be used to model a wide range of physical systems, such as the motion of objects, electrical and mechanical systems, and even population growth. Essentially, any problem that involves change over time can be solved using differential equations.

3. How do you determine the initial conditions in a differential equation for math modelling in physics?

The initial conditions are the starting values of the variables in the system. They can be determined through experimentation or by using known values from previous observations. In some cases, they may also be determined by assumptions or theoretical considerations.

4. What are some key techniques for solving differential equations in math modelling for physics?

There are several techniques for solving differential equations, including separation of variables, substitution, and using integrating factors. It is important to choose the appropriate technique based on the type of differential equation and the specific problem being solved.

5. How can we verify the accuracy of a math model using differential equations for physics?

One way to verify the accuracy of a math model is by comparing the predicted results with real-world data or experimental results. If the model closely matches the actual observations, it can be considered a valid representation of the physical system. Additionally, performing sensitivity analysis and testing the model under different conditions can also help to assess its accuracy.

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