Diracs delta equation - general interpretation

[finitefemmet]

Im really just searching for a general explanation!

If you are solving a pretty standard left hand side differential equation, but a diracs delta function on the right hand side. I am abit confused about how to interpret this.

If this is the case for the right hand side:

r(t) = Diracs (t) ,for 0≤ t<T with the period T=2∏

Think of this as an periodic outside force on a spring system, now I don't know how to interpret this. Does this mean that r(t) repeats itself, at t=0, t=2∏ and so on. Or that the diracs delta equation only excists between 0 and 2∏?

Since its a diracs delta equation, it cannot work over a longer time interval? Since its an instant impuls over an extremely small time space.

If anyone could shed some light over this, I would be most gratefull.
I am not looking for a solution, just general information on how to interpret this information with the diracs delta function

Thank you, and excuse my poor english!

 

[Office_Shredder]

They mean that the function repeats itself

 

[HallsofIvy]

For every positive integer k, let fk(t)=
0 for 2(n-1)pi+ 1/k to 2npi- 1/k,
k/2 for 2npi- 1/k to 2npi+ 1/k

for n any positive integer. The periodic "delta function" is the limit of fk(x) as k goes to infinity. Essentially, that gives a "delta function" at every multiple of 2pi.

 

[finitefemmet]

Thank you both;)

 

[blue_raver22]

The Dirac delta function, also known as the impulse function, is a mathematical tool used to describe a point-like source of energy or force. In the context of a differential equation, it represents an instantaneous change or impulse in the system.

In your example, the Dirac delta function is acting as an external force on a spring system. This means that at specific points in time (t=0, t=2∏, etc.), there is a sudden change in the system due to this external force. The Dirac delta function only exists at these specific points and has no effect on the system in between.

Think of it as a hammer hitting a nail. The force of the hammer is only applied at the moment of impact, not continuously throughout the motion. Similarly, the Dirac delta function only acts at specific points in time, not over a longer time interval.

It is important to note that the Dirac delta function is not a function in the traditional sense, but rather a distribution. It has a value of infinity at the point of impulse and is zero everywhere else. This makes it a useful tool in solving differential equations, particularly in physics and engineering applications.

I hope this helps to clarify the interpretation of the Dirac delta function in a differential equation. It is a powerful tool that allows us to model and understand systems with sudden changes or impulses.