What are some applications for the Dirac delta function in classical mechanics?

[ZetaX]

Hey community, are there some application for the Dirac delta function in classical mechanics?
Im interessted in some application of the famous delta function.
If there applications can someone explain it?

Greetings :)

 

[Noctisdark]

Hey community, are there some application for the Dirac delta function in classical mechanics?
Im interessted in some application of the famous delta function.
If there applications can someone explain it?

Greetings :)

As far as I know I has many mathematical application such as solving ODE and PDE, specially used in green function, green function are can solve ODE of the form ∇²Φ = f(x,y,z) this is the famous poisson's equation, these are very important for solving for eq of motion in electric/graviationnal field and the famous Heat equation, but if I can think of a direct application, I'd say gauss's law for electrostatics :D

 

[Dr. Courtney]

It would be an approximation, I think. Mechanical systems are inherently continuous. Zoom in enough and the appearance of discontinuity would become continuous.

 

[George Jones]

Ideal collisions are often considered in first-year physics. In these collisions, momentum is often treated as if it changes instantaneously. Since force is the time derivative of momentum, force is a delta function.

 

[ZetaX]

Ok, are there direct application in classical mechanics like potential in electrostatics or the biot savart law?

thanks :)

 

[George Jones]

What is the charge density of a point particle?

 

[ZetaX]

This can be described by the delta function for example $$\rho(\vec{r})=q\delta(x-a)\delta(y-b)\delta(z-c)$$
Are there similiear application in classical mechanics?

 

[Noctisdark]

There is gauss law for electrostatics
∫∫_∂SE.dS = ∫∫∫_S∇.EdV, E = Q/4πε₀, so ∇.E = Q/ε₀*δ³(r) thus ∫∫∫_SQ/ε₀δ³(r)dV = Q/ε₀ and you get that the electric flux around any closed surface is equal to the charge enclosed over the permeativity of (free) space :D

 

[ZetaX]

But that isn't a application in classical mechanics the electric field described by gauss law ...
Are there no straight forward allplication in classical mechanics or relativistic mechanics like the description of the moment of inertia or something else?

 

[vanhees71]

I'd say, a very good exercise is the evaluation of the Green's function for the damped harmonic oscillator for all cases of damping, including the limit of no damping at all. The equation reads
$$\ddot{G}+2 \gamma \dot{G} + \omega^2 G=\delta(t).$$
For an arbitrary force on the right-hand side you then get
$$\vec{x}(t)=\vec{x}_0(t)+\int_0^{\infty} \mathrm{d} t' G(t-t') \vec{F}(t').$$
Here $\vec{x}_0(t)$ is the general solution for the homogeneous equation, which can be used to adjust the solution to the initial conditions.

You can solve for the Green's function in (at least) two ways:

(a) just directly solve the homogeneous equation, use the retardation condition $G(t)=0$ for $t<0$ to match the two free integration constants such as to get a continuous Green's function with the right unit step in $\dot{G}$.

(b) use the Fourier transform
$$G(t)=\int_{\mathbb{R}} \mathrm{d} \omega \frac{1}{2 \pi} \exp(-\mathrm{i} \omega t) \tilde{G}(\omega),$$
solve the algebraic equation for $\tilde{G}(\omega)$ and then evaluate the integral for $G(t)$ (using contour integration in the complex $\omega$ plane).

 

[ZetaX]

Ok, delta function can be used to solve the harmonic osscilator problem. Are there more application in mechanics for example to derive the moment of inertia about an axis or the center of mass ... or projectile motion?

Thanks

 

[jasonRF]

If you have a complicated rigid body with a weird shaped blob on a moment arm, you can approximate the contribution of the blob by allowing the mass to have a delta function at the position of the center of mass of the blob, with amplitude equal to the mass of the blob. Then the contribution of the blob in the integrals for the moments of inertia are trivial (they become the kind of calculations you do in intro physics!). This is a good way to get ballpark numbers.

Also, any linear ODE/PDE with a driving term can be solved with the same technique (Green's function) Vanhees71 described. If you include fluid mechanics as part of classical mechanics then small acoustic sources are often modeled as delta functions, etc.

By the way, I know mechanical engineers that have spoken of the "hammer" test - in order to understand the resonances of a structure they can strike it with a hammer (approximating a delta function) and measure the movement of the structure (for example using lasers). They are essentially trying to measure the Green's function.

jason