Hot answers tagged linear-systems
9
The principle of superposition comes from the fact that equations you solve most of the time are made of Linear operators (just like the derivative). So as long as you are using these operators you can write something like
$$ \mathcal L\cdot \psi = 0$$
where $\mathcal L$ is a linear operator and, let say, $\psi$ is a function that depends on coordinates ...
6
It is true up to very high filed strengths. For too high strengths the field itself is not stable, it can create real pairs. It is like a limit on a field strength in a capacitor. The capacitor dielectric can break.
EDIT: Classical Maxwell equations are linear indeed so the principle of superposition is implemented into them. But break of a dielectric can ...
5
Well, surely you can compute it using matrix operations. But it won't be very natural. Let me instead provide you with a very similar solution (based on a similar matrix) that you'll hopefully find useful. It's not new at all (Kirchhoff, 1847) but I think it's not very well known. I first learned about it in this Wu's review paper of Potts model, p. 252. Let ...
4
The problem you describe is (mathematically) similar to blind deconvolution. Given a signal which is the result of blurring an image (a linear operation) and adding noise, blind deconvolution tries to estimate the blur and the image.
As described here, the blind deconvolution process consists roughly of:
Guess the blurring function (transfer function)
...
3
Within the realm of Maxwell's equations, the principle of superposition is exactly true because Maxwell's equations are linear in both the sources and the fields. So if you have two solutions to Maxwell's equations for two different sets of sources then the sum of those two solutions will be a solution to the case where you add together the two sets of ...
3
This question is very hard to answer at a fundamental level, because quantum mechanics seems to be exact so far, yet one cannot be sure in the scientific sense without confirmation that nontrivial quantum computation is possible. If this is so, then one would have to renounce any classical descriptions, at least within the bounds of scientific reason, and it ...
2
Math:
If you have an operator $D$ with
$$D(\Psi+\Phi)=D(\Psi)+D(\Phi),$$
then if $D(\Psi)=0$ and $D(\Phi)=0$, you can also conclude that $D(\Psi+\Phi)=0$. This is the case for the Schrödinger equation, as it reads
$$D(\Psi):=(i\hbar\tfrac{\partial}{\partial t}-H)\Psi=0,$$
where $H$ is linar. For example you certainly have linearity for the derivatives: ...
1
A rule of thumb would be to get rid of unwanted variables. For example, since we're only interested in $\frac{m_1}{m_2}$ and $\theta$, we can get rid of $F_T$.
$$F_T \sin \theta = m_1 g$$
$$F_T \cos \theta = m_1 a$$
Rearrange to get
$$\frac{m_1 g}{\sin \theta}=\frac{ m_1 a}{\cos \theta} \hspace{20mm}\text{ ...Eq 5}$$
I don't know whether getting ...
1
It is not completely clear what you mean by the approach 2. What one can do is to calculate the current via
$$j^{\mu}(\phi,A)=\frac{\delta S_{eff}[\phi,A]}{\delta A_{\mu}}.$$
Here the effective action $S_{eff}$ is a functional of both the source $A$ and the phase $\phi$ of the condensate in the superconductor. Imagine now that you solve the linearized ...
1
One way to think of superposition is this: If particles behave to some degree like waves in the sense that they can never be completely "squeezed down" into actual points, then the waves -- the probability functions -- can add together very much like waves on a pond. So, just as on a pond surface you could combine together large waves with crests a foot ...
1
If you can find it, this one is good: Introduction to Electroacoustics and Audio Amplifier Design by W. M Leach.
Keep in mind that microphones and loudspeakers are (electroacoustic) mechanical systems. For example, the cone has mass and the surround provides a restoring force.
In the book I linked to, an electrical analogy of the mechanical system (and ...
1
While the first part of the question has been answered satisfactorily, everybody seems to confuse the unconditional linearity of the Maxwell equations with the often observed linearity of the constitutive relations for the material law. The field of nonlinear optics is concerned with the behavior of light in nonlinear media where the constitutive relations ...
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