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Questions tagged [linear-systems]

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2
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2answers
45 views

Why some forces follow superposition principle?

Let there be a system of $n$ source charges and a test charge $Q$. When we say superposition applies to electrostatic force, we conclude that the interaction between a given source charge and the test ...
0
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2answers
61 views

Are all superposition principles related?

Are all superposition principles related? Is there a relationship between the microscopic superposition principle and the macroscopic superposition principle? Does the microscopic one lend to the ...
3
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3answers
559 views

Why do electrostatic potentials superimpose?

I've been trying to convince myself that the assertion that I've read in basic E&M books (Halliday & Resnick, Purcell), and even Griffiths, that the electrostatic potential at a point in space ...
1
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1answer
37 views

Which Hamiltonian systems are intrisically linear?

What physical properties has a dynamical system whose equation of motion are linear? When does it exist a change of coordinates which turn the equation of motions in a linear system? My teacher says ...
1
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1answer
33 views

Confused over the complex term in the simple harmonic wave equation

I am trying to derive the general equation of Lamb wave. My book says that $$y = A\exp(i(kx−\omega t))$$ is the general equation of simple harmonic wave propagating in +ve $x$ direction. but I am ...
0
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1answer
46 views

Is the space-time curvature linearly additive?

Could someone please show using equations if space-time curvature due to two bodies being linearly additive or not in general.
0
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2answers
40 views

How is the simple damper equation derived?

I know the spring is modeled as $F_{\text{elastic}} = k\cdot x$ when the displacements are small since this is empirically based, but what happens with $F_{\text{damping}}=c\cdot\dot{x}$? It is the ...
0
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4answers
169 views

How does Superposition principle follow from Maxwell's equation's linearity?

It is said that whole of electromagnetism can be completely described by the Maxwell's equations. The thing that intrigues me is that how does superposition principle follow? First, I take an ...
3
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3answers
347 views

Superposition Principle for Electric Fields

If there is a collection of charges $q_1,q_2,q_3....q_n$, and we want to calculate the total Electric Field due to all these charges at a point $P$ ,then the we sum them all up by the principle of ...
0
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1answer
32 views

Wave Superposition on a crystal

Does the principle of superposition apply for electromagnetic waves on a crystal? So I know that the principle applies for any wave but I don't understand why some books say that doesn't apply for ...
0
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0answers
71 views

When and why does the superposition principle of Coulomb's law fail to hold?

In this lecture, Professor Shankar Ramamurthi says that the superposition principle for force vectors of Coulomb's Law is experimentally observed and is not a product of logical analysis. In fact, the ...
1
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2answers
76 views

Should Hamiltonians in quantum field theory be linear operators?

The usual structure of quantum mechanics imposes that Hamiltonians are linear operators. I am not sure if this really holds in quantum field theory. Do non-linear Hamiltonian operators really make ...
0
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1answer
48 views

How do magnetic fields combine?

How do the two fields interact to give the combined field, do they superpose like in waves? And how does this field cause the force on the conductor?
0
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1answer
33 views

Application of linear constant coefficients ODE of the second order [closed]

I've asked this question in math forum. Apparently this question is not welcomed there. So maybe here I can get a proper response. Consider ODE in the form of $$y''+ay'+by=f(t)$$ where $a$ and $b$ are ...
0
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2answers
53 views

Complex solutions to an Underdamped Oscillator

In many of the books talking about damped simple harmonic motion, the underdamped oscillator is treated as follows: Newton's second law says $$m\ddot{x} + r\dot{x} + sx = 0 $$where s is stiffness ...
1
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1answer
36 views

Forced Oscillations and Complex Representation

An oscillating force $F \cos \omega t = \Re\{Fe^{i\omega t}\}$, where $F$ is real, is applied to a mass $m$ on the end of a spring with spring constant $k$. The displacement, $x$, of the particle can ...
1
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1answer
67 views

Bandwidth of a control system

Why is it said that larger bandwidth leads to better command following , better disturbance rejection and speedy response , but the practical bandwidth being limited by external noise?
1
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1answer
59 views

The superposition principle for linear waves

As far as i've seen, the proof for that principle is to show that, the equation representing linear waves has the perk of being linear, thus if y(x,t) and z(x,t) are solutions of the linear equation,...
3
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1answer
54 views

Unpredictability, per definitions of chaotic behavior

Apparently I've been confused about the meaning(s) of "chaotic behavior". I always thought it meant that infinitesimal perturbations of a system parameter would lead to large changes in the system's ...
5
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1answer
105 views

Can a linear system be chaotic?

A chaotic system is a system in which infinitesimal perturbations of a parameter can result in large changes in the behavior of the system. I thought it is not possible for a linear system to exhibit ...
3
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2answers
74 views

What is the difference between solutions to 2nd order homogeneous ODE?

I’m studying Vibrations, and we have two forms to the 2nd order homogeneous ODE: $$mx ̈+kx ̇=0$$ $$x(t)=C_1 e^{iw_n t}+C_2 e^{-iw_n t}$$ and $$x(t)=A\cos(w_n t)+B\sin(w_n t)$$ Even though I can use ...
0
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0answers
34 views

Can light of different frequencies interfere with each other?

In principle I'm aware of superposition and how it works. Nevertheless I'm not really able to answer the following: Will two light beams of different frequencies interfere with each other? More ...
0
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1answer
56 views

Why consider only real part when summing several simple harmonic motions?

I have been studying vibrations and I stumbled upon the overlapping of simple harmonic motions. Consider the case where the number of oscillators $n$ is $n \gg 1$, all of them have the same angular ...
0
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1answer
71 views

Superposition principle in electrostatics

Poisson's equation in electrostatic does not satisfy the linear superposition principle. Can I say that since Laplacian operator is a non-linear operator so it does not follow the linear superposition ...
0
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1answer
93 views

Analyticity of the generalized susceptibility in the linear response theory

In linear response theory, the generalized susceptibility $\chi(\omega)$ is defined as $$\chi(\omega)=\int\limits_{0}^{\infty}\phi(t) e^{i\omega t} dt, ~~t\geq 0\tag{1}$$ where $\phi(t)$ is the ...
0
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2answers
192 views

Assumption in the derivation of Schrödinger's equation

I read on the "derivation" (with some assumptions) of the Schrödinger equation. The idea is to start from $$T + U = E $$ wher $T$ is kinetic energy, $U$ is potential energy, and $E$ is total energy. ...
2
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3answers
141 views

Linear and non-linear systems

When I read about the superposition principle, it says that it works only on linear systems, my problem is that I cannot really understand the difference between a linear and a non-linear system. I ...
2
votes
1answer
52 views

the effects of initial condition on Green function

In literature, for proving the existence of Green function for linear systems, it is argued that if for a linear differential equation like $\mathcal{D}[y] = \sum_{n=0}^N {a_n y^{(n)}}$ $y(0)=y_0, \...
1
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0answers
17 views

What concept expresses the entropy/system_size ratio?

Being entropy an extensive property, I assume that two systems having the same absolute entropy but different sizes (e.g. particles numbers), have in fact a substantial difference regarding its ...
1
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2answers
34 views

Is the trajectory of a particle with constant velocity (though its direction can change by collisions) always non-chaotic?

Suppose we have a particle that travels with constant velocity, without heat losses by friction, and no forces acting on it except for occasionally collisions with much bigger wall-like masses than ...
11
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3answers
797 views

Where does the use of tensors to describe orientation dependence of physical phenomena arise from?

In the context of anisotropy, I have often read that the use of a rank 2 tensor is "a model". But what is the idea behind this choice? Can anyone describe in what sense the use of tensor in this ...
1
vote
1answer
58 views

Electric potentials and superposition

I had a question regarding the addition of electric potentials. Consider two positively charged particles $q_1$ and $q_2$ at distance $R$ apart. Let the charges have magnitudes $q_1$ and $q_2$. For a ...
2
votes
2answers
118 views

Beat frequency of superimposition of three sine waves

Basically we have the function $$f(t)=\sin(2π\nu_1 t)+\sin(2π\nu_2 t)+\sin(2π\nu_3 t)$$ Let $T$ be the fundamental period of $f(t)$. Beat frequency is defined as the number of peaks in intensity ...
7
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3answers
188 views

Is Light intangible to other Light? And how does all the intersecting light exist in space?

I was thinking of how light actually gets into my eyes, and thought about my light bulb shining rays to every part of my bedroom wall, and reflecting them towards me. but then i realized, i could be ...
5
votes
1answer
178 views

I tested the RLC circuit natural frequency formula, and I'm getting the exact opposite result. Why?

For a series RLC circuit, the natural frequency (angular frequency of current in the absence of a harmonic driving voltage) is given by the formula: $$\omega=\omega_0\sqrt{1-\zeta^2}\tag{1}$$ where $...
2
votes
2answers
412 views

Using complex numbers to represent waves [duplicate]

When talking about a plane wave of the form $$\vec E=\vec E_0\cos(\vec k \cdot \vec r-\omega t)$$ We can replace it by $$\vec E=\vec E_0\exp[i(\vec k \cdot \vec r-\omega t)]$$ so that it is easier for ...
1
vote
2answers
137 views

Do all waves superpose on each other?

What would happen if two waves with different frequency were to pass through 1 point in space at the same time? Would they interfere at that point in space? Are superposition and interference the same ...
-1
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0answers
38 views

Helmholtz's decomposition yields over-determined system

From Helmholtz's decomposition, $v=v_{\scriptscriptstyle IR} +v_{\scriptscriptstyle R} $ where $\nabla\times v_{IR} =0$ and $\nabla\cdot v_R=0$ when apply this to the linearized Navier-Stokes ...
2
votes
2answers
56 views

Why we can assert, in general, that physical processes have the behaviour of low-pass filter?

Consequently, why is it not allowed to produce physically some controllers for processes that are described by a transfer function that is an improper function? A simple example is the driven ...
1
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3answers
64 views

Why can the deformation of a spring at a given point of the spring be considered directly proportional to the relative distance of the point?

Hello i have been studying differential equations and in one example my professor tries to deduce the partial differential equation that describes the longitudinal displacement on a elastic, ...
2
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3answers
209 views

Why do phasors add like vectors?

Phasors are used to represent sinusoidals. So why do they add like vectors? Why is it that when I add two sinusoidals in the form of phasors and add them like vectors, I get the right phase and ...
2
votes
2answers
767 views

The sum of all external forces acting on all the particles is equal to the total external force applied to the system of particles. Why? [closed]

As per the book: $\sum_{j=1}^N f_{j}^{ext}$+$\sum_{j=1}^N f_{j}^{int}$=$\sum_{j=1}^N dp_{j}/dt$ The first term, $\sum_{j=1}^N f_{j}^{int}$, is the sum of all internal forces acting on all the ...
2
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3answers
90 views

How Newton's laws replicate themselves on a larger scale?

Now I was reading The Feynman Lectures on Physics and found this which I found somewhat peculiar and deep and thus want your assistance here. So here it goes: The theorem concerning the motion of ...
1
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0answers
282 views

Relationship between Green's function and impulse response

In my field, electrical engineering, we frequently study linear time-invariant systems of the following form: $$ a_n\frac{d^ny}{dt^n} + a_{n-1}\frac{d^{n-1}y}{dt^{n-1}} + \ldots + a_1\frac{dy}{dt} + ...
10
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4answers
887 views

Why is classical electromagnetism linear? [duplicate]

When I ask this, I mean it as in when a test charge $q$ is placed in a region that contains two fixed charges $q_1$ and $q_2$, the force acting on it is the vector sum of the forces it would ...
2
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2answers
211 views

Ladder operators for a general second-order linear differential equation

We know that Schrödinger equation for a 1D harmonic oscillator $$ \left( \hat{p}^2 + \frac{1}{2} m \omega^2 \hat{x}^2 \right) \psi(x) =\left( -\frac{\hbar^2}{2m}\frac{\mathrm{d}^2}{\mathrm{d}x^2} ...
0
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2answers
124 views

Significance of the word 'linear' in linear harmonic oscillator

In my book Advanced Acoustics there is a line- A particle undergoing SHM is called a linear harmonic oscillator If I say that the word linear is used for the 2 reasons- The motion of the particle ...
0
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1answer
42 views

Understanding feedback =1 and comparing with a system with no feedback

I was studying this system In this V its a Voltage in a circuit and V_c its the voltage in a capacitor, the circuit was not provided, then calculated the step response (well I was told the step ...
0
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0answers
26 views

time response of this system

I'm doing a little exercise, that requests to calculate the time response of the next system so I proceed the next way $r=2x$ $e=u-2x$ $m=k(u-2x)$ then $x=\frac{1}{s+1}[k(u-2x)]$ making all the ...
0
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1answer
157 views

Principle of superposition in uniform compression of bar

With the following quantities defined as follows: Normal stress along x, $\sigma_x = \frac{F_{nx}}{S}$ Strain along x, $\epsilon_x= \frac{\Delta L_x}{L_x}$ and Poisson's Law: $\epsilon_y=\epsilon_z=...