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Questions tagged [linear-systems]

A linear system is a mathematical model of a system based on the use of a linear operator. A system is linear if and only if it satisfies the superposition principle, or equivalently both the additivity and homogeneity properties, without restrictions.

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Do all nonlinear systems store energy?

I would like to clarify, this question comes from my own curiosity while solving for nonlinear differential equations. I have noticed that I lack the fundamental understanding of linearity/...
Evank800's user avatar
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2 answers
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Why are forces superimposable in Classical Mechanics? Does this also apply in higher theories like General Relativity and Quantum Mechanics?

In classical mechanics, forces are treated as vectors and are added linearly. Is this principle to be treated as an axiom or is there some underlying principle from which this is derived? And given ...
Vivek Kalita's user avatar
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What are the limitations of Nodal Analysis?

Yea, that's the question basically. What are the limitations of Nodal Analysis? Like, for example take the following case, we have to find out the net capacitance between A and B. Now I want to solve ...
Adhway's user avatar
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Why we take only the real part of a solution as the actual motion?

I am taking Analytical Mechanics, and in Goldstein's book, chapter 6 (page 241) about linear oscillations, he says the following: "... $\eta_i=Ca_ie^{-i\omega t}$ (6.11) ... It is understood of ...
A24601's user avatar
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Mirror image Electric field and potential

pg My book says the electric field due to a positive point charge at a certain distance from the surface of flat, infinite earthed conductor can be obtained by introducing a virtual negative mirror ...
Manvendra Singh Gehlot's user avatar
1 vote
1 answer
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Solutions for nonrelativistic-matter perturbations

I'm studying the nonrelativistic-matter perturbations if the expansion of the Universe is driven by a combination of components. I'm currently Following this document (The growth of density ...
merlinbluepickle's user avatar
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2 answers
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To calculate the resultant resistance of a circuit can you use all possible paths and assume they're parallel? [duplicate]

To calculate the resultant resistance of a circuit can you use all possible paths and assume they're parallel? Say I have a circuit like this: And I want to determine the resistance between x and y. ...
WilliamHarvey's user avatar
2 votes
1 answer
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How to demonstrate in a simple way that this system of differential equations form a damped harmonic oscillator? [closed]

How may I demonstrate in the most simple way that the following system of differential equation form a damped harmonic oscillator ? $$ \dot x = -\alpha_x x - \omega y \\ \dot y = -\alpha_y y + \omega ...
chmike's user avatar
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How to justify sound propagation is a linear time-invariant (LTI) system?

Background A linear time-invariant (LTI) system (black box) is one described by the system: \begin{align} \dot{\xi}(t) & = A\xi(t) + B\omega(t), \; \xi(0) = 0 \label{eq-abc-1}\\ \lambda(t)...
César VB's user avatar
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How to solve the coupled equation of motion? [closed]

there we have the EOM: \begin{align*} \alpha q_{2} + \lambda - \ddot{q}_1=0 \\ \alpha q_{1} + \lambda - \ddot{q}_2=0 \end{align*} and $q_{i}$ is the canonical coordinates. Can I use the Fourier ...
Qian-Sheng's user avatar
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Gravitational field at a point $P$ in a small hole dug in a thin spherical shell?

I am supposed to find the field at a point $P$ in a hole, I initially thought that since the field inside is $0$ initally, now it must be $GM/R^2$ but that is not so, since we cannot assume it to ...
bobby76's user avatar
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2 answers
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Why is time period same even if you give an impulse perpendicular to the spring?

It all started with this question. There are three different ways to solve this but one way is using kepler's second law. $\frac{dA}{dt}=\frac{L}{2m}.$ This applies because angular momentum is ...
evmorfia's user avatar
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Coupled oscillators and stability of equilibrium points

My question is about parts (e) and (f). I have found the matrix to equation of motion to be $\frac{d}{dt}\begin{bmatrix} x_1 \\ x_2 \\ p_1 \\ p_2\end{bmatrix} = \begin{bmatrix} 0 & 0 & 1 & ...
Dave Conkers's user avatar
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Do the solutions to Maxwell's equations form a group?

How many solutions are there for Maxwell's equations? (Or rather, is there a finite number of them?) Regardless of how many solutions to these equations exist, could we claim they form a group? If so,...
Lagrangiano's user avatar
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Possibility of complex EM waves

I'm currently studying Quantum Mechanics, and I have just been presented Schrödinger's (time dependent) equation. Of course, the first solution to said equation I've been taught is that of a (complex) ...
Lagrangiano's user avatar
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Are there any "linear" lagrangian systems of interest for which an analytic solution is not obvious?

Out of curiosity, I am interested in Lagrangian dynamical systems that can be expressed in a "linear" manner. By this, I mean that their Lagrangian can be expressed, quadratically, as $$L = \...
Meclassic's user avatar
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Interpreting non-normalized covariance matrix eigenvalues as physical Actions

Summary: Eigenvalues of a "non-normalized" covariance matrix of time-series measurements from a linear system have units of Action (energy * time). Can we interpret this to obtain ...
user3716267's user avatar
1 vote
1 answer
81 views

Why does linearity imply no communication between Everettian worlds?

Scott Aaronson said in this interview https://youtu.be/1ZpGCQoL2Rk?t=3255 that the linearity of Schrodinger's equation prevents us from communicating with other Everettian worlds. Why? Is it analogous ...
ngc1300's user avatar
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2 answers
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Why are the mechanics of different axes independent of each other?

Why are the mechanics of different axes independent of each other ? Even though the question might seem absurd, but that is how physics works, is'nt it. While solving projectile motion, why does the ...
Ayesha J.'s user avatar
3 votes
2 answers
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Can $\langle0|(\hat{a}-\hat{b})|0\rangle$ be written as $\langle0|\hat{a}|0\rangle-\langle0|\hat{b}|0\rangle$?

I am studying quantum mechanics and dirac notation, and I am wondering if, given the operator $\hat{A}=\hat{a}-\hat{b}$, the expectation value $\langle0|\hat{A}|0\rangle$ can be written as $\langle0|\...
Barney_Dinosaur's user avatar
2 votes
2 answers
336 views

Eigenfunctions of Momentum Operator [closed]

Suppose we have the 1-d wave function $\psi(x)=A\sin\left(\frac{p_0x}{\hbar}\right)$ and we want to know wheter this is an eigenfunction for $\hat{p}=-i\hbar\dfrac{d}{dx}$ The argument usually goes ...
Johann Wagner's user avatar
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5 answers
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Why we use vector sum to calculate net potential in AC circuits?

My physics professor used vector sum to find net voltage at any instant in the following $RL$ circuit and said that it is equal to vector sum of phasor vector of potential drop across Resistor and ...
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1 vote
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Why discarding the linear combination of solutions?

In Griffiths's textbook (Introduction to quantum mechanics), part I, 4.1.2, he's solving Schrodinger equation in three dimensions, after separating the variables $Y(\theta, \phi) = \Theta(\theta)\Phi(\...
Arthur's user avatar
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2 answers
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Identifying the elastic limit? [closed]

Consider this problem I understand that the elastic limit is the point at which the material no longer elastically deforms, that is it doesn't return to its original shape. However, I am struggling ...
Quin Gardiner Bax's user avatar
32 votes
9 answers
6k views

What is the most appropriate mathematical theory for electrical circuits?

What exactly are electrical circuits as mathematical objects? It seems quite intuitive to me, that they are geometric realization of some graph with some additional structure. Another thing I notice ...
Cathartic Encephalopathy's user avatar
1 vote
1 answer
89 views

Is QFT linear with respect to superposition of multi-particle states?

I saw other posts such as this one but I don't think it's quite the same question, or even if it is, the answer employs the operator formalism and I'm not sure I follow it. I'm wondering, if you have ...
Adam Herbst's user avatar
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Is unitarity equivalent to linearity plus conservation of the norm?

Unitarity is the condition that the inner product in the Hilbert space is preserved. But if you suppose that the norm of any state is already preserved, then does unitarity follow from linearity? ...
Adam Herbst's user avatar
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3 votes
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Can a non-unitary process be made unitary using a transformation or by expanding the phase space?

Suppose I have a matrix differential equation: $$ \frac{d\mathbf{x}}{dt} = A\mathbf{x}$$ The solution to this is $$\mathbf{x}(t) = e^{At}\mathbf{x}(0)$$ If $A^{\dagger}=-A$, then the time evolution ...
confusion's user avatar
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Why for motion planning of quadrators the goal is to minimize the jerk/snap?

In motion planning for quadrators the optimization goal is sometimes to minimize the (norm squared of the) jerk and more often the (norm squared of the) snap. Can someone provide an intuitive and ...
Math98's user avatar
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1 answer
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Rotating wave approximation and linear response function

Is it true that the rotating wave approximation (RWA) is only the thing for the non-linear cases, and in the linear regime it does give any benefits? Let us say we have a rotating wave, so we don't ...
freude's user avatar
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0 votes
3 answers
586 views

How to check that any linear combination of solutions is itself a solution to the time-dependent Schrödinger equation?

David Griffiths states in 'Introduction to Quantum Mechanics': The general solution is a linear combination of separable solutions. As we're about to discover, the time-independent Schroedinger ...
Rasmus Andersen's user avatar
1 vote
2 answers
118 views

Coupled-mode theory and slowly varying envelope approximation

I am facing a situation where I have the following coupled-system equation: $ \dot{U}(z) = i \; M(z) \cdot U(z) \quad ,$ where U is a N-vector and M is a NxN matrix. Now, the diagonal elements of M ...
MPdeSH's user avatar
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0 answers
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Concerete examples of physical systems that can be (approximately) modelled using a 2D triharmonic equation?

I have some experimental measurements of input-driven standing-wave resonances in a nonlinear, 2D medium. I think it's fair to assume that the dynamics are homogeneous and isotropic, and we can think ...
MRule's user avatar
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1 vote
2 answers
258 views

Superposition of Quantity

I searched the first page of search result "Superposition" the closest answer came was The Meaning of Superposition but what noted that major answers are in context of Quantum Mechanics. My ...
Sage's user avatar
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7 votes
3 answers
663 views

Why are operators in quantum mechanics always linear?

After looking around in the internet, I could not find a sufficient proof how every operator in QM has to be linear. Many sources claim that the linearity of the Schrödinger equation implies that, ...
Susp1cious's user avatar
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1 answer
50 views

Problem identifying type of equation (linear/nonlinear)

I've looked at the answer to this Math.SE question, but I still can't know the answer to my question here. The following is the equation of equilibrium: divergence of stress tensor that is the sum of ...
user134613's user avatar
1 vote
1 answer
91 views

Kinetic and Potential Energy of a multi degree of freedom (MDOF) system

Consider the following MDOF system: $M\ddot x+Kx=F$ where $M$ and $K$ are the mass and stiffness matrix respectively, and $x$ and $F$ are the displacement and force vectors. How can one determine the ...
Mark's user avatar
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1 vote
1 answer
228 views

How translation, rotation and translation plus rotation of a body can be define particle by particle?

Let use simple example, a uniform rod with center of mass (COM) at the center of rod. The rod is in free space there are no other forces acting on it. If we apply single force acting on a particle at ...
123's user avatar
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0 votes
3 answers
114 views

Mass-spring system linear equations: I don't get the last term, shouldn't it be $V=\frac{1}{2}k_3x_{\text{wall}}^2-2k_3x_{\text{wall}}x_2+k_3x_2^2$?

I don't understand the last term in setting up the linear system of equations for multiple mass-spring systems. It is about the last spring in the next example: Source: https://math24.net/mass-spring-...
bananenheld's user avatar
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8 votes
10 answers
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How can forces be added?

I know this sounds elementary; that is why it has taken me this long to ask this question. I understand how forces can be added this way (above). But I don't see how it can be added in this way (...
Tca's user avatar
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1 vote
1 answer
431 views

What is the difference between a linear and a non-linear perturbation?

Sometimes you will hear about the stability of certain solutions (black holes, solitons, etc) with respect to perturbations. Often they talk about linear vs. non-linear perturbations. What is the ...
Superbee's user avatar
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0 votes
1 answer
488 views

Do all objects in a system need to have the same acceleration? [closed]

What is the definition of a system? Could multiple objects accelerating at different magnitudes and directions still be considered a system?
nebbie's user avatar
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0 votes
1 answer
108 views

Superposition of two electromagnetic waves

If an electromagnetic wave in isolation with vector potential $A^1_{\alpha}$ satisfies the wave equation $\Box A^1_{\alpha}=0$, how do we construct the total electromagnetic wave that results from ...
MrDerDart's user avatar
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0 votes
1 answer
492 views

Imaginary numbers in AC circuits

I've heard/read multiple times that "the use of imaginary numbers in ac circuits simplyfy calculations". My questions is: how is the calculations simplified? (exaple calculations?) And what ...
Vebjorn's user avatar
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2 votes
2 answers
937 views

Are magnetic fields additive?

When considering the field around a permanent magnet or current carrying wire, would it be accurate to say the magnetic field effects of each element add linearly in space, or is the interaction ...
J Collins's user avatar
  • 139
0 votes
1 answer
185 views

Physical significance of circuit eigenvalues and eigenvectors

When solving a DC circuit (say, with resistors and voltage sources only), we can use Kirchhoff's laws to get a set of equations in the currents: $$ RI=V, $$ where $R$ is a matrix relating to the ...
Rd Basha's user avatar
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0 votes
1 answer
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$m$, $T$, $f$ and initial phase determines amplitude $A$ of a SHM?

I was trying to calculate all the stuffs of simple harmonic motion knowing the mass, frequency and initial phase. with $\omega$ and $m$ I can calculate $k$, $\omega^2m=k$, with $f$, I can calculate $T$...
ends7's user avatar
  • 139
11 votes
4 answers
3k views

Are there any nonlinear Schrödinger equations?

The 1D Schrödinger equation reads: $$\frac{\partial \Psi}{\partial t}=\frac{i\hbar}{2m}\frac{\partial^2 \Psi}{\partial x^2}-\frac{i}{\hbar}V\Psi.$$ Now, generally we have $V=V(x)$ (or it dependending ...
agaminon's user avatar
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1 vote
0 answers
49 views

The criteria for zero DC resistivity from Kramers-Kronig relation?

While studying introductory superconductor theory, Neil Ashcroft came up with a criteria for zero DC resistivity as a following: $$\lim_{w→0}w\cdot\rm{Im} \ \it{\sigma(w)}\neq\rm{0}$$ And this must ...
GeorgePhysics's user avatar
0 votes
2 answers
264 views

Why is Newtonian gravity linear and independent on the presence of other bodies?

I have read somewhere that gravitational fore is linear and does not depend on the presence of other bodies around it, what does that mean? Another important characteristic of gravity is that it is &...
Md Faiyaz's user avatar
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