The Stack Overflow podcast is back! Listen to an interview with our new CEO.

Questions tagged [linear-systems]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2
votes
0answers
99 views

Is linearity of the quantum state space a necessary postulate in the reconstruction of quantum theory?

This question is about quantum reconstruction. I am new to this topic, and I decided to read some papers on it. I selected some works which follow an "information-focused" approach. The authors of ...
0
votes
0answers
16 views

Effect of external forces on a horizontal mass dampener

I am reading this paper which models the motion of a horizontal mass dampener. They say that on adding a dampener, the equation consisting of the forces is: $ma = -cv -kx$ where $ c$ = damping ...
1
vote
2answers
43 views

Superposition principle in sinusoidal waves

In sinusoidal wave equations that produce interference we simply add their displacements by superposition principle, however superposition position principle can be applied to only linear equations. ...
0
votes
1answer
36 views

Why is resultant displacement in an composition of simple harmonic motion the sum of individual displacements?

I recently came across the concept of the composition in simple harmonic motion. A paragraph says that: If $$x_1 = A_1sin(\omega t)$$ $$x_2 = A_1sin(\omega t + \phi)$$ Then, the resultant ...
2
votes
1answer
68 views

Solution of a differential equation in physics

In physics when we solve the differential equation, in some cases we get two part of the solution, one is real and another is imaginary. Some cases we consider that the real part have some physical ...
0
votes
1answer
84 views

Linearity of Schrödinger equation and perturbation theory

So, I was studying quantum mechanics and reached the point where perturbation theory is discussed. It is my first time in this topic, and something called my attention: it was said that we need ...
4
votes
3answers
107 views

Spring force under gravity

Why do I have to use law of conservation of energy to solve problems regarding calculation of extension in spring length when a box attached to the lower end of the spring is released from rest (such ...
0
votes
2answers
58 views

What is meant by “linear” in non-equilibrium thermodynamics?

I'm trying to learn a bit about non-equilibrium thermodynamics, and am currently reading de Groot and Mazur. In it, there is a quote right in the beginning, talking about the phenomenological ...
0
votes
1answer
45 views

Step in the derivation of complex wave notation [duplicate]

I'm reading Hecht's Optics and I have a problem understanding a step in the derivation of the complex notation of waves He writes that the wave equation for a harmonic wave can be written as $\Psi(x,...
0
votes
4answers
86 views

Why is it okay to decompose forces?

I guess the title says it all, it doesn't seem intuitive that we can consider something as abstract as forces to be decomposable vectors. An example would be a mass sliding off an inclined plane, we ...
1
vote
2answers
75 views

Problem in understanding superposition principle in electrostatics

When I have a single charge, it produces a electric field and a test charge will experience a force. Now when I have two(identical,same sign) charges, they produce electric fields and when the test ...
2
votes
3answers
80 views

Linearity (or lack thereof) of the Poynting vector

Maxwell's equations are linear. If we have a solution for the electromagnetic fields $\vec{E},\vec{H}$, and another solution $\vec{E}',\vec{H}'$, then $\vec{E}+\vec{E}',\vec{H}+\vec{H}'$ is also a ...
1
vote
2answers
22 views

Basis of 2D motion analysis

The basic argument for analysing 2D motion is that if we have a projectile we can break its 2D motion into 2 1D motions along 2 perpendicular axes. The motions along these axes and their corresponding ...
0
votes
1answer
38 views

Solution as the real part of a complex exponential from simple harmonic motion

From the book entitled Classical Mechanics written by John R Taylor, chapter no 5, Simple Harmonic Motion. I'm just citing the lines. $$x(t)=\text{Re }Ce^{i\omega t}=\text{Re }A e^{i(\omega t-\...
3
votes
0answers
41 views

Initial Condition in Spaghetti Cracking

In this Paper B. Audoly, S. Neukirch - Fragmentation of Rods by Cascading Cracks: Why Spaghetti Does Not Break in Half on Page 2 (bottom), the author argues that using an integral of motion, the ...
1
vote
1answer
52 views

Why are all solutions to this system of pendulum differential equations a linear combination of the two given solutions?

I am currently trying to do a lab report for a coupled pendulums experiment in which we find the following linear system of second order differential equations (describing the position as a function ...
0
votes
0answers
39 views

Linearity of Maxwell's equations in tensor formulation

Maxwell equation in tensor formulation are $\partial_\nu F^{\mu \nu}=J^\mu $ and $\partial_{[\gamma} F_{\mu \nu]}=0$. So to show Maxwell equation are linear in vacuum is the following method correct: $...
0
votes
1answer
26 views

How to see linearity of an interaction if it's lagrangian density is known?

The Lagrangian of electrodynamics is $-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+A_\mu J^\mu$ we know that electrodynamics is linear in special relativity but when we go to general relativity it becomes non-...
14
votes
3answers
258 views

Why are so many energies represented by $\frac{1}{2} ab^2$? [duplicate]

Why are so many energies in our universe mathematically represented by the equation $\frac{1}{2}ab^2$. For example: Kinetic energy $$\frac{1}{2}mv^2$$ Energy stored in a capacitor $$\frac{1}{2}CV^2$...
1
vote
1answer
66 views

Are there different types of superposition?

In electrostatics or in gravitational, when we are talking about interaction between multiple charges or multiple masses, we say that the interaction between any two charge or mass is independent of ...
6
votes
3answers
162 views

Vibrating string as a dynamic system

It's known first order dynamical systems had one energy storage (example C, in RC circuits) these systems act as a filter but don't resonate, on the other hand a second order system had two energy-...
0
votes
0answers
22 views

Distinguishing a LTI from not with unknown inputs

Linear time invariant (LTI) systems are a staple of physics. They appear in many situations. But how do you know a system is a LTI? In particular, if you are provided with a black box which ...
3
votes
1answer
199 views

Applying Kramers-Kronig relation to a simple damped oscillator

I just discovered the Kramers-Kronig relation and am trying to apply it to a simple damped oscillator of the form subjected to an impulse at $t=0$, which is a causal system: $$m\ddot x + c\dot x + k ...
2
votes
2answers
205 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
votes
2answers
104 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
votes
3answers
595 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
vote
1answer
44 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
vote
1answer
46 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
votes
1answer
55 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
votes
2answers
161 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
votes
4answers
284 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 ...
4
votes
3answers
915 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
votes
1answer
38 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
votes
0answers
125 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
vote
2answers
118 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
votes
1answer
155 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
votes
1answer
39 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
votes
2answers
125 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
vote
1answer
46 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
vote
1answer
273 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
vote
1answer
169 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,...
2
votes
1answer
87 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 ...
4
votes
1answer
221 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
votes
2answers
79 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
votes
0answers
44 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
votes
1answer
86 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
votes
1answer
160 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
votes
1answer
119 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 ...
1
vote
2answers
455 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
votes
3answers
175 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 ...