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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Applications of Signal and System theory

I recently heard a lecture about Signals and Systems and find the subject extremely exciting. I would like to do more in this direction, so I would be interested to know in which modern research area ...
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Do retarded potentials imply homogenous solution?

I am having trouble reconciling the retarded potentials, with a possibility for a background homogenous solution to the EM field to exist. In the Lorenz gauge $$\nabla \cdot \vec{A} = - \mu_0 \...
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Why we take complex current into effect why not just the real part and similarily power formula why we use complex conjugate of current?

In a RLC circuit having a AC source . The actual current flowing in any branch will be the real part of the complex current. The imaginary component has no important role. It is just there. Now that ...
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Is there a "measure of nonlinearity" that can be measured when testing quantum mechanics?

For context, I think the comparison to tests of general relativity here is apt. There is the post-Newtonian formalism that has some well-defined parameters that can discriminate between general ...
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Solve a particular case of the Lyapunov equation for infinite dimensional matrices

Consider the matrix Lyapunov-type equation $BX + XB = \partial_R B^{-1}$ where $B$ is a (known) symmetric square matrix which depends parametrically on $R$. Can we find a symmetric matrix $X$ which ...
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What is the physical meaning of the eigenvalues of a state-space representation of a physical system?

Consider the following state-space model of a physical system. $\begin{bmatrix} \dot{{x_1}} \\ \dot{{x_2}} \end{bmatrix} = \begin{bmatrix}\ 0 & 1 \\ 0 & -c/m \end{bmatrix}\begin{bmatrix} \ ...
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Two masses connected by spring moving in a plane

Is the 2-dimensional motion of two masses connected by a spring non-linear? As far as I see it, the magnitude of the force on each mass is proportional to the spring's displacement from its ...
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How does the positional form and phase form of a body having simple harmonic motion relate to each other? [closed]

A body which is connected to a spring end, has been pressed and released. It has possessed simple harmonic motion. I was told its positional form and phase form would look something like the following ...
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Can the phase portrait of SHO rotate counter-clockwise? or is it the case that there can be no physical motion corresponding to that?

Framing the question In the case for Simple Harmonic Oscillation, we have the equation: $$\ddot{x}+x=0 \tag{1} \label{1}$$ (say, we put all the coefficients to be 1) Now, if we try to solve it in ...
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Symmetry arguments in Gauss law and Superposition [duplicate]

Right now I am working on electrostatics. One of the techniques used is to use coulombs law and superposition in a situation where one can sum the point charges and corresponding fields to get the ...
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Modelling a Simple Linear Translational Mechanical System

I am trying to model the following translational mechanical system in I/O form, however I am unable to see how the force of the damper b12 is affected the force on M2. The damper b12 reduces the net ...
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Momentum analysis by transfer functions and the extra term

I was studying system dynamics and saw one of classic questions about conservation of momentum.Problem trying to be solved by its transfer function and interesting thing is when final velocity (at t=0+...
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Normal modes of coupled oscillators

For two pendulums of mass $m_1$ and $m_2$, coupled by a spring of constant k, both suspended by strings of length $l$, the following matrix equality results from their equations of motion: $$ \omega^2 ...
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Dirac equation on a superposition

If the following is a solution to the Dirac equation $$(i\gamma^\mu\partial_\mu - m)(\psi_1+\psi_2)=0$$ Therefore after distributing $$(i\gamma^\mu\partial_\mu - m)\psi_1+(i\gamma^\mu\partial_\mu - m)\...
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I don't get why the square roots of the probabilities should transform linearly

I do get why probability should transform linearly. Consider (in classical mechanics), a system whose state is unknown to us, and is given by a probability vector $(0.3,0.2,0.5)$ where the numbers are ...
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Superposition principle in electrostatics: Experiments

I understand that the superposition principle in electrostatics is consistent with experience and therefore is not very questionable. However, is it possible to perform a direct experimental test of ...
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What do we mean by linear Simple Harmonic Motion?

I am confused by the use of "linear" in SHM. What does linear actually signify in Linear SHM. Does it mean :- That the displacement is linear? (I suppose it isn't since displacement varies ...
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Complex form for the Plane wave

It's a very well-known fact that plane waves can be represented in the complex form: \begin{equation} \mathbf{F}(\mathbf{x},t)=\mathbf{F}_0e^{i(kx-\omega t)} \end{equation} However, I've been ...
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Why must operators in QM be linear?

Why must all operators in QM be linear (and therefore able to be represented by matrices). What is the physical reasoning behind this? Is it be possible that the non-unitary nature of quantum collapse ...
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Superposition principle and fixed positions charges

I am trying to understand this sentence of my course: The superposition principle implies that the net force between any two charges is independent of the presence of other charges. This is true if ...
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Why is it sometimes acceptable to eliminate complex terms? [duplicate]

In my classical mechanics class we were going over an example of an object moving in a non-inertial reference frame from the perspective of an observer in that non-inertial frame. In the derivation ...
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How does linear combination of two solutions of a second order differential equation have complex coefficients...in case of Simple harmonic motion? [duplicate]

I have searched all the websites but couldn't get the answer...everywhere I am seeing linear combinations with real coefficients..but in case of exponential solution of second order differential ...
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How do I use dimensional analysis to find the ratio of potentials at the center and corner of a uniformly charged cube?

The problem goes like this, from Purcell's electromagnetism: Consider a charge distribution that has the constant density $ρ$ everywhere inside a cube of edge $b$ and is zero everywhere outside that ...
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Is this system linear? [migrated]

$$y(t)=\begin{cases}0 \hspace{4.86cm}x(t)<0\\ x(t)+x(t-2)\hspace{2cm} x(t)\geq0\end{cases}$$ Assuming $C>0$. If $x(t) = C$ is an input, the output will be $2C$, and the output of $-x(t)=-C$ must ...
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Are these linear system definitions different?

I often find the following two definitions of linear system during my studies. Since people often refer to linear system in a generic way I suspect these defintions are somehow connected and they ...
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Relation between $\delta$-system

I'm reading Pavel Grinfeld's book "Introduction to tensor analysis and the calculus of moving surfaces". I've reached the section where the author talks about $\delta$-systems and the ...
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On solutions of Schrödinger equation

Consider a quantum system described by the wave function $\psi({\bf x}, t)$ and subjected to a time-independent ordinary potential $V({\bf x})$. The relative Schrödinger equation takes the form: $$\...
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Is there any fundamental reason why acceleration is a linear function of external forces?

Perhaps a trivial question, but it is something which I couldn't ever grasp ever since beginning physics. Why exactly should Newton's second law be linear in application of all the external forces? ...
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Complex physical quantities

I have a question regarding complex physical quantities. Why do we consider only the real part of a complex physical quantity? Why not the modulus? Since, for $z=a+bi$, we have $|z| = \sqrt{a^2+b^2}$, ...
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Why potential due to system of charges is scalar sum?

Suppose there are three point charges $q_1,q_2,q_3$. We have to find potential at point $P$. So we do scalar sum of all potentials at that point taking one charge at a time. But I am saying that, when ...
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What is a linear system?

I was reading about Resonant Frequency, and found a lot of information about it: like its uses in daily lives, for eg: swing, pendulum. etc I also read about its advantages and disadvantages but the ...
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Addition of Potentials

Say we have a charged shell A (radius $R$) that is present within another charged shell B (radius $2R$). The charged shell A has a net charge $Q$ and the shell B has a net charge $3Q$. Taking shell A ...
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Solution to differential equation

If I have a differential equations of the form $$\frac {d^2y}{dt^2}=\alpha^2y$$ Assuming the roots of the characteristic equation is complex the solution to the differential equation is: $$y=C_1e^{j\...
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Does superposition principle of electric fields hold for conductors?

Suppose we have two identical oppositely charged spheres separated by some short $ x $ distance, then if we say that superposition principle holds, then at all points in space outside the conductors, ...
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3 answers
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Can superposition principle work with boundary conditions in electromagnetism?

Let's suppose I have two finite lines of charge, each of them at some voltage $V$. The first one would be at $x=0$ and the second one at $x=a$, and each of them has the same length $l$. If I want to ...
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Why do you add independent solutions when finding a general equation for SHM?

In my physics class, we have an assignment based on simple harmonic motion with the differential equation: $$ \frac{d^2x}{dt^2} + a\frac{dx}{dt} + a^2x = 0 $$ Different parts of the question help us ...
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What is proof of principle of superposition of waves on a string?

I was studying waves on a string and there was this 'Principle of Superposition of Waves' which states that the net displacement of a point is equal to the vector sum of displacements caused to the ...
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16 votes
4 answers
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In classical physics (classical electrodynamics), why linearity of Maxwell's equations prevent interaction of electromagnetic waves?

In classical physics (classical electrodynamics), electromagnetic waves don't interact. In quantum mechanics, they could. In this article on light-by-light scattering: https://arxiv.org/abs/1702.01625 ...
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Why does the dependence of Coulomb force on the medium not conflict with the superposition principle?

Why does the dependence of Coulomb force on the medium not conflict with the superposition principle? As I have been told (and checked myself on the web including this website), the Coulomb force ...
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1 vote
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RL circuit with AC source transient

I have a simulation of RL circuit connected with an ac source. Any one can tell me please that why the peak of the current during the first positive half cycle is higher than the peak of current ...
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Transient in RL circuit

Why the transient current will die off in series RL circuit with AC source? Every time the ac voltage source reverses direction so does the current and the current has to pass through the same ...
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Do linear theories have infinitely many solutions?

It is said that, in a linear theory, you can add any number of solutions to still find a solution. So, say I initially found three solutions to a linear theory, and I call them a1, a2, and a3. Now I ...
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2 votes
1 answer
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How does the linearity of the Schroedinger equation reflect the interactions?

There is a common lore that linear equations describe non-interacting systems, why non-linearities correspond to non-trivial interactions. My (loose) question is how is that compatible with the ...
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Throwing a baseball at a car and it staying there [closed]

I was recently asked the following question in a physics contest: For some odd reason, you decide to throw baseballs at a car of mass $M_i$, which is free to move frictionlessly on the ground. You ...
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Direction of current using superposition method

I have to calculate the current I and the voltage between a and b using the superposition theorem. I can solve this directly by calculating the voltage between points, but I have to do it with that ...
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Three masses with 2 springs in 1D

Consider three arbitrary masses attached by two different springs in vacuum, starting at arbitrary initial positions with no initial velocities. Is this system chaotic? Is this system analytically ...
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Can concepts like "critical damping" or "resonant frequency" be applied to more complex systems than just a spring and damper in parallel?

I am trying to do some modeling analysis by representing materials with parallel systems of springs and dampers. In the simplest case with just one spring parallel to a damper, we have the traditional ...
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Does Lagrangian follow superposition principle?

In other words: is the Lagrangian of a composite system the sum of the Lagrangians of its components? It shouldn't be, since it's energy. But then we lost one of the most important principles in ...
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Are all systems really non-linear?

A few years ago I had a course on control and systems engineering. In one of the first chapters of the book we used, it was stated that "all physical systems are nonlinear". When discussing ...
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Why linear optical response is absent in a non-centrosymmetric system with time reversal symmetry?

In this paper, it is mentioned that a non-centrosymmetric system with time-reversal symmetry doesn't have a linear response. It is actually a consequence of the Onsager reciprocal theorem. But I didn'...
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