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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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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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(\...
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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 ...
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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 ...
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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 ...
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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? ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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How is the chain operator as defined in Griffiths "Consistent Quantum Theory" is a linear operator on the space of histories?
on page 138 of Griffiths "Consistent Quantum Theory" (2004) he defines a chain operator $K$ s.t. $$K(F0*F1....Fn)=FnT(jn,jn-1)Fn-1....F1T(j1,j0)F0$$
Where $T$ is the unitary Time operator.
...
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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 ...
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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, ...
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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 ...
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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 ...
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Why do we consider the whole process, taking a single moment? Universality of the vector equation
I have considered a simple model for describing the elastic force vector of a spring. First, I chose a reference frame in an arbitrary way, then I drew the necessary vectors, we get a ratio that ...
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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 ...
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Can Gaussian states be implemented with a classical system?
Gaussian states for $n$ quantum harmonic oscillators can be fully described by a $2n \times 2n$ matrix $\Theta_{ij} = \langle \delta \alpha_i \delta \alpha_j\rangle$ (and a displacement vector which I ...
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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-...
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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 (...
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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 ...
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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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 $T$...
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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 ...
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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 ...
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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 &...
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Superposition principle for the electric field
I had read somewhere that superposition principle is valid for linear functions, but the electric field is not a linear function, then why is the superposition principle valid for electric field?
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Applications of Signal and System theory [closed]
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}$$
(say, we put all the coefficients to be 1)
Now, if we try to solve it in ...
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Symmetry arguments in Gauss law and Superposition [duplicate]
Right now I am working on electrostatics. One of the techniques used is to use coulombs law and superposition in a situation where one can sum the point charges and corresponding fields to get the ...
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Normal modes of coupled oscillators
For two pendulums of mass $m_1$ and $m_2$, coupled by a spring of constant k, both suspended by strings of length $l$, the following matrix equality results from their equations of motion:
$$ \omega^2
...
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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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