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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Modelling of oxygen content in dynamic tank
I have a closed tank with an outlet at the bottom where I want to control the water level and oxygen concentration.
For the level control I have a inlet pump where I can control the speed. For the ...
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Validity of linear superposition in solving Maxwell's equations
I'm currently reading the book "Electromagnetic Waves and Antennas" by S. J. Orfanidis (publicly available at his own website) and a doubt came to me.
I'm referring in particular to the ...
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Are the Kramers-Kronig relations valid for time-translation variant systems?
For definiteness, consider a linear response theory context. Generically, we have a linear response function
$$\chi(t,t') = \Theta(t-t')f(t,t').$$
Suppose the system is not time-translation invariant, ...
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Calculation of charge by superposition
Consider this example from page 149 in Electricity and Magnetism by Purcell and Morin. The charge on each plate is calculated by superposition, by first assuming that plate 1 has a potential $\phi_1$ ...
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Computing real part of AC conductivity via linear response theory
I am attempting to use linear response theory to compute the AC conductivity in fourier space.
Schematically, the AC conductivity $\sigma(\omega)$ is defined by
\begin{equation}
\langle J_i \...
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Canonical transformation along the eigenvectors
From this:
Normal mode oscillations. If the Hamiltonian turns out to be a quadratic function of coordinates and momenta for a system of $N$ objects, e.g. $$H=\sum_{ij} M_{ij} q_i q_j + \sum_{ij} M_{...
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Spring system in two-dimensions based on graph network matrix representation
Wikipedia page "Spring system" includes an illustration of modelling a mass-spring network, based on a matrix representation. The example uses an oriented incidence matrix, which derives ...
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Conjecturing the homogeneous solution of a $2^{nd}$-order constant coefficients ODE - why conjecturing a $3$ D.O.F. solution in this course?
In this video (MIT $8.02$ course titled "Electricity and Magnetism", video number $208$, taught by Pr. Peter Dourmashkin) professor solves the undriven $RLC$ circuit ODE ($2^{nd}$-order ...
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Driven $RLC$ circuit why is the permanent signal sinusoidal and why two degrees of freedom (amplitude and phase)?
In a second-order ODE with constant coefficients, with a sinusoidal RHS term (such as the ODE of the driven $RLC$ circuit), how do we know:
that the particular solution (i.e. the permanent signal) is ...
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Why can forces be split into components? [duplicate]
Is there some physical understanding to it or we just know that nature works that way?
I used to think of it as was that if a force acts on an object in a certain direction producing some acceleration ...
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Why do people say the dynamics of quantum mechanics is always linear?
This statement seems false. An example of a non-linear equation governing the dynamics of a quantum system is the Gross-Pitaevskii equation.
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Understanding linearity of Maxwell's equation compared to non-linarity of GR
In this post, it is mentioned that a linear equation means that the solutions 'do not interact with each other' or 'do not know' about each other. But we know that Maxwell's equations are linear ...
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Complex representation of electric field
Hi I have been struggling for a while. Let's say I represent my complex field as $E_1=Ae^{i\omega t}$. Let's say this is the input to a component that takes in two inputs and is supposed to split the ...
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Confusion when utilizing complex electric field to do calculations
I've been struggling using complex exponentials. From my understanding using complex exponentials can simplify calculations without the use of sinusoidal waves. The thing that has been bothering me is ...
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Non-linear time and concurrent perceptions of reality [closed]
I am asking about what the fact that a photograph and a physical space can exist, in what we perceive to be different moments in linear time (with the photo being made from what we regard as our ...
