# 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.

289 questions
Filter by
Sorted by
Tagged with
48 views

### 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/...
340 views

### 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 ...
114 views

### 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 ...
• 374
1 vote
138 views

### 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 ...
• 13
90 views

### 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 ...
1 vote
79 views

### 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 ...
49 views

### 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. ...
51 views

• 175
84 views

### 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 ...
• 121
1 vote
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 ...
• 284
1 vote
95 views

### 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 ...
125 views

• 11
65 views

### 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 ...
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 ...
1 vote
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 ...
• 2,475
59 views

### 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? ...
• 2,475
136 views

### 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 ...
45 views

### 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 ...
• 31
74 views

### 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 ...
• 1,725
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 ...
1 vote
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 ...
• 13
42 views

### 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 ...
• 101
1 vote
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 ...
• 19
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, ...
• 185
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 ...
1 vote
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 ...
• 15
1 vote
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 ...
• 310
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-...
• 2,035
2k views

### 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 (...
• 157
1 vote
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 ...
• 661
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?
• 59
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 ...
• 272
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 ...
• 169
937 views

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 ...
• 139
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 ...
• 2,141
51 views

### $m$, $T$, $f$ 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 $T$...
• 139
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 ...
• 1,765
1 vote
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 ...