Questions tagged [laplace-transform]
Use for the Laplace integral transformation.
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Laplace Transform of time evolution of Amplitudes involving correlation functions
The correlation function is given by
$$
G_{ij}(t-t')= g_{i} g_{j} e^{(i\omega_{i0}-\frac{\Gamma_{0}}{2})t-(i\omega_{j0}-\frac{\Gamma_{0}}{2})t'}
$$
With the time evolution of amplitudes being given by
...
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What is the difference between scaling property when scaling factor is $-1$ and the time reversal property of Laplace transform?
I came across this question when one of my students asked me about it.
Suppose that $L$ denotes the Laplace transform and that $L\{F(t)\}=f(s)$. By scaling property, we have $L\{F(-t)\}=-f(-s)$. The ...
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Laplace transform: How to evaluate partial derivative in the denominator of a fraction?
I am solving a differential equation using the Laplace transform. However, to evaluate it I need to evaluate some strange terms. Specifically, I have a partial derivative in the denominator of the ...
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First passage time for state transitions expressed in Laplace space?
I'm struggling to understand the reasoning between moving between two steps in a biophysical reaction scheme for a paper I am reading. For this (from the description), the probabilities over different ...
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Unique distribution of charge on a conductor
If i place some charge on a conductor then it will distribute itself in such a way that electric field everywhere inside is zero. My text book says that only one kind of such charge distribution is ...
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PI controller - output calculation [closed]
The propeller of an airship is connected to a DC motor. The motor is being controlled using a PI controller. The PI controller has a proportional gain of Kp = 2, and an integral gain of Ki = 0.5, and ...
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Fourier vs. Laplace transforms
Electronics books often use Laplace to analyze circuits, while in physics we use Fourier, most of the times... if not always: from complex impedances to electromagnetism, quantum mechanics, Green ...
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What is the physical meaning of sigma in the Laplace transform?
Let’s use a simple harmonic oscillator as an example. When we calculate the Fourier Transform (a special case of the Laplace transform) of that system we get a function that shows which frequencies of ...
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What is the intuitive interpretation of the transfer function of this system?
If I have the system that could be observed in the next Image:
I want to know the transfer function, where the external force $f$ is the entry and $x_1$ is the output. The direction and positive ...
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Harmonic oscillator boundary condition issue for an impulse force
I am trying to solve an equation of an underdamped harmonic oscillator with a damping, and I get a weird boundary condition that perplexes me. Let me precise the issue, the equation is :
\begin{...
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Finding the Frequency and Damping Ratio of a Mass Spring Damper excluding Air Resistance
In summary, I need the damping ratio and damped frequency of a system excluding air resistance, as the system will be operating in a vacuum. The system can be modeled as a mass-spring-damper, and the ...
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Using laplacian transform to derive a general solution to the quantum harmonic oscillator
I'm unhappy with the often used approach of guessing the form of the solution and working from there. In the case of the quantum harmonic oscillator, once the form
\begin{equation}\frac{d^2 \psi(x)}{...
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Transfer function of system of coupled 2nd order ODE
I am wondering how to calculate transfer function $H(s)$ of system described by 3 coupled differential equations. The pourpose of work is to calculate "Bodedx" diagram ($|H(i\omega)|(\omega)$...
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Temperature distribution using Laplace transform
Consider room that occupies quarter space That is formed by two walls.
One wall is fully insulated and has a constant temperature $0$.
Another wall has a window of length L that with one edge at the ...
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Laplace transformation with mass
I am doing some QFT calculations with the fermion mass term, and it is making the calculation much more challenging.
Say I have
$$F(t,m) = \int_0^1 dx \frac{1}{1-x}\frac{1}{(t ~(1-x)+m(1-x)^2)^\...
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How to determine the forced frequency response using Fourier transform?
I would like to determine the forced frequency response of the following vibrating system using the Fourier transform method:
$$m\ddot{x} + c\dot{x} + k = f_0\sin(\omega t) \tag{1}$$
So by obtaining ...
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What is the Laplace transform of a Linear Time-Varying system?
The Problem
I have the following damped mass-spring system in the form of a Linear Time-Varying (LTV) system:
$$\mathbf{M}(t)\mathbf{\ddot{x}}(t) + \mathbf{C\dot{x}}(t) + \mathbf{Kx}(t) = \mathbf{f}(t)...
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Fourier and Laplace transforms (Forster's Hydrodynamics book)
In Forster's hydrodynamics textbook, he defines the following Fourier transform and (one-sided) Laplace transform
\begin{align}
S(k,\omega) =& \int_{-\infty}^{\infty} dt e^{i\omega t} S(r,t) \\
\...
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How to find the steady state response of a Multi-Degree of Freedom (MDOF) system?
The Problem
I currently have a Multi-Degree of Freedom (MDOF) system with the following equation:
$$\mathbf{M\ddot{X}}+ \mathbf{D}(t)\mathbf{\dot{X}}^2 + \mathbf{C\dot{X}} + \mathbf{KX} = \mathbf{F}(t)...
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How to find frequency response of a damped spring mass system using the Laplace transform
I would like to find the frequency response of a spring mass system of multiple degrees of freedom by using the Laplace transform. I think I know how to do this with one mass oscillating, however I ...
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Modelling of ball and beam system
I'm trying to model a basic ball and beam system using Euler-Lagrange Equation. My system looks something like this:
I have come up with this final Euler-Lagrange Equation:
$$\left(\frac{J_B}{r^2}+ m ...
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Inverse Laplace transform for Debye function
Given $$D_n(x)=\frac{n}{x^n}\int _{0}^{x}\frac{t^n}{e^t-1}dt,$$
the Debye function. Wikipedia says that it computes the Heat capacity of the Debye model.
Mathematically, we are interested in the ...
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What are the real world applications involving Laplace transforms? [closed]
Laplace transform is really interesting. Speaking about Fourier transform, there are many real world applications like we use in removal of noise and Laplace transform is again the extension of ...
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Laplace transform for damped waves
In Zeidler's book on QFT, page 94 there is a definition for a Laplace transform that reads
\begin{equation}
(\mathcal{L} f)(\mathcal{E}) = \int_0^\infty e^{i\mathcal{E}t/\hbar}f(t) dt,
\end{equation}
...
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Is time constant the same for linear system for all time?
I have heard that if you take a measurement $T_1$ and wait, then take another measurement $T_2$ and then find $\Delta T = T_2 - T_1$. Then 63% of $\Delta T$ + $T_1$ will represent the measurement when ...
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RLC - CLR - RCL Transfer function
I have problem because I don’t understand the difference between CLR RLC and LCR, they are the same no ? They have the same composant just placed in different ways, I did already found the Transfer ...
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Diffusion equation with time-dependent boundary condition
I was trying to solve this 1D diffusion problem
\begin{equation}
\dfrac{\partial^2 T}{\partial \xi^2} = \dfrac{1}{\kappa_S}\dfrac{\partial T}{\partial t}\, , \label{eq_diff_xi}
\end{equation}
with ...
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Density of states for 3D simple harmonic oscillator
I have the thermal partition function and the density of states for the 3D simple harmonic oscillator, which are given below
$$
Z(\beta) = \frac { 1 } { \left( 2 \sinh \left( \frac { \beta \omega } { ...
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Laplace transform on thermodynamics equation
I'm trying to create a simple model of a single-flash geothermal plant which consist of 3 main parts (flash-separator, turbine, and condenser). Is there any way to create a transfer function using ...
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Why is the Laplace Transform essentially never used when dealing with problems involving resonance?
Both the Laplace Transform and the Fourier Transform can be applied to a PDE, for example the wave equation, and used to derive a solution to the equation.
But I never see the Laplace Transform used ...
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Laplace Transform Density of States & Partition function
I am currently going through Pathria's Statistical Mechanics text , and under the Canonical Ensemble description, the author stresses that the partition function of a continuous system is the Laplace ...
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Fourier Transforms of Harmonic Functions
Suppose you are presented with the equation ($D=3$)
\begin{equation}
\nabla^2 A(x) = \nabla^2 B(x).
\end{equation}
Decompose $A$ and $B$ into their Fourier components,
\begin{equation}
A(x) = \int d^...
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Why is there a sudden change in current between $t=0^{-}$ and $t=0^{+}$ when an active inductor is connected in series with a relaxed inductor?
Let us take the following question as an example:
For the above question I drew the corresponding Laplace transform diagram, as follows (didn't draw the switch since it basically open circuit after $...
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Difference between Fourier and Laplace transforms in analyzing data
I have a set of displacement-time graphs from an experiment to convert to the frequency domain. Both the Fourier and Laplace transform seem to do this, so what's the difference between them (...
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What is this equation, written on a wall? [closed]
I've also asked this in MathOverflow, but since the equation in question is related to acceleration and physical phenomenon, I figure this is also a good place.
I was asked to ID the following, but ...
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Question about the Radiative Transfer Equation (RTE) Phase function term
I am new to this subject. I am trying to understand what is the purpose of the phase function
$$
∂u/∂t=-v∘∇u-v( μ_{a}+μ_{s})u+∫vμ_s p_{s} (Ω^{'}→Ω)u(r,Ω^{'},t)ⅆΩ^{'}+q(r,Ω,t)
$$
Were
u= angular ...
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Why is the impedance of a resistor the same in time and frequency domains if the Laplace transform of a constant is not the same constant?
I'm sure this is a silly question, but if the Laplace transform of a constant is not a constant, e.g.
$$\mathfrak{L}[1] = \frac{1}{s}$$
then how come the impedance of a resistor is the same in the ...
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Solving LR Circuit with Laplace Transform
I have a RLC circuit where the capacitor is connected in parallel with a resistance and inductance in series. The battery is connected "in parallel" with the capacitor and the RL branches. At t=0 the ...
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Express Laplace transform of voltage across a capacitor in terms of charge
In Charge Tunneling Rates in Ultrasmall Junctions section 2.1, the authors consider the problem of charge relaxation in a simple circuit shown in Figure A.
The implicitly use an assumption made about ...
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What is the Laplace transform of a single mass?
I'm trying to get a transfer function of $F=ma$ in the Laplace domain. This should be simple, but yet I'm confused. The transfer function is displacement over force. So, I have two approaches.
First ...
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