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0
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1answer
38 views

Minimizing a damping constant in order to minimize the amplitude of oscillations

How can I determine the damping coefficient that minimizes the amplitude of vibrations? This is an extension of Coupled ODEs that model a quad rotor \begin{align} ...
1
vote
1answer
134 views

Heat equation with heat radiation and heat transfer

If I want to calculate steady temperature distribution on a one-dimensional stick, and I need to consider both the heat radiation and heat transfer, then my equation will be in the form: $$ \frac{\...
2
votes
0answers
52 views

Locus of a moving mass point

Two very small mass particles $m_1$, $m_2$ are connected by a $2l$ long, infinitely soft and inelastic thread without mass. The initial condition of the system before being freely released is as in ...
4
votes
1answer
213 views

field solutions for covariant derivative of vector field constrained to zero

Question: What do the solutions of $\nabla_\mu A^\nu = 0 $ look like? And is it possible for spacetime curvature to somehow restrict the solution to $A^\nu = 0$? Here is my current ...
20
votes
1answer
1k views

Does the heat equation violate causality?

I've ran across the idea that, besides simply writing partial differential equations in covariant form, they need to be hyperbolic with all characteristic speeds less than the speed of light. A ...
1
vote
1answer
187 views

Heat Equation with In-Depth Radiation Exact Solution [closed]

I am looking to solve the heat conduction equation in a semi-infinite solid with in-depth radiation on the domain $-\infty < x < 0$. The governing equation of this problem is: $$\rho c \...
8
votes
3answers
181 views

What physical phenomena are modelled by Chebyshev equation?

What physical phenomena are modeled by Chebyshev equation? The equation is below $$(1-x^2) {d^2 y \over d x^2} - x {d y \over d x} + p^2 y ~=~ 0 .$$ I could not find it in Wikipedia or in Google (at ...
3
votes
1answer
127 views

Non-deterministic particle system

This question is in the spirit of Norton's dome, an example of an apparently non-deterministic system in Newtonian mechanics. Under certain restrictions, the Picard–Lindelöf theorem guarantees the ...
1
vote
1answer
339 views

Describe Ising model dynamics in stochastic differential equation or stochastic process

The Ising model is described by the Hamiltonian $$ H(\sigma) = - \sum_{<i~j>} J_{ij} \sigma_i \sigma_j -\mu \sum_{j} h_j\sigma_j, $$ and is treated extensively by equilibrium statistical ...
34
votes
6answers
903 views

Motion described by $m \frac{\mathrm{d}^2 x}{\mathrm{d}t^2}=-k\frac{\mathrm{d}^{\frac12 }x}{\mathrm{d}t^{\frac12}}$

What kind of motion would a (preferably dimensionless for simplicity) body do if the force acted on it was proportional to the semi-derivative of displacement, i.e. $$m \frac{\mathrm{d}^2 x}{\mathrm{...
2
votes
2answers
191 views

What happens to the position function when an oscillator is overdamped and does not have angular frequency?

My question is simple: What happens to the behavior of the position function, $x(t)$, when an oscillator is overdamped and $\omega$ does not exist? Here's the background on why I'm confused: For an ...
1
vote
1answer
52 views

Modeling wall's behaviour

Sorry if the quesiton is inconvenient, but I judged the physics forum would be the best place to go. My house is divided in two parts by a wall, and there's some tree pushing it, so the wall is about ...
2
votes
1answer
44 views

Simple modelling of seasonal variation of temperature?

I'm really curious about this: What is the simplest (or most simplified) differential equation that accounts for the variations of temperature throughout the year at some point on the northern ...
3
votes
2answers
137 views

Solving differential equations without approximations?

In physics, many problems start with a mathematical relationship of the physical phenomenon at hand, and then, in many occasion, always only leave whatever in the first order to get a nice and ...
3
votes
0answers
96 views

Non-linear Wave Equation - Numerical Methods

Motivation: I'm working with a highly non-linear spherical wave-like equation (second order PDE). The equation can be written on the form $$\ddot{u} = f(t, \dot{u},\dot{u}',u',u'')$$ where $'=\frac{...
11
votes
6answers
1k views

Why is it natural to look for solutions involving dimensionless quantities?

While studying the Heat Equation, I got stuck in a statement in my book. It says: We have seen that the combination of variables $\displaystyle \frac{x}{\sqrt{Dt}}$ is not only invariant with ...
8
votes
1answer
221 views

Physical interpretation related to a non-linear partial differential equation

I am doctoral student in pure mathematics working on a particular problem. My question is if this problem has applications to real world phenomena. I will try to explain the direct problem starting ...
6
votes
2answers
725 views

WHY does the “order” of a differential equation = number of “energy storage” elements in a system?

OK. in all engineering courses there comes a point when they introduce you to systems theory and modeling of systems (for eg. via the impulse response) and then the Laplace transform. The modern ...
3
votes
3answers
329 views

Proper and rigourous derivation of Gaussian beam?

Gaussian beams are known solutions to the Paraxial Wave Equation: $$ \frac{\partial^2 \Psi(x,y,t)}{\partial^2 x} + \frac{\partial^2 \Psi(x,y,t)}{\partial^2 y} = 2ik\frac{\partial \Psi(x,y,t)}{\...
4
votes
0answers
209 views

Naive questions on the classical equations of motion from the Chern-Simons Lagrangian

Consider a Chern-Simons Lagrangian $\mathscr{L}=\mathbf{e}^2-b^2+g\epsilon^{\mu \nu \lambda} a_\mu\partial _\nu a_\lambda$ in 2+1 dimensions, where the 'electromagnetic' fields are $e_i=\partial _0a_i-...
3
votes
1answer
198 views

Is there any physically relevant example of constructing series solution about infinity of an ordinary differential equation?

I was reading about how to test if a given second order ordinary differential equation has singularity at infinity from Arfken and Weber. I understood the steps mathematically but I could not find its ...
1
vote
1answer
371 views

Differential Equations for Block Diagram of Satellite Attitude Control System

I am trying to understand the procedure to setup differential equations from a block diagram. The enclosed example is about the attitude control of a satellite. The ultimate goal is to find a state-...
0
votes
2answers
163 views

$R = dV/dI$ for varying temperature

I'm trying to do my prelab for an E&M course, and am asked if, for plotting $V$ vs $I$ with a varying temperature, I should expect a linear slope. I know that both $V$ and $I$ depend on $R$, and ...
5
votes
2answers
402 views

Time-dependent Schrödinger equation with $V=V(x,t)$

I was wondering about the following: If you have the time-dependent Schrödinger equation such that $$i \hbar \frac{\partial\psi(x,t)}{\partial t} = - \frac{\hbar^2}{2m} \frac{\partial^2\psi(x,t)}{\...
1
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0answers
46 views

Is there a second order differential wave equation that only allows a finite set of discrete eigenvalues?

I tried constructing a second order differential wave equation that only allows a finite set of discrete eigenvalues by using the power series expansion such as \begin{align} A_{j+2} = \dfrac{j-m}{(j+...
5
votes
1answer
132 views

Is the mathematical form of the acoustic diffusion equation present in other fields of physics?

We are working in the field of High Performance Computing and we have developed a very efficient parallel implementation for solving the Acoustic Diffusion Equation as described below: $$ \frac{\...
2
votes
2answers
452 views

What does it mean to “solve an equation”?

I don't understand what is meant by there being a "solution" to an equation. For example, what does a solution to the wave or heat equation represent, and what are we solving for? Of course, we can ...
2
votes
1answer
100 views

product solutions for PDEs, physical motivation

Given a boundary value problem with independent variables $x_1,x_2, \dots , x_n$ and a PDE say $U(x_i, y, \partial_j y,\partial_{ij} y, \dots )=0$ we typically begin constructing a general solution by ...
2
votes
1answer
505 views

Help with Modeling a Liquid Vortex. (Related to General Fusion)

I want to model liquid lead swirling in a sphere. This is connected to General Fusion’s fusion machine. A 55 million dollar, Jeff Bezos funded, 60 person company trying to change the world with ...
2
votes
2answers
158 views

Why fundamentally does classical mechanics lead to second order dynamics? [duplicate]

What's so special about second order equations in classical mechanics? I have a basic understanding of the Lagrangian and Hamiltonian formulations of classical mechanics, so I'm not looking for ...
1
vote
3answers
270 views

Solving the simplest coupled nonlinear ODES for chemical kinetics [closed]

I am just trying to get the integrated form for the kinetics of the reaction $A + B \rightarrow C + D$ characterized by: $$ -\dfrac{d[A]}{dt} = -\dfrac{d[B]}{dt} = k[A][B] \; . $$ As you note, ...
-1
votes
1answer
60 views

Curves satisfying this functional [closed]

This is a problem in Hartle's "GRAVITY": Consider the functional $$S[x(t)]= \int_{0}^{T} \left[\left(\frac{dx(t)}{dt}\right)^2 + x^2(t)\right]\text{ }dt$$ Find the curve $x(t)$ satisfying the ...
3
votes
1answer
97 views

solution of pendulum equation [closed]

I have the pendulum expression $$\ddot{\theta}+\omega_{o}^{2}\sin(\theta)=0,$$ where I used a Taylor expansion for the sine term: $$\ddot{\theta}+\omega_{o}^{2}\left(\theta-\frac{1}{6}\theta^3\...
1
vote
1answer
165 views

Normal mode of a coupled pendulum: why the constant $\psi_1$, $\psi_2$

I need to solve a problem that tells me to find out the motion of both the pendulums that appear in the first 45 seconds of this video I think this kind of motion is described by a system of ...
2
votes
2answers
185 views

How do you integrate an expression over a variable in the limit of an integral?

I am trying to follow the steps to solve the integro-differential equation that arises from a plasma sheath problem given in this paper. This is the step I can't follow: $$\epsilon_o\frac{d}{d\...
2
votes
1answer
527 views

A general solution to continuity equation

Let us write the standard continuity equation $$\frac{\partial \rho}{\partial t} + \vec{\nabla} \cdot \vec{\jmath} = 0.$$ Should the relation $\vec{\jmath} = \rho \vec{v}$ be considered as a general ...
0
votes
0answers
41 views

Is there any general position function $x(t)$ that gives the solution to $x''(t) = k/x(t)^2$, where k is a constant? [duplicate]

In physics class, I often come across various inverse square law equations like the following: $F_G= G\frac{m_1m_2}{r^2}$ $F_E = k_e\frac{q_1q_2}{r^2}$ Specifically, we are typically given ...
15
votes
2answers
3k views

First-order wave equation: Why is its presence not common?

The (one-dimensional) wave equation is the second-order linear partial differential equation $$\frac{\partial^2 f}{\partial x^2}=\frac{1}{v^2}\frac{\partial^2 f}{\partial t^2}\tag{second order PDE}$$ ...
1
vote
1answer
146 views

The source of gravitation in a spacetime without matter

In a discussion concerning: Physical meaning of non-trivial solutions of vacuum Einstein's field equations there were a number of answers claiming that the flatness of the Ricci space (Rµv=0) ...
0
votes
0answers
635 views

Coupled mass spring system with damping and initial values

After researching through the web, I can't figure out how to express into a differential equation a coupled mass spring system with damping and initial values. Two masses and two springs, no external ...
1
vote
0answers
86 views

Derivation of cylindrical line heat source problem?

I have a line heat source embedded inside a cylinder and I am trying to find the temperature distribution T(r,t). By using the similarity variable, the solution to the differential equation is $$\...
4
votes
1answer
343 views

Research problems in application of Lie groups to differential equations [closed]

Are there any open problems in physics involving Lie groups and differential equations for a phd theses. Some applications are say, Noether's theorem in classical or quantum field theory. But I am ...
17
votes
5answers
1k views

Conservation of Mathematical Constraints when deriving Energy and Momentum from $F=ma$

Background: Starting from $F = ma$, integrating with respect to time, and using basic calc, one can derive $\int Fdt = m (v_f - v_i)$ Starting from $F = ma$, integrating with respect to distance, ...
2
votes
2answers
441 views

Stick and slip motion: mass and spring inside a box model

I am trying to determine a set of differential equation which can describe the motion of a mechanical system as below. Here, at the bottom we have a plate, and a box on top of it. Inside the box, ...
5
votes
2answers
184 views

What information is lost in the symmetrization necessary to derive the BBGKY hierarchy?

The book on Kinetic theory I'm reading derives the BBGKY hierarchy after introducing the reduced distribution functions $f_s(q^1,p_1,q^2,p_2,\dots,q^s,p_s):=\int\ \rho\ \ \mathrm d q^{s+1} \mathrm d ...
4
votes
1answer
159 views

Solution of QM tasks by using asymptotics

When we solve QM tasks by solving the Schrödinger equation, such as tasks about a particle in a Morse potential, a Poschl-Teller potential and many others, we usually find approximations (lets call ...
2
votes
1answer
103 views

Solving 2nd-order ODEs

I was reading this PDF REF per request: ...
0
votes
1answer
82 views

Showing specififc internal energy $e$ is a perfect differential

I am reading a book on Gas dynamics and there is a small section on thermodynamics before the conservation laws of mass momentum and energy are introduced. The book says $$ p = R \rho T$$ where $...
0
votes
1answer
1k views

Problem involving 1st law of thermo and ideal gas law

Problem: $1.0 \text{ kg}$ of air at pressure $10^6 \text{ Pa}$ and temperature $398 \text{ K}$ expands to a five times greater volume. The expansion occurs such that in every instance the added ...
0
votes
0answers
110 views

Boundary conditions for 2D helical waveguide

I'm interested in looking at standing wave solutions for the wave equation on a 2D annulus, with the twist that the annulus is "streched" in to a helix in 3D, but so that the rings themselves are 2-...