Tagged Questions
0
votes
1answer
33 views
Finding the Extremals of a Functional J.
The functional $J$ is defined on smooth functions $y \colon [a,b] \to \mathbb{R}$ satisfying $y(a) = u$, $y(b) = v$ and is given by
$$J[y]=\int_a^b \sqrt{y} \sqrt{1+(y')^2}\, dx.$$
I have found ...
1
vote
1answer
26 views
Differential equations basic problem
I know this is a basic Physics problems but somehow I can't solve it.
We have the differential equation: $2x''x^2 - 4 x^2x' - 2 x^3 = 0$
We have to conclude that the system:
$x' = y $
$y' = 2y + ...
3
votes
2answers
34 views
Up and Down Motion (Two objects meeting in time?)
PROBLEM:
Suppose than an object is thrown upward with an initial velocity of 200ft/sec and that another one is thrown upward 5 seconds later with an initial velocity of 300ft/sec. When and where do ...
1
vote
2answers
75 views
Resonance Frequencies of Oscillator
I understand that resonance is when the force term increases the natural oscillation of the system.
In the next equation the oscillator has a natural frequency $\omega_0=\sqrt{\frac{k}{m}}$. But I ...
0
votes
1answer
90 views
Nonlinear Second-order ODE BVP with 4 boundary conditions
My Lagrangian comes out in this form when I impose spherical symmetry:
$$ φ''(ρ)+{3\overρ} φ'(ρ)+{4μ^4\over M^2} φ(ρ)-{4μ^4\over M^4} φ^{3}(ρ)-{μ^4\over2M} ϵ=0 $$
The following boundary conditions ...
1
vote
1answer
28 views
Are there solutions when the boundary conditions are particle positions at 2 different times instead of positions and speeds at an initial time?
Is it possible to find solutions for a dynamic system when the boundary conditions are particle positions at 2 different times instead of positions and speeds at an initial time?
The question is ...
0
votes
1answer
63 views
Find all solutions of the 1-D heat equation of a specific form
I'd like to find all solutions of $u_t$ = $u_{xx}$ of the form
$$u = \left(\frac{1}{\sqrt{t}}\right)v\left(\frac{x}{2\sqrt{t}}\right).$$
I know that this problem reduces to solving a second order ...
3
votes
0answers
198 views
Solve a differential equation using Fourier series
Assume I have a second order differential equation $\ddot{x} = F(x,\dot{x})$ (or an equivalent equation of first order) and that I know there is a periodic solution to it (for simplicity's sake, ...
4
votes
1answer
98 views
Is this an undergraduate-level proof of conservation of energy, or an arbitrary manipulation of symbols that happens to give the right answer?
This is a slightly farcical question, for which I apologise.
An undergraduate tutee of mine was faced with the following problem:
Q. A particle of mass $m$ moving along a line is subject to a force ...
1
vote
0answers
40 views
Planar circular restricted 3-body problem
Hi and sorry for my bad English, it's not my first language. I'm trying to find the equations of motion of the planar circular restricted 3-body problem. I did the gravitational force, but I have some ...
4
votes
3answers
138 views
Physics notation justified
Sometimes in physics they do things like this one:
If $dq=f\left(x\right)\cdot dr$ then $\frac{dq}{dt}=f\left(x\right)\cdot \frac{dr}{dt}$
Which mathematically is a wrong deduction. Is there any ...
0
votes
1answer
258 views
1
vote
2answers
28 views
Trigonometry in the DE
Just a simple undergrad physics student asking them mathematicians.
I have a very simple 2nd order homogeneous DE of the form:
$$y''=-a^2y$$
So, a solution will be of the form $$y(t)=c_1 ...
0
votes
1answer
71 views
Finding the slope for Euler's method of evaulating differential equations
The change in the belocity of a body falling at a relatively slow speed over a short distance is given by $\frac{\mathrm{d}v}{\mathrm{d}t}= g - kv$, where $g$ is the acceleration due to gravity and ...
3
votes
1answer
97 views
Mass Spring Oscillator Interpretation
I’m having some trouble predicting the behavior of ODE’s using the mass-spring analogy. For example, consider the second order IVP listed below:
$$y’’ – \space 6y’ + 8y = 0, \space \space \space ...
2
votes
1answer
73 views
Differential equation for propotional absorbtion of light depending on intensity at a point
The rate (per foot) at which light is absorbed as it passes through water is proportional to the intensity, or brightness, at that point.
a. Find the intensity as a function of the distance ...
0
votes
1answer
37 views
Help with a differential word problem
When the electromotive force (emf) is removed from a circuit containing inductance and resistance but no capacitors, the rate of decrease of current is proportional to the current. If the initial ...
4
votes
1answer
65 views
Existence Energy of Wave Equation
I was just going trhough some properties of the wave equation, including the energy of the wave equation given by $E(t)=\int_{-\infty}^{\infty}u_t^2+c^2u_x^2 dx$, i.e the sum of kinetic and potential ...
2
votes
0answers
77 views
How to convert a hologram into an image?
Suppose one knows in full detail the phase and intensity of monochromatic light in a plane. This is basically what a hologram records, at least for some section of a plane. By using this as the ...
14
votes
1answer
210 views
How does one parameterize the surface formed by a *real paper* Möbius strip?
Here is a picture of a Möbius strip, made out of some thick green paper:
I want to know either an explicit parametrization, or a description of a process to find the shape formed by this strip, as ...
0
votes
2answers
89 views
Show that the following cycle has a limit cycle
By direct calculation show that (using polar coordianted) that
$$
\dot x=x-y-x(x²+y²)
$$
$$
\dot y=x+y-y(x²+y²)
$$
Show that this has a limit cycle
I need help understanding how to test whether it ...
0
votes
1answer
83 views
Which of the following is gradient/Hamiltonian( Conservative) system
The question that I have to solve is found below. However, I do not know how to start the solution since I am unsure about the defintion of a Gradient/Hamiltonian System. What must I check first to ...
3
votes
3answers
448 views
Cat Dog problem using integration
Consider this equation :
$$\sqrt{\left( \frac{dy\cdot u\,dt}{L}\right)^2+(dy)^2}=v\,dt,$$
where $t$ varies from $0$ to $T$ , and $y$ varies from $0$ to $L$.
Now how to proceed ?
This equation ...
0
votes
2answers
331 views
How do you know if an equation of spring motion is overdamped?
Looking at an equation, how can you know if if it overdamped, critically damped, or under-damped?
For example:
How can you tell that the equation $c_1e^{2x} + c_2e^{-2x}$ is overdamped?
How ...
0
votes
2answers
169 views
Determining diameter of parachute to obtain specific landing speed of a body, with Differential Equations
An example on a physics assignment asks this question:
NASA is preparing a probe to send to Mars. The probe weighs $40kg_f$ on Earth. As it approaches Mars, the gravitational field of Mars, which ...
0
votes
1answer
83 views
Three ball-spring system
So here is a crazy problem for you all. Imagine there is a system of three balls in a line. The first and last balls have a larger mass M and the middle ball is a smaller mass m. Inbetwen the two ...
0
votes
1answer
1k views
Solving differential equation for simple harmonic motion. Finding k?
A 1 lb weight is suspended from a spring. Let y give the deflection (in inches) of the weight from its static deflection position, where “up” is the positive direction for y. If the static deflection is ...
0
votes
1answer
489 views
Finding the period and frequency for simple harmonic motion
A 1 lb weight is suspended from a spring. Let y give the deflection (in inches) of the weight from its static deflection position, where “up” is the positive direction for y. If the static deflection is ...
0
votes
1answer
55 views
Related Rates of perpendicular motion
Two objects A and B are connected by a rigid rod that has a length L. The objects slide along perpendicular guide rails. If A slides to the left with a constant speed v, what is the velocity of B when ...
1
vote
0answers
91 views
damped harmonic oscillator driven by a stochastic momentum (not force)
Could you give references for solutions or solutions to the following problem:
Given: damped harmonic oscillator driven by stochastic force of very short duration (= stochastic momentum).
Find: ...
1
vote
0answers
57 views
Differential equations with different constants for different sub-domains
I remember that when I was studying differential equations, there was an example with solutions of the form $f(x) + C_1$ for $x>0$ and $f(x)+C_2$ for $x<0$ where $C_1$ and $C_2$ may be different ...
1
vote
1answer
76 views
What does it mean mathematically to set some of the integration constants in the general solution to a linear differential equation, equal to zero?
I'm trying to calculate the position of a particle in a quadrapole magnet depending on the entry position $x_0$ and the combined (constant) physical parameters $k$.
Given an equation
$$x(t) ...
2
votes
1answer
82 views
solution for ODE problem
I was trying to simulate a physical system which lead me to this equation. I don't know if it has any solution or not, but I guess you guys can help me find the answer. $$v'(t) = a + s * ...
0
votes
1answer
61 views
Yet another differential equation
Hello,
I would appreciate any help solving the following equation:
$$\begin{align}
y''[t] + \dfrac{d}{m}y'[t] + \dfrac{k}{m}y[t] = G \\
\end{align}$$
subject to:
$$t[0] = t0$$
$$t'[0] = 0$$
This is ...
3
votes
2answers
106 views
Help solving differential equation
I want to solve the following differential equation:
$y[t]$ : vertical position (height) of the object at time t
$y_c$ : height of the ceiling
$y_e$ : equilibrium point, the height at which the ...
4
votes
2answers
68 views
What do I need to know to simulate many particles, waves, or fluids?
I've never had a numerical analysis course so I don't know what I need to know. I'm just wondering what kind of books I should get to make me able to simulate these things. I'm wanting to simulate ...
3
votes
1answer
154 views
How come in classical mechanics we can get away with writing $a=v(dv/dx)$, treating $v$ as a function of $x$?
In classical mechanics we often use the relation $a=v(dv/dx)$ to help solve differential equations. I assume when we write $dv/dx$, we really mean $dV/dx$, where $V$ is a function defined so that ...
3
votes
1answer
970 views
Simple simple Euler Lagrange Equation
Just starting a course on Lagrangian Mechanics and I'm just wondering what about the Euler-Lagrange equation, and more specifically what I'm meant to be trying to do ..
One of the questions from my ...
3
votes
2answers
61 views
Approximated solution to differential equation in the form $f(u)u'^2+(u-u_0)^2=k$
I'm trying to solve the following differential equation, that arises from conservation of energy in a physical problem. $R,k$ are constants.
$$(1+R^4u^4)u'^2+(u-u_0)^2=k$$
Now, according to my book I ...
0
votes
2answers
101 views
Determine the flow and amplitude equation for thermal energy (with Del operator)
It is a question vector calculus and Maxwell's laws. I put it this way. Let's say, we are working in a $3$-Dimensional space ( e.g $x\cdot y\cdot z = 4\cdot3\cdot2$, a certain room/class of that size ...
7
votes
1answer
436 views
How does a harmonic oscillator with nonlinear damping behave?
It is well known that for a harmonic oscillator with linear damping,
$$\ddot x+c\dot x+x=0$$
with positive $c$, the amplitude of the oscillations decays exponentially when $c<2$. If it is higher ...
2
votes
0answers
95 views
Check my solution - Modelling of a spring with Differential Equation
I am doing some work with differential equations. I have solved the following problem but am uncertain if I'm doing it correctly. Could someone look over it for me and check if I'm doing something ...
48
votes
3answers
63k views
Teenager solves Newton dynamics problem - where is the paper?
From Ottawa Citizen (and all over, really):
An Indian-born teenager has won a research award for solving a
mathematical problem first posed by Sir Isaac Newton more than 300
years ago that has ...
2
votes
1answer
325 views
Vector ODE: how to solve?
In developing a particle system simulator, I ended up with this apparently innocuous vectorial differential equation:
$m \vec v\,' = -\mu \vec v + \alpha \vec v / |\vec v|$
Where $\vec v = \vec ...
1
vote
1answer
79 views
Probabilistic interpretation of $\sum_{n \geq 0}\mathrm P_{n}(t)= 1$
The following is a problem from Spiegel's Applied Differential Equations:
The probability $\mathrm{P}_{\mathrm n}(t)$ that a counter (such as a Geiger counter) will register exactly $\mathrm n$ ...
1
vote
1answer
106 views
How to construct and oscillation with exponentially growing period times?
I'm searching for the (maybe even smooth) "oscillating" function
$$f(t)=A\sin{\left(g(t)\right)},$$
such that there are zeroes at times $t_n=T^n$ for some fixed number $T$. So this will not really ...
0
votes
1answer
127 views
Why is $I$ = $\int_0^L R^2 \, dm$ considered a differential equation?
Earlier today in my physics course, we were talking about rotational inertia and how to calculate it for various shapes using calculus. Having not taken diff eq. yet, can anyone explain why this ...
2
votes
3answers
150 views
Non-uniqueness of the solution of the equation for a plucked string
I'm a bit confused about what is written in this PDF (in page 2). The author asserts that the differential equation $y'' +y = 0$ with boundary conditions $y(0)=0=y(\pi)$ has infinitely many solutions. ...
4
votes
2answers
363 views
General solution for $y^{iv}+ 2y''+y=\cos x$
Here is another problem from Mathews and Walker that has given me some trouble.
1-18. Find the general solution of
$y^{iv}+ 2y''+y=\cos x$.
Note: Thanks, everyone, for clearing up the ...
1
vote
1answer
286 views
How can I put the “3 body problem” mathematically?
I'm trying to put the 3 body problem mathematically. But I don't know how. I always get something reasonable, but I get something that is wrong.
