Questions related to mathematical physics which include application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories

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36 views

Finite difference scheme for the continuity equation

I am currently trying to solve a system of PDE's numerically, one being the equation; $$ (1)\quad \partial\rho/\partial t + \partial(\rho v)/\partial x = 0 $$ I have been reading up on ...
-1
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3answers
63 views

How to show that show that $\frac{v+u}{1+ uv/c^2}=c$ when $u=c$?

I am trying to show that $\dfrac{v+u}{1+\dfrac{uv}{c^2}}=c$ when $u=c$. Context It's needed for a physics proof that I'm working on. This is the formula for relative velocity, $u$ represents the ...
3
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1answer
25 views

Simulating elastic collision

I wrote a simple program where i can move around some objects. Every object has a bounding box and I use hooke's law to apply forces to the colliding objects. On every tick, I calculate the forces, ...
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1answer
24 views

Physics differential equations problem

I am told that a torpedo is fired and has an initial velocity of 60 km/hr. After 1 km travelled, its velocity falls to 30 km/hr. We know that the drag force acting on the body is proportional to the ...
3
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3answers
164 views

General Solution for the Gravity Between Two 3D Triangles

I would like to find the general solution for the gravity between two (flat) triangles in 3D, including the location $(x,y,z)$ where this force should be applied (in order to later account for ...
2
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3answers
100 views

It takes 80J of work to stretch a spring 0.5m from its equilibrium position. How much work is needed to stretch it an additional .5m?

It takes $80\,\textrm{J}$ of work to stretch a spring $0.5\,\textrm{m}$ from its equilibrium position. How much work is needed to stretch it an additional $0.5\,\textrm{m}$? Here's what I have: ...
2
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0answers
55 views

Equipartition of energy

Let $u$ solve the initial-value problem or the wave equation in one dimension: $$\begin{cases}u_{tt}-u_{xx}=0 & \text{in } \mathbb{R} \times (0,\infty) \\ u = g, u_t = h & \text{on } ...
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1answer
32 views

Not locally flat space

In physics, and particularly in general relativity, we use the notion of manifold to describe space-time. In this way we have a space that locally looks like $\mathbb{R}^n$, a "flat" space. Are there ...
2
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3answers
204 views

“Methods of Theoretical Physics for Mathematicians”

I read in the Princeton Companion to Mathematics that even pure mathematicians should know some theoretical physics. However, I see that there are many reference books of mathematical methods for ...
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1answer
148 views

Solving Collision Problem (Momentum conserved) Systematically in more than 2 dimensions

I know that the equations for conservation of momentum and energy $m_1v_{i1}+m_2v_{i2} = m_1v_{f1}+m_2v_{f2},\;\frac{1}{2}\epsilon(m_1v_{i1}^2+m_2v_{i2}^2) = \frac{1}{2}(m_1v_{f1}^2+m_2v_{f2}^2)$, ...
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1answer
176 views

How to find the integral curves that are orbits of one-parameter groups?

Consider $\mathbb{R}^2$ with standard symplectic structure and inner product. Consider a Hamiltonian $$H=(x,y)A(x,y)^t$$ where $$A=\begin{pmatrix} \alpha & \beta \\ \beta & \delta ...
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0answers
37 views

Stress Volume of Revolution

A bar with circular cross-sections is supported at the top end and is subjected to a load of $P$ as shown in Figure below. The length of the bar is $L$. The weight density of the materials is $ρ$ ...
1
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1answer
62 views

Help with a 3-body problem

If I have three particles with masses $ m_1, m_2, m_3$ with their respective position vectors $ x_1, x _2, x_3 $ and their speeds $ v_1, v_2, v_3 $ how could I find a parametric function that would ...
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2answers
65 views

How to solve for multiple unknowns using substitution?

$R_1$, $R_2$, $R_3$, $R_4$, $R_5$ and $V_6$ suppose to be 'known' values. $$\frac{V_{n_1}}{R_1} + \frac{V_{n_1}-V_{n_3}}{R_2} + i_6 = 0$$ $$ \frac{V_{n_2}-V_{n_3}}{R_4} + \frac{V_{n_2}-V_{n_4}}{R_3} ...
2
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2answers
206 views

I want to learn math from zero

I finished high school 2 years ago and now I'm stuck in a university in Turkey. I am interested in learning precalculus, discrete mathematics, physics and chemistry. Question: I need to learn math ...
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1answer
36 views

Objects sliding on a frictionless surface

An object with the mass 2 kg slides on a frictionless surface. When the velocity of the object is h, the object is subjected to a force (air resistance) that's $5v^2$ N. Apparently the equation is ...
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39 views

Stadium billiard reflection angles

Given a boundary and a massless particle with constant velocity with a certain direction, a billiard consists of an experiment where the particle collides with the walls conserving its velocity ...
0
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1answer
46 views

Partial Differentiation in Statistical Mechanics

I am damn struggling with basics in here. I know that $U=U(N,V,T)$ and $z=z(N,V,T)$ so that $N=N(z,V,T)$. Now, I want to do partial differentiation using chain rule involving three variables so that I ...
2
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1answer
128 views

Physical or geometric meaning of the trace of a matrix

The geometric meaning of the determinant of a matrix as an area or a volume is dealt with in many textbooks. However, I don't know if the trace of a matrix has a geometric meaning too. Is there ...
9
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3answers
300 views

Examples of useful, insightful, and interesting hand-waving [closed]

It seems to me that some hand-waving (by which I mean some arguments that aim at giving some form of intuition on the problem even at expenses of complete rigour [and not mnemonics for high-schoolers ...
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1answer
55 views

Interchange of derivatives

Given Euler-Lagrangian equation $$\frac{d}{dt}\frac{\partial L}{\partial \dot q}-\frac{\partial L}{\partial q}=0$$ Can I equivalently write as $$\frac{\partial \dot L}{\partial \dot q}-\frac{\partial ...
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2answers
67 views

How much thrust is required to move a boat of 120 kg / 265 pounds to a speed of … [closed]

How much thrust is required to move a boat of 120 kg / 265 pounds to a speed of 10 km / 6 miles per hour in 7 seconds. I found the following: http://en.wikipedia.org/wiki/Thrust-to-weight_ratio ...
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3answers
160 views

Work done to fill up a spherical tank

A spherical tank of radius $12$ feet is $40$ feet above the ground. How much work is done in pumping water into the tank until it is full? I obtained $$ w= \int_{16}^{40}[12^2-(40-y)^2y] \, dy. ...
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1answer
62 views

mechanics piston problem involving rotational motion.

The above figure shows a piston driving a crank OP pivoted at the end $O$. The piston slides in a straight cylinder and the crank is made to rotate with constant angular velocity $ \omega $. Find ...
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3answers
125 views

Geometric Algebra/ Calculus for Physics

I don't know if this would be a better question for physics.SE, but I'll try here first: There is at least one good book on classical mechanics using the geometric algebra/ calculus (GA): New ...
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2answers
86 views

How did they solve for a here?

Consider the following algebraic steps: $$ F - (M_1 a + \mu_k M_1 g) - \mu_k M_2 g = M_2 a $$ $$ F - \mu_k M_1 g - \mu_k M_2 g = (M_1 + M_2) a $$ $$ a = \frac{F - \mu_k M_1 g - \mu_k M_2 g}{(M_1 + ...
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0answers
42 views

Solving 2 equations (Projectile motion)

The question is from Physics, but all I need is help on solving maths. So basically, i am trying to find out the optimal angle for projectile motion from a certain height and I end up with these two ...
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0answers
22 views

How to describe the motion of a mass point?

Consider a mass point moving around a fixed point on a circle with radius $r$ with constant angular velocity $ω$. At a certain moment of time, the connection is removed, and the point mass is flying ...
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2answers
71 views

How to find $\theta$ at which $d$ is the maximum possible?

I have an equation: $$d=\dfrac{v\cos \theta}{g}\left(v \sin \theta + \sqrt{v^{2} \sin^{2}\theta + 2gh} \right),\ g≈9.81 \dfrac {m}{s^{2}}$$ How to find $\theta$ at which $d$ is the maximum possible? ...
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1answer
103 views

Dirac Gamma matrix identity

In my library's (old -- 1996) copy of Peskin and Schroeder, there's an identity I'm struggling to prove. In my copy it occurs on page 42, between equations 3.28 and 3.29, but I don't know how well ...
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1answer
36 views

How do I calculate the centripetal force?

In this experiment, a brown rubber weighs 42.2 grams and was spun at a velocity of 4.66m/s with a radius of 40cm and masses of 200g...the answer should be about 1.96N and I got 2.3. Can someone ...
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1answer
14 views

$\int_C (\alpha x, -\alpha y) . dr = 0$ where C is the unit circle

Circulation is given by $$\int_C u . dr$$ I want to show that the circulation around the unit circle is $0$ for $u = (\alpha x, \alpha y)$. Ie. $$\int_C (\alpha x, -\alpha y) . dr = 0$$ How would ...
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2answers
145 views

Is “mixed math” a useful way to learn math?

I was reading a book about how mathematics was taught in Cambridge in the 19th century, and it struck me how much physics was included in the syllabus, and it wasn't optional but everyone had to learn ...
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1answer
122 views

Work required to pump water out of tank in the shape of a paraboloid of revolution

This is the problem I have been assigned: A water tank has the shape of a paraboloid of revolution: its shape is obtained by rotating the parabola $y=x^2/4$, for $0\le x\le 4$, around the ...
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39 views

intersecting point of two lines

The circle has R radius and and ellipse is intersecting the circle. I need to findout $x_c$ and $y_c$, which is the midpoint of the 2 intersected point of ellipse.Line 3 is the tangent of the ...
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1answer
45 views

A formal justification for this “physicism”?

I gave a presentation for a seminar class yesterday on Fourier analysis, and introduced the sawtooth function as a counterexample, for a function whose Fourier series is not termwise differentiable. ...
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0answers
18 views

Proving susceptibility in Lorentz Model satisy Kramers-Kronig relations

My instructor asked me to prove that the real and imaginary parts of the electric susceptibility derived from Lorentz Model satisfy the Kramers-Kronig relations using the residue theorem. The problem ...
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0answers
16 views

Potentials and Markov Processes

Given a resistive electrical circuit $G$, i.e. a graph with nonzero weights attached to each edge whose reciprocal we call the "resistance," we can define a reversible markov chain on the graph, ...
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3answers
61 views

$\vec{r} \times (\vec{\omega}\times \vec{r})=r^2\vec{\omega}-(\vec{\omega}\cdot\vec{r})\vec{r} $

Show (in cartesian coordinates) that $\vec{r} \times (\vec{\omega}\times \vec{r})=r^2\vec{\omega}-(\vec{\omega}\cdot\vec{r})\vec{r} $ I am not really sure how to calculate this. Do I just assume ...
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0answers
29 views

Ode with Piecewise function

We can write this $$12x"+36x'+48x=f(t)$$ my main problem is how to solve this non-homogeneous ODE I know how to do this as 2 different ode unfortunately its not in a syllabus which doesn't use ...
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1answer
47 views

Heat equation fundamental solution

The following is from a book of PDEs and I have cannot seem to figure out a particular step in it with regard to the derivation of the fundamental solution of the heat equation. I have highlighted it ...
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0answers
55 views

Converting a boolean expression into CNF and DNF

Is there any systematic way to convert the following boolean expression (QUBO) into CNF or DNF? Here, $x_1, \ldots, x_n \in \{0, 1\}$, $a_1, \ldots, a_n \in \mathbb{Z}$ and $b_{1,1}, \ldots, b_{n,n} ...
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0answers
69 views

All $f(x)$ on $[0,1]$ such that center of mass of the function (uniform density) is on its graph

So, as the title describes, I'm trying to find a way to express all $y=f(x)$ differentiable on $[0,1]$ such that the center of mass of the function, assuming it has uniform density, will be a point on ...
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1answer
21 views

Angular momentum superposition

I am doing a space simulation. I have a spaceship and this spaceship has engines that don't push the spaceship through its center of gravity. These engines can therefore give the spaceship angular ...
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1answer
44 views

Equations of Motion of a Satellite

I'm working on a control problem where I need to know the equations of motion for a satellite orbiting the earth about a central axis. Using Newton's second law, I was able to show that ...
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1answer
41 views

Calculation of a spread light on a surface

Suppose we have a light of power $P$ distributed on a plane $(x,y)$. The distribution of the power is of the form: $$P=f(x,y)$$ If we have a lens conjugating every point of the plane $(x,y)$ in ...
2
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1answer
81 views

Solve $\int_0^T f(t) dt =1$ for T.

I have to solve this equation for a physics problem and I don't know where to start: $$\int_0^T f(t) dt =1 \quad\text{and}\quad f(T)=C$$ Where $T>0$, $C>0$ and $f(t)>0$ we can suppose that ...
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0answers
48 views

Approximation for uniform load on parabolic cable along its arc length

I am doing analysis for cable structures. Let's say that the cable stretches from point A to point B and carries a vertical ...
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2answers
54 views

Is it possible to build a fiber bundle whose fibers are different? (Or we should not call it a fiber bundle?)

Suppose there is a fiber bundle $E$. The base space is $M$ so that $\pi:E\rightarrow M$ is the projection. By the definition, the bundle has a typical fiber $F$ such that the local trivialization over ...
6
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1answer
38 views

Is there always an equilibrium point in a field?

For instance, considering a set of planets represented as point masses that create a gravitational field, will there always, no matter what set of points, be a place where I can stand with no net ...