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Questions tagged [mathematical-physics]

This tag is intended for questions on methods used e.g. in quantum mechanics or general relativity at an advanced undergraduate or graduate level. For questions about more basic mathematical problems originating physics from use the (physics) tag.

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0answers
29 views

Euler field for magnectic symplectic structure in $T^*Q$

Let $Q$ be a differentiable manifold, $\pi\colon T^*Q\to Q$ denote its cotangent bundle, and $B \in \Omega^2(Q)$ be a closed form. I'm playing around with some things, and I'm not sure whether it ...
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0answers
8 views

Word problem on resistance and required effort. [on hold]

1m crowbar w/ resistance arm of 3cm - how much effort at end crowbar to lift cover which exerts 445 N on bar? What is IMA of the crowbar? Sorry, I am new here so if this breaks rules please be kind. ...
0
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0answers
21 views

Good introductory book on Mechanics

I have to learn mechanics from a mathematical perspective for by graduate courses. I have a basic idea about Newtonian mechanics from school and under-grad mathematical background . Could some one ...
0
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0answers
16 views

Is there a proof that a function which satisfies these axioms of relativity must be linear?

Let us only consider the 2 dimensional case where the two frames of reference coincide at the origin. For each $v\in (-c,c)$, let $f_v:\mathbb{R}^2\to\mathbb{R}^2$ be a smooth function satisfying the ...
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0answers
19 views

Calculating Steering Angle for Curve

I am a vehicle trying to get from point $A$ to point $B$. $A$ and $B$ are on the same line, but the vehicle needs to travel along a curve outside the line, how do I calculate what angle the vehicle ...
0
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0answers
20 views

Stability criterion for leapfrog in relativistic physics.

I am doing a 2D MD simulations of charge carriers in graphene using the Leapfrog algorithm. I am trying to prove that, in some specific cases (when distance between particles is small), the method is ...
1
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3answers
51 views

Tensor equation [on hold]

If you have two antisymmetric tensors $A_{\mu \nu}$ and $B_{\mu \nu}$, and for every anti symmetric tensor $\epsilon^{\mu \nu}$, $\epsilon^{\mu \nu} A_{\mu \nu} = \epsilon^{\mu \nu} B_{\mu \nu}$ Is ...
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0answers
19 views

High speed flow [on hold]

a) The inlet of a supersonic fighter is designed to raise the pressure of the air entering engine. Determine the maximum pressure that can be achieved in an inlet if it is mounted on an aircraft ...
4
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1answer
51 views

Is the set $\{\delta_x\}_{x \in [a, \ b]}$ a basis for the set of distributions on $C^{\infty}_c([a, \ b])$?

$\def\braket#1#2{\langle#1|#2\rangle}\def\bra#1{\langle#1}\def\ket#1{#1\rangle}$ Is the set $\{\delta_x\}_{x \in [a, b]}$ a basis for the set of distributions on $C^{\infty}_c([a, \ b])$? Below are ...
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0answers
39 views

What is Omega(Ω) for in maths? [on hold]

I looked up the meaning of Ω on google but it just said that it signifies the last item of something or something like that but I would like a more clear explanation. I recently got this equation in ...
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0answers
15 views

Value of integral of Omnes function

I want to get the value of this integral: $$\int_s^\infty \left(\frac{a}{z(z-b)} \right) dz$$ where a and b denote real constant , and b > s so there is a singularity and I don't know how to get the ...
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0answers
19 views

Modified equation for KdV using 2.Order Discretization scheme

The modified equation of linear PDEs can be found in a systematic manner (https://www.sciencedirect.com/science/article/pii/0021999174900114). However, it does not seem to be that easy for nonlinear ...
1
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1answer
29 views

What is the definition of a bounded operator in an infinite dimensional Hilbert Space?

I am struggling to understand the meaning of a bounded operator in a Hilbert Space. Does a bounded operator simply means that if it acts on an element of the Hilbert Space, the "result" is bounded?
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0answers
10 views

How do u determine the stagnation temperature and pressure in the combustion chamber? [on hold]

A rocket engine is powered by liquid hydrogen and oxygen. THe combustion gas is accelerated to supersonic speed through a convergent divergent nozzle. It is designed to produce an exhaust gas velocity ...
0
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1answer
79 views

Find acceleration time for u,a,t,s

A car moves from point $A$ to $B$ in $t_{\text{total}}$. $$v_i=0\ m/s\\ a= 0.65\ m/s^2\ (acceleration)\\ d=-0.65\ m/s^2\ (deceleration)\\ v_{max} = 1 m/s\\ t_{\text{total}}=10.0\ s\\ s_{A\to B}(\...
4
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1answer
54 views

Differential Geometry in Hamiltonian Mechanics

I have following question about the Hamiltonian mechanics from differential geometrical viewpoint: We start with a physical system parametrized by generalized (position) coordinates $(q^i)$ providing ...
0
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1answer
62 views

How to find the free fall time of a particle starting at rest at a distance R from a mass M.

This is in-fact a physics question but seems as how I only want to know how to treat it mathematically I thought I would ask it here(Note: this is not a homework question, I'm just studying for an ...
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1answer
31 views

How do I solve this trajectory of projectile equation? [closed]

$$y\ =\ x\tan\theta\ -\ \frac{1}{2}\cdot g\cdot\frac{x^2}{v_0^2.\cos^2\theta}$$ Where $$g\ =\ 3;$$ & $$y\ =\ 3\ @ \ x\ =\ 3,$$ $$y\ =\ 0\ @\ x\ =\ 6,$$ $$y\ =\ 0\ @\ x\ =\ 0,$$ Solve For: $$\...
0
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2answers
39 views

High School Mathematical Research Project Ideas [closed]

i am currently looking out for some possible topics i could study for my research project in high school. Algebra, trigonometry, Pythagoras’ theorem, geometry, circles and their properties, etc. and ...
1
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2answers
54 views

Delta distribution and the Schrödinger equation

While studying the lecture notes of my quantum mechanics course I came across something that seemed a bit odd. There we want to solve the Schrödinger equation for the potential $V(x)=V_0 \delta(x)$, ...
1
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1answer
49 views

How can the shape in this picture be descibed mathematically?

I'm reading an article on Boltzmann that contains the following figure: It is obviously related to harmonic oscillators (since the article says so), but how could I describe this figure ...
0
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1answer
34 views

Lie algebra generator that does not appear on rhs of the algebra

Is it possible that in a lie algebra one may have a generator that does not appear on any of the commutators? If the above line is too physicisty for you, maybe I can try to translate to a more ...
0
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0answers
16 views

Problem in calculating the gradient of a function?

I am trying to understand the gradient calculation of a formula, which is an optimization function and trying to maximize the GDL (Generalized Dice Loss) where, w_l...
3
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2answers
63 views

Proving that a solution to a differential equation is monotonic

The answer that ws given on a previous question of mine, stated that the solution to this DE: $$x(t)\cdot r+x'(t)\cdot l+a\cdot\ln\left(1+\frac{x(t)}{b}\right)=0\space\Longleftrightarrow\space x(t)=\...
2
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0answers
21 views

Decay properties of the Dirac equation in Witten's positive energy proof

In his 1981 paper E. Witten provides a proof of the positive energy theorem by considering the "Dirac" equation $\not D \epsilon \equiv h^{ab}\gamma_a \nabla_b \epsilon=0$ on a spacelike hypersurface ...
0
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2answers
41 views

Can someone help me simplify this boolean algebra [closed]

ABCD + AB(CD)' + (AB)'CD when i used basic rule it becomes weird but boolean calculator shows something else The Question is to simplify the expression using the boolean algebra so My solution was ...
5
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1answer
51 views

Definition of conjugate momentum on a manifold

I have trouble understanding this definition: Let $Q$ be some manifold and $L: TQ \to \mathbb{R}$ a smooth function. Then for some local coordinates $(q, \dot{q})$ on $TQ$ the conjugated momentum ...
2
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1answer
52 views

Solving an DE involving a logarithm

I've the following DE, describing a physical phenomenon. And the prupose is to solve that DE: $$x(t)\cdot r+x'(t)\cdot l+a\cdot\ln\left(1+\frac{x(t)}{b}\right)=0\space\Longleftrightarrow\space x(t)=\...
2
votes
1answer
68 views

Is it possible to take square root of the operator $e^{a\frac{d}{dx}}$?

Is it possible to take square root of the operator $e^{a\frac{d}{dx}}$ where $a$ is a real or complex constant? Actually in physics one can take the square root of $(\Box +m^2)$ operator associated ...
2
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0answers
66 views

Integrals over the space of Riemannian metrics on $M$

Let $M$ be a closed smooth $n$-manifold. In the Hartle-Hawking proposal of Euclidean quantum gravity, the authors consider functional integrals of the form $$ I_M=\int_{\mathcal{Met}(M)}e^{-S(g)}\;\...
1
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1answer
56 views

Black Holes and Diagrams of Different Coordinate Systems

In General Relativity by Woodhouse there are the three following diagrams in Chapter 9 about Black Holes. Despite a (very brief) description of these diagrams in the book itself, I am struggling to ...
0
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1answer
28 views

Complex zeroes of Error Function and Parabolic Cylinder Function.

Does there exist EXACT zeros of Error Function (Erfc(z)) and Parabolic Cylinder Function ($D_v(z)$)(http://functions.wolfram.com/HypergeometricFunctions/ParabolicCylinderD/). Here (https://dlmf.nist....
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0answers
38 views

How does one use Green's function of the operator to get the solution of the arbitrary boundary value problem?

Assume I've been given an operator $L$ and its Green's function $G(s, s')$. This is the function that solves the following: $$ L G(s, s') = \delta(s-s'), G(a,s') = G(b, s') = 0 $$ I know how to get a ...
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0answers
31 views

How does one obtain Green's function on practice? Is it used besides than a solution representation nowadays?

If it is used, than how to numerically compute it. It seems to me that it would take a great deal of time to get it with a reasonable precision. And as the function is used as accumulator, the ...
3
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2answers
89 views

Black Holes and the Schwarzschild Solution

From Section 9.1, in General Relativity by Woodhouse: For a normal star, the Schwarzschild radius is well inside the star itself. As it is not in the vacuum region of space-time, the Ricci tensor ...
1
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1answer
31 views

If we use a sinusoidal signal as an input signal to a linear transmission path, then we always get out a sine wave of the same period/frequency

An Introduction to Information Theory: Symbols, Signals and Noise, by John R. Pierce, says the following: With the very surprising property of linearity in mind, let us return to the transmission ...
2
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1answer
49 views

Viscous Fluids at a Slope (Navier-Stokes)

An in-compressible viscous fluid flows down a flat slope of angle θ to the horizontal under the force of gravity, with g the acceleration due to gravity. What are the boundary conditions for the fluid ...
1
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1answer
50 views

Energy and the Schwarzschild Metric

In the Schwarzschild metric, the geodesic Lagrangian is $$L=\frac{1}{2} \left[ \left( 1-\frac{2m}{r} \right) \dot{t}^2-\frac{\dot{r}^2}{1-2m/r}-r^2(\dot{\theta}^2+\sin^2\theta\dot{\psi}^2) \right]$$ ...
0
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1answer
21 views

Stationary Observers Question

An observer in a fixed location relative to our coordinate system has a worldline with constant $r, \theta, \phi$, and thereofre has four velocity $U$ with only the first component non zero. Because $...
0
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1answer
64 views

Further deriving the weak field limit in general relativity

Consider the motion of a slow moving particle with worldline $x^a = x^a(t)$. We have $$\frac{dx^0}{dt}=1, \frac{dx^1}{dt}=u_1,\frac{dx^2}{dt}=u_2,\frac{dx^3}{dt}=u_3$$ where $(u_1,u_2,u_3)$ ...
1
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1answer
17 views

The weak field limit metric setup

Assume that $$g_{ab}=m_{ab}+h_{ab}$$ where $m_{ab}=\text{diag}(1,-1,-1,-1)$ is the Minkowski space metric in an inertial coordinate system $x^a$, and $h_{ab$} is small and slowly varying. $1)$ I ...
4
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1answer
105 views

The Metric Tensor, A Body of Mass m and Minkowski Space

By saying the body has mass m, we mean that the metric approaches that of Minkowski space for large r and that $$g_{00} \sim 1-2m/r$$ This was under a section in General Relativity by Woodhouse ...
13
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1answer
149 views

In the Physicists' definition of the path integral, does the result depend on the choice of partitions?

The standard definition of the path integral in Quantum Mechanics usually goes as follows: Let $[a,b]$ be one interval. Let $(P_n)$ be the sequence of partitions of $[a,b]$ given by $$P_n=\{t_0,\dots,...
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0answers
48 views

Tidal forces in General Relativity

In local inertial coordinates in which the observer is instantaneously at rest, $V = (1,\textbf{0})$ and $Y = (0,\textbf{y})$, where y is the position vector of the second particle. The acceleration ...
2
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1answer
39 views

How to solve this functional equation involving hyperbolic functions?

I'm reading this (physics) book. They have the recurrence relation (book eq. 14.2.14) $$f(K_1,0)=-\frac{1}{2}\ln\{2\sqrt{\cosh(2K_1)}\}+\frac{1}{2}f(\ln\sqrt{\cosh(2K_1)},0).\qquad(1)$$ They give ...
2
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1answer
66 views

General Relativity and the Wave Equation

Suppose that u is a function on space-time. We can define a vector field with components $\triangledown^a u = g^{ab}\partial_b u$. Why have they chosen $g^{ab}$ and not $g_{ab}$ here? The wave ...
2
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1answer
38 views

Uniqueness of representations of matrix product states

Let $\{A_k\}_{k=1\dots N}$ and $\{B_k\}_{k=1\dots N}$ be two sets of $d\times d$ matrices over the complex numbers such that for any length $L$ and any sets of indices $\{j_1,j_2,\dots j_L=1\dots N\}$ ...
1
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2answers
33 views

Series expansion upto linear order

I want to do a series expansion of the function given below around $a=0$ and keep the terms only upto $O(a)$. The function is $$ f(a)=\frac{a \pi^2\sin^2\theta}{a^2-\pi^2\cos^2\theta} $$ Using $f(0)+...
0
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1answer
56 views

functional equation in renormalization group theory

In the Renormalization Group Theory, a key step is the derivation of the so called scaling equation, which in general is in the form of a functional equation of the kind: $$ g(\mu(\lambda)x,\nu(\...
1
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2answers
43 views

Minimizer of square root operator norm

Let $A:D(A) \to \mathcal H$ be a positive self-adjoint operator and $\sqrt{A}$ defined by via the spectral theorem on $D(\sqrt{A}) = Q(A)$ where $Q(A)$ is the quadratic form domain. Let $$E=\inf\{\...