0
votes
1answer
26 views

How does the hermitian conjugate of an unitary operator act?

I know that $$ A| v \rangle = \sum _n e^{i\alpha n} | n \rangle $$ where $A$ is an unitary operator, and $ \left \{ |n\rangle \right \} $ is an orthonormal complete basis. In that case, is it true ...
0
votes
0answers
17 views

About diagonalizing a matrix for a quadratic expression (with the goal of uncoupling mixed terms)

my question is originated from a physical problem. I will try to present the problem as simple as possible, but I fear it will still be long since I'm bad at expressing myself briefly. It starts with ...
3
votes
1answer
49 views

How to mathematically determine if the magnitude of a cross product is up/down(positive/negative?)?

So, I'm a newbie at complex vector math. I'm working on a 2D physics engine, and my issue is, with angular acceleration from torque, is it supposed to be positive or negative? I understand the right ...
38
votes
15answers
6k views

Why learn to solve differential equations when computers can do it?

I'm getting started learning engineering math. I'm really interested in physics especially quantum mechanics, and I'm coming from a strong CS background. One question is haunting me. Why do I need ...
1
vote
1answer
39 views

Transpose of a differential operator

Let $H$ be a diagonalizable matrix (not necessarily Hermitian). Then, it induces a biorthogonal left and right vectors, such that $$ ...
1
vote
1answer
61 views

Programming constraints in video game. How are these two equations equal?

I'm currently working on programming a game that uses a physics engine (NAPE). Inside of that engine there are constraints that you can program. In order to program those you need a somewhat ...
2
votes
1answer
161 views

Determining the ratios needed in gear reduction

I am trying to work out the math behind building a gear box for turning a gear a specific RPM from a small motor. Given that a typical DC hobby motor turning at 200 RPM, and a target in the final ...
1
vote
0answers
46 views

Cross Product - Moments :: Dynamics 2

This problem is related to Cross Product - Moments :: Dynamics Please look at that link for the background on the problem I am faced with right now, I have linked a pdf of the book that I am using ...
0
votes
1answer
45 views

Linear algebra and special relativity

I'm going over an exam I had a couple months back, over the exercises I didn't manage to get right and I'm kinda stuck with the following subtask: Let $\xi$ be a 4-vector with the Minkowski scalar ...
0
votes
1answer
65 views

Elliptical polarisation

In physic context one find the curve with parametrisation in t, $x=x_0\cos(t)$ and $y=y_0\cos(t+\varphi)$ with is an ellipse with equation ...
2
votes
2answers
92 views

Square vs non-square tensors?

In mathematics, tensors are objects that operates on vector space. In physics or engineering, tensors usually operates on one vector space and its dual space: $V^{*} \times V^{*} \times V^{*} \times ...
4
votes
2answers
140 views

Should isometries be linear?

Question Suppose $V$ is a (finite-dimensional) vector space over $F$ ($\operatorname{char }F\neq2$, due to user1551) equipped with a non-degenerate quadratic form $Q$, and $T$ is a ...
2
votes
0answers
48 views

Fock Subspaces and Weight Vectors

I've got an assignment due in a few hours, and I'm at a complete loss as to how to even start it, really. I haven't encountered any Dirac notation before, so I'm having a lot of trouble attempting the ...
0
votes
0answers
30 views

Determine whether intersecting sphere moves towards cuboid?

I am programming a physics simulation in which I check every frame of a sphere intersects a cuboid. If it intersects, I want to check if the sphere moves "towards" the cuboid in a sense. If it does, ...
0
votes
1answer
168 views

Understanding quaternions & gradient descent in a paper on inertial / magnetic sensor arrays

I hope this question is appropriate here! I and a friend at work are trying to understand Sebastian Madgwick's paper, "An efficient orientation for inertial and inertial/magnetic sensor arrays" ...
1
vote
3answers
554 views

Velocity vectors and trigonometry

I am trying to learn about velocity vectors but this word problem is confusing me. A boat is going 20 mph north east, the velocity u of the boat is the durection of the boats motion, and length is ...
17
votes
2answers
381 views

Angular distribution of lines passing through two squares.

Let's say I've got two squares with side length $d$ that are held parallel at a distance $m$ apart. Suppose that particles are randomly falling from above in such a way that the polar angle ...
2
votes
3answers
365 views

What is the relation between vectors in physics and algebra?

Vector math is something I find very interesting. However, we have never been told the link between vectors in physics (usually represented as arrows, e.g. a force vector) and in algebra (e.g. ...
3
votes
2answers
128 views

Quadratic Equation with “0” coefficients

Let's say I have two objects $x$ and $y$ whose position at time $t$ is given by: $$ x = a_xt^2+b_xt+c_x \\ y= a_yt^2+b_yt+c_y $$ And I want to find which (if any) values of $t$ cause $x$ to equal ...
4
votes
1answer
487 views

Derivative of a bra?

I understand that $$ \frac{\mathrm d}{\mathrm dt} \langle\psi|\psi\rangle =\left[\frac{\mathrm d}{\mathrm dt} \langle\psi|\right]|\psi\rangle + \langle\psi|\left[\frac{\mathrm d}{\mathrm ...
6
votes
2answers
194 views

Vector Components - Superposition of Forces

I understand that by inspection of the given figure, $F_{1,x}$ (the x-component of $F_1$) must $< 0$, so $F_{1,x} = 250\color{red}{\cos{53^{\circ} }}$ can't be right. But I don't see how $F_{1,x} ...
2
votes
0answers
92 views

Solving Generalized Eigenvalue Problem perturbatively

Let me formulate the problem to convey my notation. I have a matrix $A$ which is hermitian - and is diagonalisable by a transformation $$ U_A A\,\,U_A^{-1} = A_{diag}$$ Now the matrix is changed, ...
10
votes
2answers
1k views

Cross product and pseudovector confusion.

So called pseudovectors pop up in physics when discussing quantities defined by cross products, such as angular momentum $\mathbf L=\mathbf r\times\mathbf p$. Under the active transformation $\mathbf ...
0
votes
1answer
42 views

What is $\left(\delta_{ab}\right)^{-1}$?

I have an expression that involves the Wigner 3j coefficient: $$\left(\matrix{a&b&0\\0&0&0}\right)^{-1}$$ This simplifies to: ...
0
votes
1answer
47 views

A group of matrices satisfying a particular constraint(definition) of the group

Suppose that one wants to have a group of matrices that satisfy some constraints. (As for a similar example, Pauli matrices satisfy some particular constraints.) The constraint goes like following: ...
1
vote
1answer
115 views

Eigenstate and quantum mechanics position opperator

Quantum mechanics math question: Suppose that there is eigenstate $|q \rangle$ where $q$ is position observable . The question is, 1) What is eigenstate? How is this different from eigenvector? ...
4
votes
4answers
464 views

Gram-Schmidt and zero vector

I have a problem concerning the orthogonalization of a coordinate system; this is necessary in the context of a normal mode analysis of molecular vibrations. I am working on H2O, giving me a ...
1
vote
1answer
69 views

How $v=(-\sin\frac{\alpha}{2}\sin\frac{\beta}{2},\cos\frac{\alpha}{2}\sin\frac{\beta}{2},\sin\frac{\alpha}{2}\cos\frac{\beta}{2})$ is derived from…

In the book Quarternion and Rotation Sequences, I can't seem to work out how the final equation (colored in $\color{red}{red}$) is derived from the original equation (colored in $\color{blue}{blue}$). ...
5
votes
1answer
221 views

A particular (functional) determinant calculation

One wants to calculate the quantity, $\det'(\frac{\partial}{\partial t} - i [\alpha, ])$ where the prime on the "det" means that one wants to do a product over only non-zero eigenvalues of the ...
3
votes
1answer
422 views

Rotation matrix from an inertia tensor

I have a set of weighted points in 3D space (in fact, a molecule) and I'm trying to align the principal axes of this set with the $x$, and $y$ and $z$ axes. To do so, I've first translated my points ...
3
votes
2answers
326 views

Practical physics problem about distance between lines.

I am a software engineer and I’m developing a soccer game. I have a solution for this problem based on Newton law and I’m using Newton method to solve the equation I got. I’m here because I think ...
3
votes
1answer
871 views

Prove Pythagoras theorem through dimensional analysis

I've recently become acquainted with Buckingham's Pi theorem for the first time . Then I've found an excercise that says: Use dimensional analysis to prove the Pythagoras theorem. [Hint: Drop a ...
2
votes
3answers
244 views

Applying a linear transformation to time sequences to separate interfering oscillations

This is an applied problem, which arises from the problem of reorienting of a sensor axes according to particle displacement directions: Consider a sensor which is located inside the solid substance. ...
2
votes
3answers
527 views

Scale Operator $Uf(x)=f(kx)$

I am looking for an operator $U$, that can do this to a function: $$Uf(x)=f(2x).$$ In particular I am happy if there is an $U$ for the general case: $Uf(x)=f(kx)$. Does such an operator exist for ...
6
votes
1answer
677 views

Fitting a parameter dependent matrix to its eigenvalues

The essence of my question is, if I have a Hermitian matrix that is linearly dependent on a set of parameters and I have an estimate of its eigenvalues, is there a "simple" way to determine the values ...
6
votes
1answer
734 views

How can angular velocity or angular momentum be a vector?

Rotations in 3 dimensions are not commutative; however they are in the plane. In classical mechanics, are we allowed to say that angular momentum is a vector because particles only rotate along a ...