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1 vote
Accepted

Formula for predicting next position does not include maximum velocity. Is there a formula that does?

The original formula you gave is obtained by integrating the velocity: $x(t) = x_0 + \int_{0}^{t} v(t) dt = x_0 + \int_{0}^{t} (v_0 + at) dt = x_0 + v_0 t + \frac{1}{2} at^2$ Where we computed $v(t)$ ...
0 votes

Finding Height In Gerstner Wave Function In World space.

Solution that I use: $$ y=\frac{a}{2}\sin\left(\lambda x-a\cos\left(\lambda x-a\cos\left(...\right)\right)\right) $$ with: $a$: wave effect strength $p$: wavelength $\lambda$: wave number, $\lambda=\...
1 vote

Following/Seeking a Moving Position with Velocity & Sin/Cos

You might have a better time if you avoid the trig functions altogether. Let $\Delta x = \mathbf{targetX} - \mathbf{projectileX}.$ Let $\Delta y = \mathbf{targetY} - \mathbf{projectileY}.$ Let $\Delta ...
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1 vote
Accepted

Following/Seeking a Moving Position with Velocity & Sin/Cos

It doesn't specify what language you are using, but atan2 almost universally takes the $y$ coordinate first and the $x$ coordinate second, and then why are you ...
  • 10.4k
1 vote
Accepted

Product of two spherical harmonics as a linear combination of spherical harmonics

It is (3.8.72) and background sections (3.6, 3.8) of Modern Quantum Mechanics, by Sakurai & Napolitano (2010 Addison Wesley), ISBN-13: 978-0805382914 . The point is $$ \langle \theta, \phi| l,m\...
4 votes

Integral of an exponential multiplied by a line

Hint Start completing the square $$Ar^2+Br=A\left(r+\frac{B}{2 A}\right)^2-\frac {B^2}{4A }$$ Now, let $r=x-\frac{B}{2 A}$ to make $$\int r\,e^{Ar^2+Br}\,dr=\frac{e^{-\frac {B^2}{4A } } }{2A }\int (B-...
3 votes
Accepted

Why do physicists label irreducible representations of su(2) with half integers?

To my mind, the explanation is at the level of Lie groups, not Lie algebras, since the Lie algebras preserve slightly less information than the Lie groups. Here, while $su(2)\to so(3)$ is an ...
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1 vote
Accepted

Initial-value problem with a forcing function that has step discontinuity

I thought that it might be instructive to present an alternative approach that recasts the problem using a nascent Dirac Delta [1]. To that end we proceed. Suppose $x_n(t)$ satisifies the ...
  • 168k
1 vote
Accepted

On the spectral decomposition of a positive semidefinite matrix

We want to show the following two statements are equivalent: For any positive semidefinite matrix $\rho$ there exists an orthonormal basis $|i\rangle$ and nonnegative $\lambda_i$ such that $$ \rho = \...
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2 votes

Initial-value problem with a forcing function that has step discontinuity

Conservation of energy is the way to go. Since you mention Lagrangian mechanics, the discontinuous potential $H(x)$ means you'd either need to formulate the equations of motion in a weak sense, or let ...
  • 3,651
3 votes

Initial-value problem with a forcing function that has step discontinuity

One way to solve the equation is to first multiply with $\dot x(t)$: $$ m\,\ddot{x}\dot{x} = - \left[V_2 - V_1\right]\delta(x)\dot{x}. $$ Both sides can then be written as derivatives: $$ \frac{d}{...
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0 votes

Collision time of 2 circles with friction

Your equation is correct, the time where the objects collide is the smallest solution of the equation. Unless this is a special case, that equation can't be simplified any better since if you're able ...
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0 votes

differentiating an expression in differential form

The issue isn't with the mathematics but with the physical implications following from the expression I am working with. For the benefit of future audiences: Note that (some included for completion): $...
0 votes

differentiating an expression in differential form

I think it follows from the fact that $pv = \frac{PV}{m}$ so if you consider the differential of that, you get, using the product rule that $ d(pv) = d(\frac{PV}{m}) = dP \cdot \frac{V}{m} + \frac{P}{...
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2 votes

How does the pushforward of the inverse metric relate to the inverse of the pullback metric for an embedding?

As pointed out in the comments, the subspace projection in $TM$ needs to account for the metric $g$. I have put together a proof on matrix level to see how the pieces come together: If we obtain the ...
1 vote

The vector A whose magnitude is 1.72 units makes equal angles with the coordinates axes. Find Ax,Ay,Az

The results are different because you took the liberty to claim that the angle was $45^\circ$. Whereas the angle must be $\cos \frac{-1}{\sqrt3}\simeq54^\circ$ (approx). Which can be found by the ...
0 votes

Question on gauge fields "acting on different representations"

To answer via the second suggestion, $\rho_{R^*}$ is $0$ on $\mathfrak{su}(2)$ because it is effectively just the trivial action on $\mathbb{C}$: $$\rho_{R^*}(W+B)(X\otimes Y) = \rho_1(W)X \otimes Y + ...
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1 vote
Accepted

How to solve the differential equation of the form $a\ddot{x}-f(t)\dot{x}^{-1}+b \dot{x}^2+c=0$?

Let $x(t)=k^{-1}\ln y(t)$, $k=\frac{\kappa}{m}$ and $f(t)=\frac{\tau(t)}{mrk}-\frac{c}{mk}$. Your second equation may be written $$\tag{1} y''-fy=0 $$ This linear second order equation has no closed ...
  • 3,651
1 vote

What does the "3" in $d^3x_1d^3p_1\ldots d^3x_Nd^3p_N$ mean here?

The explanation may be found on page 2 of the document. Here is an image: It's the same number as the 6N here: You have $N$ particles with 3 position and 3 momentum values each.
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0 votes

What does the "3" in $d^3x_1d^3p_1\ldots d^3x_Nd^3p_N$ mean here?

It's just an index indicating that you are considering a volume (3D) integral. Each of the $x_j$ and $p_j$ is a vector with $3$ components.
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3 votes
Accepted

Confusion About Physical Interpretation of Complex Numbers

Often in physics we want the real solutions to an equation which, due to linearity, has the following property: if $x$ is an in general complex solution, so are $x^\ast$ and the real-valued linear ...
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