All Questions
Tagged with calculus classical-mechanics
61
questions
0
votes
1
answer
177
views
Acceleration as the second derivative of $e^{-\frac{1}{t^2}}$ [duplicate]
If we have, say, a material point with a zero velocity at the time $t=0$, and this point starts moving at a time $t>0$ , then we look at the force impressed on the point by inspecting the second ...
- 139
0
votes
1
answer
328
views
Looking for a calculus based physics book [duplicate]
Ive gotten through University Physics by freedman et al and I realised i really enjoy the calculus problems in them which involve integrating infinitismal element (moment of inertia, charged rods, ...
1
vote
3
answers
2k
views
Integration over arbitrary domains
In mathematical physics, we sometimes encounter situations where we have integrals of the forms:
$$\text{(case 1):}\ \ \ \ \int\limits_{D} f(x,y,z) dx dy dz =k$$
$$\text{(case 2):}\ \ \ \ \int\...
- 632
0
votes
1
answer
2k
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What is the curl of $k\hat{r}/r^n$?
I'm trying to find the curl of ${\bf F}(r) = k \hat{r}/r^n$. I think that this converts to:
$$
k\left(\frac{\hat{x}}{r} + \frac{\hat{y}}{r} + \frac{\hat{z}}{r}\right)\frac{1}{(x^2 + y^2 + z^2)^{n/2}}
...
- 13
1
vote
0
answers
258
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Partial derivative of $v$ w.r.t. $x$ in Lagrangian dynamics [duplicate]
In Lagrangian dynamics, when using the Lagrangian thus:
$$
\frac{d}{dt}(\frac{\partial \mathcal{L} }{\partial \dot{q_j}})-
\frac{\partial \mathcal{L} }{\partial q_j} = 0
$$
often we get terms such ...
- 11
3
votes
2
answers
847
views
Moment of inertia integral has mass, not radius differential?
We've been learning about the derivation for moment of inertia as:
$$\int r^2 dm$$
However, for me, this looks like it's a bit backwards. As a first year calc student, I see the differential in the ...
- 1,177
1
vote
2
answers
585
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Expansion in $\epsilon$ and $v^2$ dependence of the Lagrangian - Landau & Lifshitz's Mechanics [duplicate]
On page 4 of Landau & Lifshitz's Mechanics they say
$$L\left({v^\prime}^2\right) = L\left(v^2 + 2\bf{v \cdot} \bf{\epsilon} + \epsilon^2\right).$$ Expanding this expression in powers of $\...
- 11
1
vote
2
answers
758
views
Composing integrals in physics?
OK... so this problem isn't really specific... it's more of a conceptual puzzle.
I've recently started using integrals while solving problems in physics (specifically Newtonian Mechanics and other ...
- 105
4
votes
1
answer
1k
views
Landau's derivation of a free particle's kinetic energy- expansion of a function?
I was reading a bit of Landau and Lifshitz's Mechanics the other day and ran into the following part, where the authors are about to derive the kinetic energy of a free particle. They use the fact ...
- 1,593
7
votes
1
answer
236
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What's the proper interpretation of canceling infinitesimals? [duplicate]
In most textbooks of physics I've found this demonstration of work-kinetic energy theorem:
$$\begin{align}
W &= \int_{x_{1}}^{x_{2}} F(x)\ dx \tag{1}\\
&= \int_{x_{1}}^{x_{2}} m\cdot a\ dx \...
- 233
0
votes
1
answer
110
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Showing $ \textbf{F} \cdot d\textbf{s} = -dV$ is equivalent to $ F_s = -\frac{\partial V}{\partial s}$
Can someone please explain how the following
$$ \textbf{F} \cdot d\textbf{s} = -dV$$
is equivalent to
$$ F_s = -\frac{\partial V}{\partial s}$$
using some intermediate steps. I don't follow this ...
- 2,251