All Questions
Tagged with calculus classical-mechanics
60
questions
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130
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Working with infinitesimal quantities and the motivation behind it
So in my freshman physics class, in classical mechanics the homework was (it's solved already, this isn't a homework thread) the following:
"A thin, spinning ring is placed on a table, that divides ...
- 163
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2
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239
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$\int (f(x+\delta x) - f(x)) dx = \int \left ( \frac{df(x)}{dx} \delta x \right) dx$
From Landau and Lifshitz's Mechanics Vol: 1
$$
\delta S= \int \limits_{t_1}^{t_2} L(q + \delta q, \dot q + \delta \dot q, t)dt - \int \limits_{t_1}^{t_2} L(q, \dot q, t)dt \tag{2.3b}$$
$$\Rightarrow ...
user208739
0
votes
1
answer
55
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Change of variable in function
Suppose I have a function $h(\theta)$ measuring the height of a piston, with $\theta = \omega t$. I would like to know the vertical acceleration of this piston as $\omega$ changes at the point $\theta ...
- 1,356
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2
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2k
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Derivation of generalized velocities in Lagrangian mechanics
So I know that: $$r_i = r_i(q_1, q_2,q_3,...., q_n, t)$$
Where $r_i$ represent the position of the $i$th part of a dynamical system and the $q_n$ represent the dynamical variables of the system ($n$ =...
user63248
0
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1
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177
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Acceleration as the second derivative of $e^{-\frac{1}{t^2}}$ [duplicate]
If we have, say, a material point with a zero velocity at the time $t=0$, and this point starts moving at a time $t>0$ , then we look at the force impressed on the point by inspecting the second ...
- 139
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1
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328
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Looking for a calculus based physics book [duplicate]
Ive gotten through University Physics by freedman et al and I realised i really enjoy the calculus problems in them which involve integrating infinitismal element (moment of inertia, charged rods, ...
1
vote
3
answers
2k
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Integration over arbitrary domains
In mathematical physics, we sometimes encounter situations where we have integrals of the forms:
$$\text{(case 1):}\ \ \ \ \int\limits_{D} f(x,y,z) dx dy dz =k$$
$$\text{(case 2):}\ \ \ \ \int\...
- 632
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1
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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:
$$
k\left(\frac{\hat{x}}{r} + \frac{\hat{y}}{r} + \frac{\hat{z}}{r}\right)\frac{1}{(x^2 + y^2 + z^2)^{n/2}}
...
- 13
1
vote
0
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258
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Partial derivative of $v$ w.r.t. $x$ in Lagrangian dynamics [duplicate]
In Lagrangian dynamics, when using the Lagrangian thus:
$$
\frac{d}{dt}(\frac{\partial \mathcal{L} }{\partial \dot{q_j}})-
\frac{\partial \mathcal{L} }{\partial q_j} = 0
$$
often we get terms such ...
- 11
3
votes
2
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845
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Moment of inertia integral has mass, not radius differential?
We've been learning about the derivation for moment of inertia as:
$$\int r^2 dm$$
However, for me, this looks like it's a bit backwards. As a first year calc student, I see the differential in the ...
- 1,177
1
vote
2
answers
584
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Expansion in $\epsilon$ and $v^2$ dependence of the Lagrangian - Landau & Lifshitz's Mechanics [duplicate]
On page 4 of Landau & Lifshitz's Mechanics they say
$$L\left({v^\prime}^2\right) = L\left(v^2 + 2\bf{v \cdot} \bf{\epsilon} + \epsilon^2\right).$$ Expanding this expression in powers of $\...
- 11
1
vote
2
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755
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Composing integrals in physics?
OK... so this problem isn't really specific... it's more of a conceptual puzzle.
I've recently started using integrals while solving problems in physics (specifically Newtonian Mechanics and other ...
- 105
4
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1
answer
1k
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Landau's derivation of a free particle's kinetic energy- expansion of a function?
I was reading a bit of Landau and Lifshitz's Mechanics the other day and ran into the following part, where the authors are about to derive the kinetic energy of a free particle. They use the fact ...
- 1,593
7
votes
1
answer
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What's the proper interpretation of canceling infinitesimals? [duplicate]
In most textbooks of physics I've found this demonstration of work-kinetic energy theorem:
$$\begin{align}
W &= \int_{x_{1}}^{x_{2}} F(x)\ dx \tag{1}\\
&= \int_{x_{1}}^{x_{2}} m\cdot a\ dx \...
- 233
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1
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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