All Questions
16 questions
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2
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Why must the total time derivative only be a linear function of velocity? [duplicate]
I'm hung up on page 7 of Landau & Lifshitz Course on Mechanics. They claim,
$$L(v'^2) = L(v^2)+\frac{\partial L}{\partial v^2}2\textbf v\cdot \epsilon \tag{p.7}$$
The second term on the right of ...
1
vote
1
answer
65
views
"To order $n$ of" arguments
Often one finds in physics textbooks that arguments will be made "to order $n$". I am not sure on what the procedure or argument ought to be when we have some denominator dependence though. ...
- 1,271
0
votes
2
answers
79
views
Approximation of Small Perturbation [closed]
From Morin's Classical Mechanics, on the chapter of Small Oscillations in Lagrangian Mechanics, he does this approximation on the last equality, I don't understand what happened there.
I get the first ...
- 109
4
votes
2
answers
196
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In physical calculations, is the elimination of higher-order small quantities an approximation or a strict equality in mathematics?
Physics sometimes uses a technique called the method of differentials, which seems magical and not very systematic. This makes me unsure which variable I should take the differential of, and sometimes ...
- 119
-1
votes
1
answer
66
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Approximation entropy at $T + dT$ [closed]
I have seen this approximation for a little change of temperature, but I don't understand how we got this result. I don't understand how we did this approximation mathematically.
$dS = \frac{-dQ}{T+dT}...
- 37
0
votes
2
answers
117
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What problems would arise in physics by treating infinitesimals as ~1 (in units much smaller than the measurement precision) rather than ~0?
I have a hopefully simple/ignorant question.
The difference quotient, where $h$ is an "infinitesimally" small value:
$$f'(x) = \frac{f(x + h) - f(x)}{h}$$
What problems (if any) would arise ...
- 900
0
votes
3
answers
79
views
Function Values Surrounding Stationary Points
Taylor, in his widely read book "Classical Mechanics," writes on page 218 that
When $df/dx = 0$ at a point $x_0$, but we don't know which of the 3 possibilities obtains, we say that $x_0$ ...
- 463
6
votes
1
answer
640
views
Infinitesimal and approximations in physics
I'm a first year student studying physics. Solutions of many physics problems, which I've seen so far, are achived through solving this problem for infinitesimal part of problem's subject (some curve, ...
- 61
2
votes
3
answers
1k
views
Why is this Taylor Expansion, Leading to the Boltzmann Distribution, Acceptable?
In Stephen Blundell's "Concepts in Thermal Physics" chapter 4 he derives the Boltzmann distribution. The equation that leads to the Taylor expansion is the following:
$$P_s(\epsilon) \propto ...
- 393
1
vote
3
answers
3k
views
Second derivative of energy as frequency of oscillations [closed]
Is there a way to algebraically see why when I take the second derivative of a potential energy in a point where it is minimal (force is zero), I generally get the frequency (squared) of the ...
- 1,048
4
votes
1
answer
263
views
Taylor Series of a logarithmic function
I was reading Intro to Modern Statistical Mechanics by David Chandler, on page 63. He states the following:
we can expand $\ln\Omega(E-E_v)$ in the Taylor series $$\ln\Omega(E-E_v) = \ln\Omega(E) - ...
- 358
-2
votes
3
answers
104
views
Changing derivative to difference quotient
Can differential be changed to Delta or difference? In high school education, in the acceleration section of Newton's formula 2, acceleration is a change velocity (velocity difference) divided by a ...
- 19
-1
votes
2
answers
88
views
Change in areal element
Example 1.7
Calculate the surface integral of $\mathbf{v}=2xz\hat{\mathbf{x}}+(x+2)\hat{\mathbf{y}}+y(z^2-3)\hat{\mathbf{z}}$ over five sides (excluding the bottom) of the cubical box (side 2) in Fig. ...
- 1
4
votes
2
answers
419
views
What does it mean when we say 'The difference between two quantities is of first order'?
This question is about the explanation below Eq.(6.19) of Modern Quantum Mechanics by Sakurai Nepolitano (2nd edition)
Let ${\bf j}(dx)$ be an operator that translates a point $x$ to $x+dx$.
jf(x) = ...
- 397
1
vote
1
answer
79
views
Quick question - infinitesimals proofs [duplicate]
In a few of my courses in mechanics certain statements/equations have been proved by assuming that two infinitesimals multiplied by each other are zero.
For instance in the equation : $dx + dy + dx^2 ...
- 121
3
votes
1
answer
759
views
Neglecting second order differentials
I am currently doing some Lorentz invariance exercises considering infinitesimal Lorentz transformations, and have been told to neglect second order differentials.
It's not the first time I have come ...
- 6,137