All Questions
Tagged with classical-mechanics differentiation
12
questions with no upvoted or accepted answers
2
votes
1
answer
62
views
Implications of Galilei-Invariance on a time-independent potential
I'm trying to compute a result shown in my classical mechanics lecture on my own. Namely, consider that a system composed of $n$ particles follows a law of force
$m_k\ddot{\vec{x_k}} = \vec{F_k}(\vec{...
- 324
2
votes
3
answers
397
views
Potential energy with constraints moving body
I know that for conservative forces $\vec{F}=-\nabla{U}$. Let's consider the case of gravitational potential energy, I know that $U=mgy$. Just to check: $\vec{F}=-\nabla{U}=(0,-mg)$: perfect!
Now, let'...
1
vote
2
answers
100
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Why must a constraint force be normal?
If we impose that a particle follows a holonomic constraint, so that it always remains on a surface defined by some function $f(x_1,x_2,x_3)=0$ with $f:\mathbb{R^3}\rightarrow\mathbb{R}$, we get a ...
- 101
1
vote
1
answer
93
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How to define differentiation of a time-dependent vectors with respect to a specific reference frame in a coordinate-free manner?
It is usual in classical mechanics to introduce the derivative of a time-dependent vector with respect to a reference frame. This is accomplished through the use of a basis that is fixed with respect ...
- 245
1
vote
1
answer
143
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Question regarding Energy Interaction of two particles
https://i.sstatic.net/LUsKX.jpg
To give a context as to what I'm asking here ,I am talking about the energy of a two particle system (section 4.9 Taylor's Classical Mechanics) .
My question is what ...
- 367
1
vote
0
answers
50
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Functional derivative of a symmetrized field
I'm confused whether a symmetrisation/antisymmetrization of a function with respect to its arguments, i.e., $$F(x_1,x_2,...,x_n)=\frac{1}{\sqrt{N!}}\sum_{\pi}\textrm{sgn}(\pi)~f(x_{\pi(1)},...,x_{\pi(...
- 1,323
1
vote
1
answer
621
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Determining the change in radius of water flowing from a faucet - more general question on differentiation
I've been outside of the academic world for several years now, and I'm forcing myself to go back through old textbooks and resources and work through the information in there. I can tell I'm losing ...
- 11
0
votes
1
answer
76
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In Lagrangian mechanics, do we need to filter out impossible solutions after solving?
The principle behind Lagrangian mechanics is that the true path is one that makes the action stationary. Of course, there are many absurd paths that are not physically realizable as paths. For ...
- 50.2k
0
votes
0
answers
82
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Cartesian coordinate velocity and generalized coordinate velocity
use $x_k$ to denote the kth component of cartesian coordinate, and $q_k$ to denote the generalized coordinate.
Taking the derivate of $x_k(q_1,q_2,q_3,t)$ w.r.t. time, we have
$$\frac{d x_k(q_1,q_2,...
- 103
0
votes
1
answer
33
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Suppose for all value of $r$ expression for Effective Potential Energy $U_{eff}$ is zero, does that mean $F(r)$ is zero?
Suppose for all value of $\textbf{r}$ expression for Effective Potential Energy ($U_{eff}$) is zero, does that mean $F(\textbf{r})$ is zero?
- 11
0
votes
0
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55
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Math question about point transformations
I am trying to prove the classic problem to showcase Lagrangian's scalar invariant property.
Namely, that if you have $x_i = \{ x_1, ...., x_n; t \}$ , you can then represent $L(x_1,....,\dot{x_1},.....
0
votes
0
answers
345
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Different subscripts for $\nabla$ operators while deriving force on system of many particles
Consider a system of 4 particles in an external conservative field. So force acting on each particle is derived from potential energy $U(x,y,z)$of the particle+field system:
Total (external) force on ...
- 1,034