All Questions
17 questions
1
vote
2
answers
105
views
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 ...
- 131
1
vote
1
answer
34
views
Derivatives of the lagrangian of generalized coordinates [closed]
I know that
$$U= \frac{1}{2} \sum_{j,k} A_{jk} q_j q_k \quad \quad T= \frac{1}{2} \sum_{j,k} m_{jk} \dot{q}_j \dot{q}_k $$
and the lagrangian is
$$ \frac{\partial U}{\partial q_k} - \frac{d}{dt} \...
- 11
1
vote
1
answer
58
views
Energy change under point transformation
How do the energy and generalized momenta change under the following
coordinate
transformation $$q= f(Q,t).$$
The new momenta: $$P = \partial L / \partial \dot Q = \partial L / \partial \dot q\times ...
- 990
4
votes
1
answer
2k
views
How do total time derivatives of partial derivatives of functions work?
Say im trying to prove $\frac{\partial \dot{T}}{\partial \dot{q}^i} - 2\frac{\partial {T}}{\partial {q^i}} = - \frac{\partial {V}}{\partial {q^i}}$ from the Lagrangian equation: $L = T - V$, and the ...
- 75
2
votes
1
answer
244
views
Is Goldstein's matrix formalism to Hamiltonian mechanics necessary? [closed]
I am trying to see whether the matrix formalism of the Hamiltonian formalism (used in Goldstein's textbook) is truly necessary to solve problem in this framework.
It appears so based on the problem I'...
- 1,081
1
vote
0
answers
50
views
Functional derivative of a symmetrized field
I'm confused whether a symmetrisation/antisymmetrization of a function with respect to its arguments, i.e., $$F(x_1,x_2,...,x_n)=\frac{1}{\sqrt{N!}}\sum_{\pi}\textrm{sgn}(\pi)~f(x_{\pi(1)},...,x_{\pi(...
- 1,363
1
vote
2
answers
160
views
Why $\sum\limits_{i} \frac{\partial L}{\partial \dot{q_i}} \dot{q_i} = \sum\limits_{i} \frac{\partial T}{\partial \dot{q_i}} \dot{q_i} = 2T$? [closed]
From Landau and Lifschitz's "Mechanics"; section 6.
I understand up to this point
$$E \equiv \sum\limits_{i} \dot{q_i}\frac{\partial L}{\partial \dot{q_i}} - L $$
Then the author states:
Using ...
- 33
2
votes
2
answers
207
views
Take derivative to a cross product of two vectors with respect to the position vector [closed]
I'm doing classical mechanics about Lagrange formulation and confused about something about vector differentiation.The Lagrangian is given:
$$\mathcal{L}=\frac{m}{2}(\dot{\vec{R}}+\vec{\Omega} \times \...
- 103
2
votes
2
answers
627
views
What conditions are required for the derivative of kinetic energy to be F.v?
In Ch. 1 Derivation 1 of Goldstein's mechanics, we have
Show that for a single particle with constant mass the equation of motion implies
$$
\frac{dT}{dt} = \vec{F}\cdot\vec{v}
$$
The first step ...
- 647
0
votes
1
answer
2k
views
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:
$$
k\left(\frac{\hat{x}}{r} + \frac{\hat{y}}{r} + \frac{\hat{z}}{r}\right)\frac{1}{(x^2 + y^2 + z^2)^{n/2}}
...
- 13
1
vote
1
answer
671
views
Determining the change in radius of water flowing from a faucet - more general question on differentiation
I've been outside of the academic world for several years now, and I'm forcing myself to go back through old textbooks and resources and work through the information in there. I can tell I'm losing ...
- 11
0
votes
1
answer
91
views
In central-force mechanics, how do we substitute $ξ=\frac{1}{r}$?
I have taken a look at central-force mechanics in the past, but I still cannot understand how $ξ=\frac{1}{r}$ is substituted to find $\frac{d^2r}{dt^2}$ in terms of ξ.
So I know from $F=ma$ that:
$$(...
- 21
0
votes
1
answer
126
views
Clarification about some steps in the derivation of the Lie derivative (mechanics)
First of all, this question may seem to be undefined, because I'm not sure how to connect this (to me) newly introduced concept with the abstract notion of the Lie derivative. I'm not even sure if I ...
- 912
1
vote
1
answer
211
views
Lagrangian formalism (demonstration)
My question is about the multiplicity of the Lagrangian to a Physics system.
I pretend to demonstrate the following proposition:
For a system with $n$ degrees of freedom, written by the Lagrangian ...
- 1,092
0
votes
1
answer
72
views
Show $\frac{\partial T}{\partial \dot q_j} = m_i \dot r_i^T\frac{\dot r_i }{\partial \dot q_j} $ [closed]
This is a basic result in lagrangian mecanics. Let $T$ be the kinetic energy, $r_i$ be the position of the $i^{th}$ particle in the system I need to show $$\frac{\partial T}{\partial \dot q_j} = \frac{...
- 1,754
4
votes
2
answers
1k
views
Trouble with Landau & Lifshitz's expansion of the Lagrangian with respect to $\epsilon$ and $v$ [duplicate]
Hello I have a quick question on what I have been reading in Landau & Lifshitz's book on classical mechanics. I am in the very beginning of the book and I am having trouble with his derivation on ...
- 43
2
votes
1
answer
598
views
Clarification on a Goldstein formula steps (classical mechanics)
At page 20 of Classical Mechanics' Goldstein (Third edition), there are these two steps given between eqs. (1.51) and (1.52):
$$\sum_i m_i \ddot {\bf r}_i \cdot \frac{\partial {\bf r_i}}{ \partial ...
- 1,143