All Questions
Tagged with differentiation classical-mechanics
126 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 ...
0
votes
1
answer
90
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Derivative of the product of a scalar function and a vector valued function
According to Berkeley Physics Course, Volume 1 Mechanics,
The time derivative of a vector valued function can be derived from the formula:
$$
\mathbf{r}(t) = r(t)\mathbf{\hat{r}}(t)
$$
where the ...
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votes
3
answers
130
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When computing the Euler–Lagrange equations, why do we assume the coordinates do not depend on time?
I've just started to learn lagrangians through this video and I'm a bit confused. The setup has that $L = T-V$. With $T=\tfrac{1}{2}mv^2$ and $V=mgx$. So, $L= \tfrac{1}{2}m(dx/dt)^2-mgx$. This is all ...
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3
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In equation (3) from lecture 7 in Leonard Susskind’s ‘Classical Mechanics’, should the derivatives be partial?
Here are the equations. ($V$ represents a potential function and $p$ represents momentum.)
$$V(q_1,q_2) = V(aq_1 - bq_2)$$
$$\dot{p}_1 = -aV'(aq_1 - bq_2)$$
$$\dot{p}_2 = +bV'(aq_1 - bq_2)$$
Should ...
1
vote
3
answers
113
views
The conservative force [closed]
I read about the definition of the curl.
It's the measure of the rotation of the vector field around a specific point
I understand this, but I would like to know what does the "curl of the ...
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1
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2
answers
105
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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 ...
- 131
1
vote
1
answer
69
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Material to Study the Definition, Algebra, and Use of Infinitesimals in Physics [closed]
This is going to be a rather general question about suggestions on best supplementary material to properly explain the use of infinitesimals (or differentials?) for the purposes of integration or ...
0
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1
answer
86
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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 ...
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0
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1
answer
98
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Derivation of lagrange equation in classical mechanics
I'm currently working on classical mechanics and I am stuck in a part of the derivation of the lagrange equation with generalized coordinates. I just cant figure it out and don't know if it's just ...
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4
votes
4
answers
440
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Variation of a function
I'm studying calculus of variations and Lagrangian mechanics and i don't understand something about the variational operator
Let's say for example that i got a Lagrangian $L [x(t), \dot{x}(t), t] $ ...
- 321
1
vote
1
answer
82
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Sufficient condition for conservation of conjugate momentum
Is the following statement true?
If $\frac{\partial \dot{q}}{\partial q}=0$, then the conjugate momentum $p_q$ is conserved.
We know that conjugate momentum of $q$ is conserved if $\frac{\partial L}{\...
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2
votes
1
answer
103
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Time derivative of a "general" vector $\vec A$ in an accelerating frame: what about e.g. velocity $\vec v$?
According to Morin "Classical Mechanics" (Section 10.1, page 459), the derivative of a general vector $\vec A$ in an accelerating frame may be given as
$$\frac{d\vec A}{dt}=\frac{\delta \vec ...
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2
votes
5
answers
348
views
Why does $\vec{r}\cdot\dot{\vec{r}}=r\dot{r}$?
Why is $$\vec{r}\cdot\dot{\vec{r}}=r\dot {r}$$ true? Before saying anything, I have seen the proofs using spherical coordinates for $$\dot{\vec {r}}= \dot{r}\vec{u_r}+r\dot{\theta}\vec{u_\theta}+r\sin\...
- 69
2
votes
6
answers
274
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Lagrangian - How can we differentiate with respect to time if $v$ not a function of time?
In the Lagrangian itself, we know that $v$ and $q$ don't depend on $t$ (i.e - they are not functions of $t$ - i.e., $L(q,v,t)$ is a state function.)
Imagine $L = \frac{1}{2}mv^2 - mgq$
Euler-Lagrange ...
- 535
1
vote
2
answers
133
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Lagrangian total time derivative - continues second-order differential
In the lagrangian, adding total time derivative doesn't change equation of motion.
$$L' = L + \frac{d}{dt}f(q,t).$$
After playing with it, I realize that this is only true if the $f(q,t)$ function has ...
- 535
1
vote
1
answer
48
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Lagrangian for 2 inertial frames where only Speed is different by small amount
In Landau & Liftshitz’s book p.5, they go ahead and writes down lagrangians for 2 different inertial frames. They say that Lagrangian is a function of $v^2$.
So in one frame, we got $L(v^2)$.
In ...
- 535
0
votes
2
answers
150
views
Does the gradient of potential energy exist independent of coordinates?
Potential energy $U(\vec{r})$ of a conservative force field $\vec{F}$ is defined as a function whose variation between positions $\vec{r}_A$ and $\vec{r}_B$ is the opposite of the work done by the ...
- 587
1
vote
2
answers
268
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Why the $\Delta$ in the definition of pressure? (fluid mechanics)
I'm an engineering student (first year) studying Physics 1 (now an introduction to fluid mechanics).
Q1
In my physics textbook, the "medium pressure" is defined as:
$$p_m = \frac{\Delta F_{\...
- 141
0
votes
2
answers
79
views
Approximation of Small Perturbation [closed]
From Morin's Classical Mechanics, on the chapter of Small Oscillations in Lagrangian Mechanics, he does this approximation on the last equality, I don't understand what happened there.
I get the first ...
- 109
1
vote
2
answers
181
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Coordinate basis vectors on tangent bundle (extrinsic view)
Short Version: when we say that $(\pmb{q},\pmb{u}):TQ_{(q)}\to\mathbb{R}^{2n}$ are local coordinates for the tangent bundle of $Q$, which can be viewed as an embedded submanifold of a higher ...
- 477
2
votes
1
answer
383
views
Having trouble deriving the exact form of the Kinematic Transport Theorem
The Kinematic transport theorem is a very basic theorem relating time derivatives of vectors between a non rotating frame and another one that's rotating with respect to it with a uniform angular ...
- 3,358
1
vote
1
answer
34
views
Derivatives of the lagrangian of generalized coordinates [closed]
I know that
$$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
0
votes
1
answer
57
views
What does this vertical line notation mean?
Here is the definition of the Noether momentum in my script.
$$I = \left.\frac{\partial L}{\partial \dot{x}} \frac{d x}{d \alpha} \right|_{\alpha=0} = \frac{\partial L}{\partial \dot{x}} = m \dot{x} = ...
- 85
1
vote
1
answer
113
views
How to define differentiation of a time-dependent vectors with respect to a specific reference frame in a coordinate-free manner?
It is usual in classical mechanics to introduce the derivative of a time-dependent vector with respect to a reference frame. This is accomplished through the use of a basis that is fixed with respect ...
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1
vote
1
answer
170
views
Is the order of ordinary derivatives interchangeable in classical mechanics?
I am having trouble with a term that arises in a physics equation (deriving the Lagrange equation for one particle in one generalized coordinate, $q$, dimension from one Cartesian direction, $x$).
My ...
- 111
0
votes
0
answers
75
views
Deriving Euler-Lagrange equation [duplicate]
I have derive the Euler-Lagrange equation which is equation (2) for a condition in which generalised velocity is independent on the generalised coordinate but when generalised velocity is dependent on ...
1
vote
1
answer
164
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Question regarding Energy Interaction of two particles
https://i.sstatic.net/LUsKX.jpg
To give a context as to what I'm asking here ,I am talking about the energy of a two particle system (section 4.9 Taylor's Classical Mechanics) .
My question is what ...
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1
vote
2
answers
168
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Questions regarding the derivation of Euler-Lagrange Equation from Taylor's Classical Mechanics
I'm self-studying classical mechanics from Taylor's book. I saw his derivation of the Euler-Lagrange Equation and I'm confused about something, he created a 'wrong' function $$Y(x) = y(x)+\eta(x)\tag{...
- 115
0
votes
3
answers
480
views
Goldstein: derivation of work-energy theorem
I am reading "Classical Mechanics-Third Edition; Herbert Goldstein, Charles P. Poole, John L. Safko" and in the first chapter I came across the work-energy theorem (paraphrased) as follows:
...
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2
votes
1
answer
64
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{...
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6
votes
7
answers
255
views
Is every $dm$ piece unequal when using integration of a non-uniformly dense object?
When we want to find the total charge of an object or total mass, usually we start off with a setup such as:
$$ m = \int dm \:\;\:\text{or} \:\;\:q = \int dq$$
in which we then use (and to keep it ...
- 63
2
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1
answer
742
views
Proof that the Euler-Lagrange equations hold in any set of coordinates if they hold in one
This is a question about a specific proof presented in the book Introduction to Classical Mechanics by David Morin. I have highlighted the relevant portion in the picture below.
In the remark, he ...
anon
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votes
1
answer
134
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Velocities - Equation 1.46 of Goldstein 3rd edition
In his derivation of the Euler-Lagrange equations from D'Alembert's principle, Goldstein
uses the parametrization (equation 1.45')
$$\displaystyle{\vec{r_i}=\vec{r_i}(q_1,q_2, ..., q_n, t)}\tag{1.45'}$...
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0
votes
1
answer
82
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Rigorous treatment for continuous mass systems
I would like to ask if anyone knows an accessible, yet rigorous way of passing from a discrete system of mass-points to a continuous mass system.
For instance, we clearly know how to define the ...
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0
votes
1
answer
89
views
Step in derivation of Lagrangian mechanics
There is a step in expressing the momentum in terms of general coordinates that confuses me (Link)
\begin{equation}
\left(\sum_{i}^{n} m_{i} \ddot{\mathbf{r}}_{i} \cdot \frac{\partial \mathbf{r}_{i}}{\...
- 295
1
vote
1
answer
58
views
Energy change under point transformation
How do the energy and generalized momenta change under the following
coordinate
transformation $$q= f(Q,t).$$
The new momenta: $$P = \partial L / \partial \dot Q = \partial L / \partial \dot q\times ...
- 990
0
votes
0
answers
85
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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,...
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3
votes
2
answers
155
views
How to prove that $ \delta \frac{dq_i}{dt} = \frac{d \delta q_i}{dt} $? [duplicate]
During the proof of least action principle my prof used the equation $ \delta \frac{dx}{dt} = \frac{d \delta x}{dt} $. We were not proved this equality. I was curious to know why this is true so I ...
2
votes
1
answer
1k
views
What does a Umlaut (double dot) above an angle mean?
I'm reading a paper on double pendulums and there is an equation of motion that contains a double dot (Umlaut) above an angle. What does this mean / is this a standard notation in equations of motion?...
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1
vote
2
answers
2k
views
Derivation of Lagrange's equation form d'Alembert's Principle
Im studying Mechanics form Goldstein. I cross this equation in "Derivation of Lagranges equation from d'Alembert's Principle",section 1.4. I have two questions from this derivation.
The ...
user310684
0
votes
1
answer
55
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Which is the differential $\text{d} p_i$ of a generalized momentum?
I want to get a partition function, but I introduce a generalized momentum, my doubt is about, when I define a differential respect to $p$, it means $\text{d} p$, which is the correct form to get it?
...
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1
vote
2
answers
145
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Is $\frac{dE}{dt}=0$ in an accelerating particle’s instantaneous rest frame?
My special relativity book uses an argument that involves $\frac{dE}{dt}=0$ in an accelerating particles rest frame (to show a force parallel to a particles velocity is parallel in all frames).
...
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1
vote
0
answers
21
views
Do partial derivation respect to velocity and total derivation respect to time commute? [duplicate]
Imagine we have a function of position $x^i$ and velocity $v^i$ $f(x,v)$. Position and velocity are both functions of time $t$. If the function doesn't depend explicitely on time, then we have the ...
- 4,737
1
vote
3
answers
324
views
Derivative with respect to vector of a function depending on vectors
I've been trying to understand this concept for hours without any success. I found similar questions on this forum (Derivative with respect to a vector is a gradient?) but I still don't understand.
...
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2
votes
2
answers
172
views
Conjugate momentum notation
I was reading Peter Mann's Lagrangian & Hamiltonian Dynamics, and I found this equation (page 115):
$$p_i := \frac{\partial L}{\partial \dot{q}^i}$$
where L is the Lagrangian. I understand this is ...
- 41
1
vote
2
answers
461
views
Total time derivatives and partial coordinate derivatives in classical mechanics
This may be more of a math question, but I am trying to prove that for a function $f(q,\dot{q},t)$
$$\frac{d}{dt}\frac{∂f}{∂\dot{q}}=\frac{∂}{∂\dot{q}}\frac{df}{dt}−\frac{∂f}{∂q}.\tag{1}$$
As part of ...
0
votes
1
answer
34
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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?
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5
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4
answers
1k
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Partial derivative in Newtons Second law
Newton's second law states Force is the time derivative of momentum. But is it a total derivative or partial derivative? What is the reason behind it?
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2
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78
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Translation of coordinates to generalised coordinates
The translation form $r_i$ to $q_j$ language start forms the transformation equation:
$r_i=r_i (q_1,q_2,…,q_n,t)$ (assuming $n$ independent coordinates)
Since it is carried out by means of the ...
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1
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1
answer
80
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What is $\dfrac{\partial x}{\partial t}$ in a progressive wave?
I actually divided the velocity of a particle in a progressive wave $\dfrac{\partial y}{\partial t}$ to $\dfrac{\partial y}{\partial x}$ and got $\dfrac{\partial x}{\partial t}$. Which is equal to $\...
