All Questions
Tagged with calculus classical-mechanics
60
questions
6
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3
answers
1k
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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
2
answers
127
views
On Landaus&Lifshitz's derivation of the lagrangian of a free particle [duplicate]
I'm reading the first pages of Landaus&Lifshitz's Mechanics tome. I'm looking for some clarification on the derivation of the Lagrange function for the mechanical system composed of a single free ...
1
vote
1
answer
64
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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
votes
1
answer
75
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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 ...
- 41
2
votes
5
answers
261
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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 $$\dot{\vec {r}}= \dot{r}\vec{u_r}+r\dot{\theta}\vec{u_\theta}+r\sin\...
- 69
1
vote
2
answers
119
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Lagrangian total time derivative - continues second-order differential
In the lagrangian, adding total time derivative doesn't change equation of motion.
$$L' = L + \frac{d}{dt}f(q,t).$$
After playing with it, I realize that this is only true if the $f(q,t)$ function has ...
- 525
1
vote
1
answer
48
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Lagrangian for 2 inertial frames where only Speed is different by small amount
In Landau & Liftshitz’s book p.5, they go ahead and writes down lagrangians for 2 different inertial frames. They say that Lagrangian is a function of $v^2$.
So in one frame, we got $L(v^2)$.
In ...
- 525
-1
votes
1
answer
159
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Recommended physics book(s) that uses calculus and have difficult problems [closed]
What physics book(s) uses calculus and has complex problems (undergrad/olympiad level)?
Context:
I've read "Fundamentals of Physics by Halliday and Resnick" and I found the problems to be ...
0
votes
2
answers
129
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Does the gradient of potential energy exist independent of coordinates?
Potential energy $U(\vec{r})$ of a conservative force field $\vec{F}$ is defined as a function whose variation between positions $\vec{r}_A$ and $\vec{r}_B$ is the opposite of the work done by the ...
- 577
0
votes
2
answers
63
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Approximation of Small Perturbation [closed]
From Morin's Classical Mechanics, on the chapter of Small Oscillations in Lagrangian Mechanics, he does this approximation on the last equality, I don't understand what happened there.
I get the first ...
- 109
0
votes
0
answers
111
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Question on non-holonomic constraints (This is different to the others)
I know there are many posts on non-holonomic constraints and also many on this exact one but I feel that there is still some confusion on it.
"Consider a disk which rolls without slipping across ...
- 283
1
vote
1
answer
33
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Derivatives of the lagrangian of generalized coordinates [closed]
I know that
$$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
54
views
What does this vertical line notation mean?
Here is the definition of the Noether momentum in my script.
$$I = \left.\frac{\partial L}{\partial \dot{x}} \frac{d x}{d \alpha} \right|_{\alpha=0} = \frac{\partial L}{\partial \dot{x}} = m \dot{x} = ...
- 85
0
votes
1
answer
162
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The derivative of rotational kinetic energy in terms of period gives me the wrong answer. Why should I use the product rule? [closed]
This is my first question here so I hope I do it correctly. I've tried to solve this, and google it, but I can't find the answer to this particular question. This equation comes from Carroll and ...
5
votes
2
answers
922
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Rigorously prove the period of small oscillations by directly integrating
This answer proved that
$$\lim_{E\to E_0}2\int_{x_1}^{x_2}\frac{\mathrm dx}{\sqrt{2\left(E-U\!\left(x\right)\right)}}=\frac{2\pi}{\sqrt{U''\!\left(x_0\right)}},$$
where $E_0:=U\!\left(x_0\right)$ is a ...
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0
votes
1
answer
167
views
Time to travel a set distance given variable acceleration
Trying to solve a problem for the acceleration of an automated shuttle car at my work, been a while since I studied this stuff so thought I'd reach out for help.
I have a shuttle car that is tasked ...
-1
votes
1
answer
62
views
Derivative of distance [duplicate]
I know that $speed = |\frac{\vec{dr}}{dt}|$
and first derivative of distance with respect time will be $\frac{d\vec{|r|}}{dt}|$
These 2 expressions don't seem to represent the same thing. But when I ...
1
vote
0
answers
37
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Is $n_{cr}=\frac{60}{2\pi}\sqrt{g\frac{\Sigma m_iy_i}{\Sigma m_iy_i^2}} =\frac{60}{2\pi}\sqrt{g\frac{\int y_idx}{\int y_i^2dx}}\quad ?$
I have a question about this formula used to calculate the first critical speed of a drive shaft.
$$
n_{cr}=\frac{60}{2\pi}\sqrt{g\frac{\Sigma m_iy_i}{\Sigma m_iy_i^2}} \tag {1} \quad .$$
It is the ...
- 11
0
votes
3
answers
426
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Goldstein: derivation of work-energy theorem
I am reading "Classical Mechanics-Third Edition; Herbert Goldstein, Charles P. Poole, John L. Safko" and in the first chapter I came across the work-energy theorem (paraphrased) as follows:
...
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6
votes
7
answers
228
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Is every $dm$ piece unequal when using integration of a non-uniformly dense object?
When we want to find the total charge of an object or total mass, usually we start off with a setup such as:
$$ m = \int dm \:\;\:\text{or} \:\;\:q = \int dq$$
in which we then use (and to keep it ...
- 63
2
votes
2
answers
298
views
Solving for the radius of a sphere as a function of time
I have tried to realistically model the famous game Agar.io, which can described as the following: A sphere of initial mass $m_0$ expels part of its mass at a given rate ($\frac{dm_l}{dt}$) for thrust ...
- 300
0
votes
1
answer
43
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Doubt in derivation of bending of beam, It's about derivatives and intergration
Radius of curvature of the beam in above picture is given as:
$$ \frac{1}{R} = \frac{d^2 y}{dx^2}$$
Please help me two points used as steps of a derivation in my book:
How was the radius of ...
- 71
0
votes
0
answers
41
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Minimum seperation of moving objects doubt
Let there be $2$ objects $P_1$(initial velocity $u$ $ms^{-1}$ & acceleration $a$ $ms^{-2}$) & $P_2$ (initial velocity $U$ $ms^{-1}$ & acceleration $A$ $ms^{-2}$) initially separated by ...
1
vote
2
answers
129
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Time derivative of unit velocity vector?
Let's say I have some parametric curve describing the evolution of a particle $\mathbf{r}(t)$. The velocity is $\mathbf{v}(t) = d\mathbf{r}/dt$ of course. I am trying to understand what the expression ...
1
vote
2
answers
223
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Confused about the solution to the pendulum differential equation
So I’ve learned how to derive the exact solution to the pendulum differential equation (in respect to its period), $\ddot{\theta} + \frac{g}{l}\sin\theta=0$, where $g$ is gravitational acceleration ...
- 113
1
vote
1
answer
114
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Brachistochrone Problem without Trigonometric Substitution
I'm trying to numerically reproduce the cycloid solution for the brachistochrone problem. In doing so, I eventually ended up with the following integral:
$$ x = \int{\sqrt{\frac{y}{2a-y}} dy} $$
...
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11
votes
3
answers
2k
views
Why do we ignore the second-order terms in the following expansion?
Consider the expansion done for the kinetic energy of a system executing small oscillations as done in Goldstein:
A similar series expansion can be obtained for the kinetic energy. Since the ...
2
votes
2
answers
674
views
Equilibrium and the derivative of potential energy
In his Classical Mechanics popular lectures ( Lecture 3, at the beginning) , Susskind illustrates the idea of a stationary quantity using an example involving the notion of equilibrium.
Link : https:/...
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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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0
votes
1
answer
25
views
Issue with a derivation in Marion's Dynamics [closed]
I was solving problem 2-14 in Marion's "Classical dynamics of particles and systems" edition 5. In this problem we calculate the range of a trajectory to be $d=\frac{2{v_0}^2\cos{\alpha}\sin{...
- 93
0
votes
2
answers
98
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Contact surface between circle and curved floor?
First, consider an inelastic circle on a hard, flat floor, the shared contact surface between the two is some infinitesimally small length dx.
Consider a second case, where if the floor had some ...
2
votes
2
answers
825
views
Necessary and Sufficient Conditions for an Equilibrium to be Stable
In the 4th section The condition that convection be absent of the book Fluid Mechanics by Landau and Lifshitz, they give the following statement:
For the (mechanical) equilibrium to be stable, it is ...
- 41
0
votes
1
answer
44
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Can a position variable have an infintesimal in it?
I've been pondering unstable systems, such as a perfectly round rock atop a smooth hill. At the top of the hill is a metastable point where the rock could roll either way after an arbitrary amount of ...
- 50.1k
9
votes
7
answers
2k
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What does it mean to integrate with respect to mass?
I've encountered many integrals that seem to integrate functions of distance with respect to mass, for example, $\int_0^Mr^2dm$ for the moment of inertia of continuous mass distribution.
I'm not sure ...
- 313
0
votes
2
answers
76
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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
vote
2
answers
193
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Multivariable chain rule in classical mechanics; example of physical system [closed]
I'm a teaching assistant in calculus and my students who are studying mechanical engineering asked me to explain the multivariable chain rule. So I thought it could be fun if I could give an example ...
- 127
1
vote
2
answers
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Can anyone suggest a math review book for someone interested in beginning physics study as a hobby?
Good day. This is my first post and I was not sure whether to post here or on Math StackExchange. Since the end product of my goal results in ultimately understanding some basic math in physics, I ...
0
votes
4
answers
479
views
Time quantization [closed]
There is no evidence to support that time is quantized. So wouldn't the use of discrete values like $dt$ in calculus suggest time is quantized and comes in discrete durations of $dt$?
- 303
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vote
3
answers
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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 ...
- 1,028
0
votes
2
answers
419
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The $\delta$ notation in Goldstein's Classical Mechanics on the calculus of variation
In Goldstein's classical mechanics (page 36) he introduces the basics of the calculus of variation and uses it to effectively the Euler-Lagrange equations. However, there is a step in which the $\...
- 6,963
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votes
1
answer
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The use of $x_\varepsilon (t) = x(t) + \varepsilon (t)$ and $x_\varepsilon (t) = x(t) + \varepsilon \eta (t)$ in proving Hamilton's principle
The following Wikipedia page uses $x_\varepsilon (t) = x(t) + \varepsilon (t)$ in the proof.
https://en.wikipedia.org/wiki/Hamilton%27s_principle#Mathematical_formulation
But in my mechanics book (by ...
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3
votes
3
answers
878
views
Newton's Second Law in vertical launch of a rocket
Consider a rocket being launched vertically.
Let $T(t)$ denote the thrust from the engine and $M(t)$ be the total mass of the rocket at time $t$.
At $t=0$, $T(0)=M(0)g$ (so that the normal force due ...
- 1,047
1
vote
2
answers
2k
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Acceleration as a function of position and time
I know if you have an acceleration as a function of $t$, $a(t)$, to find the velocity you simply integrate $a(t)$ with respect to $t$. Moreover, if the acceleration was a function of position, $a(x)$, ...
- 33
0
votes
1
answer
242
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Is there a textbook for learning physics and multivariable calculus at the same time?
I am a student who took single variable calculus and algebra physics.
I want to learn either mechanics or thermodynamics or electromagnetism with multivariable calculus, matrices, lagrange ...
9
votes
4
answers
557
views
How do physicists know when it is appropriate to use $\mathrm dx$ as if it is a number? [duplicate]
I'm trying to teach myself calculus of variations when I came across a worked example about the shortest distance between two points in a plane. This is a question about the mathematics but I don't ...
- 193
0
votes
1
answer
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
0
votes
2
answers
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
0
votes
2
answers
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
votes
1
answer
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 ...
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