All Questions
36
questions
1
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0
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38
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Weird sign in EOM: Centripetal vs. centrifugal term [duplicate]
Something goes wrong when I was deriving the equation of motion in Kepler's probelm, as below,
Angular momentum conservation $L = Mr^2\dot{\theta}^2$.
And Lagrangian is $L = \frac{1}{2}M(\dot{r}^2 + ...
- 529
0
votes
1
answer
142
views
Correct Lagrangian for classical central force problem?
Wikipedia gives the following Lagrangian for central force problem:
$$\mathcal{L}=\frac12 m \dot{\mathbf{r}}^2-V(r)$$
where $m$ is the mass of a smaller body orbiting around a stationary larger body. ...
- 103
0
votes
0
answers
80
views
Substituting the conservation of angular momentum into the Binet formula results in contradiction [duplicate]
Background Information
The lagrangian of a particle in a central force field $V(r)$ is
$$
L=\frac12m(\dot r^2+r^2\dot\theta^2+r^2\sin^2\theta\dot\varphi^2)-V(r).
$$
The particle must move in a plane, ...
- 695
0
votes
1
answer
149
views
Problem 6.3 from David Morin (classical mechanics) [closed]
I get the lagrangian for the system as
$$
\begin{align}
\mathscr{L} = \frac{m}{2}(\dot{x}^2 + l^2\dot{\theta}^2 + 2l\dot{x}\dot{\theta}\cos \theta) + mgl\cos\theta
\end{align}
$$
Where $\theta$ is the ...
- 23
0
votes
2
answers
176
views
Lagrangian of inverted physical pendulum with oscillating base
An inverted physical pendulum is deviated by a small angle $\varphi$ and connected to an oscillating base with oscillation function $a(t)$. The pendulum's mass is $m$ and its center of mass is $l$ ...
1
vote
1
answer
149
views
Lagrangian formalism for non-inertial reference frames
I was solving the exercise where the massless ring with radius $R$ is rotating around axis (shown in the picture) with angular velocity $\omega$. On the ring is a point-object with mass $m$ which ...
3
votes
2
answers
121
views
Why are you allowed to omit the $V^2$ term in the non-inertial frame?
I'm trying to find trying to find the Lagrangian and Hamiltonian for a particle in a non-inertial frame, but when I try to do so, I always get a quadratic term, which textbooks like Landau & ...
- 57
0
votes
1
answer
420
views
Kinetic Energy of pendulum with moving support
I am trying to calculate the kinetic energy of a pendulum with moving support. I have come across two ways that could be used to calculate the kinetic energy, and although I know that the first of ...
- 29
0
votes
2
answers
521
views
Decomposing Lagrangian into CM and relative parts with presence of uniform gravitational field
Most problems concerning two-body motion (using Lagrangian methods) often only consider the motion of two particles subject to no external forces. However, the Lagrangian should be decomposable into ...
- 343
0
votes
1
answer
102
views
Special relativity v.s. "homogeneous time" within an inertial reference frame
I am asking a conceptual question.
As we learned from classical mechanics, say Lagrangian formulation, as stated in Chap 7.9 of Classical Dynamics book by Thornton-Marion (5th Ed) p.260:
in our ...
- 4,336
2
votes
3
answers
2k
views
Lagrange Equations for Non-Inertial Frame of Reference
I am trying to expand my limited knowledge of Lagrange's equations for evaluating motion. Regarding the Lagrangian in a rotating coordinate system, the text Mechanics by Symon states "...we use ...
- 9,381
0
votes
3
answers
195
views
Having trouble taking derivative of a cross product when finding Lagrangian to find force equation for rotating non-inertial frame
I've been working on a problem for my classical mechanics 2 course and I am stuck on a little math problem. Basically, I am trying to prove this equation of motion with a Lagrangian:
$$m\ddot{r} = F + ...
- 17
3
votes
1
answer
1k
views
Why is total kinetic energy always equal to the sum of rotational and translational kinetic energies?
My derivation is as follows.
The total KE, $T_r$ for a rigid object purely rotating about an axis with angular velocity $\bf{ω}$ and with the $i$th particle rotating with velocity $ \textbf{v}_{(rot)...
user86425
4
votes
4
answers
542
views
Is the numerical value of the Lagrangian conserved, when moving between inertial reference frames?
I am doing a course on Lagrangian mechanics and the instructor mentioned that the numerical value of the Lagrangian is conserved when I shift between two inertial reference frames, even though their ...
- 593
1
vote
0
answers
378
views
Rewriting the Lagrangian in terms of the constant(s) of motion doesn't work. Why? (spherical pendulum) [duplicate]
I am trying to solve for the equations of motion to simulate a spherical pendulum. I decided to use the spherical coordinates. The Lagrange equation is,
$$
L=T-V=\frac{1}{2}m\left(l\dot\theta\right)^2+...
- 317
2
votes
1
answer
332
views
Reference-frame transformation for the Lagrangian of a charged particle
The Lagrangian of a charged particle in a magnetic field reads:
$$
L=\frac{m}{2}\dot{\bf{r}}\cdot \dot{\bf{r}} + q\bf{A}\cdot \dot{\bf{r}}
$$
This is the Lagrangian in the reference frame $Oxyz$.
...
- 1,232
3
votes
1
answer
419
views
If you have a conserved quantity, why can't you use it to eliminate a variable in the Lagrangian? [duplicate]
Suppose, for example, we take a particle in polar coordinates $(r, \theta)$ with a central force, so $U = U(r).$ The Lagrangian is $$\mathcal{L} = \dfrac12 m (\dot{r}^2 + (r\dot{\theta})^2) - U(r).$$
...
- 53.2k
1
vote
1
answer
1k
views
Lagrangian, central forces and conservation of angular momentum [duplicate]
When studying central forces it is possible to propose the Lagrangian:
$$ L = T-U=\frac{1}{2}m \dot{r}^2+\frac{1}{2}mr^2 \dot{\theta}^2 - U(r)$$
Then we can solve the equation of motions for $\...
- 980
8
votes
1
answer
347
views
Lagrangian in non-inertial frame of reference
I'm in trouble with comprehending derivation of Lagrangian of a particle in non-inertial , translational and rotational frame of reference by Landau's Mechanics.
More precisely I don't understand why ...
- 111
0
votes
1
answer
86
views
Velocity of particle in non-inertial frame [closed]
Can velocity of the free particle remain constant in non-inertial frame as contrast to free particle in an inertial frame?
I know the answer is straightforward yes but taking a different perspective ...
- 135
2
votes
1
answer
724
views
How to deal with no-slip non-holonomic constraints in Lagrangian?
I'm solving a dynamical system of a ball of mass $m$ and radius $R$ rolling on a rotating platform ("turntable") for which I found the Lagrangian to be:
$$L=\frac{1}{2} m (\dot{x} - \Omega y)^2 + \...
- 145
10
votes
3
answers
3k
views
Lagrangian equations of motion for ball rolling on turntable
The equations governing the motion of a ball of mass $m$, radius $R$ rolling on a table rotating at constant angular velocity $ \Omega $ which are derived using Newton's laws are: (I present these for ...
- 145
0
votes
1
answer
333
views
Lagrangian of a Heavy Symmetrical Top - Inertial or Non-inertial Frame?
I'm having some confusion with the analysis of a symmetrical top (specifically, a heavy top, but this is not very important for the question).
Following Landau and Lifshitz's Mechanics, on page 110 ...
- 3
3
votes
2
answers
566
views
Deriving effective potential energy from the Lagrangian of a two-body system [duplicate]
I'm having some issues understanding how the effective potential energy of a two-body system is derived from the Lagrangian of the system. Specifically my issue is with one step...
Suppose we are ...
0
votes
0
answers
404
views
Re: Susskind and Hrabovsky: Should the Lagrangian of a particle referred to a rotating frame omit the velocity dependent "potential"?
My question pertains to Lecture 6: Exercise 4 in The Theoretical Minimum by Leonard Susskind and George Hrabovsky. A suggested solution has been posted here: http://www.madscitech.org/tm/slns/
The ...
- 2,015
0
votes
1
answer
82
views
Question regarding the definition of generalized coordinates
In Classical Mechanics, John R. Taylor defines generalized coordinates like so:
Consider now an arbitrary system of $N$ particles, $\alpha = 1, \dots , N$ with positions $\boldsymbol{r}_a$. We say ...
user113773
1
vote
1
answer
441
views
Can we consider non-inertial frames in Lagrangian dynamics formulated through d'Alembert's principle?
When we derive Euler-Lagrange equations from an action principle, there is no explicit mention of a reference frame, so I assumed that the formulation is correct even in non-inertial frames (is this ...
- 588
-1
votes
1
answer
184
views
Lagrangian for Non-inertial Frame
Context
Let us consider two reference frames: $S$ and $S'$. $S'$ is rotating with respect to $S$ with an angular velocity $\vec{\omega}$ about a rotation axis $MOM'$. The origins of $S$ and $S'$ are $...
- 3,023
4
votes
1
answer
1k
views
Lagrangian of rotating springs
I'm trying to construct the Lagrangian for the following scenario. A turntable of radius $R$ is rotating at angular velocity $\omega$, maintained by a motor. Two springs with Hooke's constant $k$ are ...
user138458
0
votes
1
answer
2k
views
Significance of centrifugal potential
While dealing with central forces (purely using newtonian mechanics) I've came across this result:
$$U_\text{eff}(r)=\frac{l^2}{2\mu r^2}+ U(r) \, .$$
I'm not at all fluent with the lagrangian ...
2
votes
2
answers
188
views
Take derivative to a cross product of two vectors with respect to the position vector [closed]
I'm doing classical mechanics about Lagrange formulation and confused about something about vector differentiation.The Lagrangian is given:
$$\mathcal{L}=\frac{m}{2}(\dot{\vec{R}}+\vec{\Omega} \times \...
- 103
17
votes
2
answers
7k
views
Lagrangian of an effective potential
If there is a system, described by an Lagrangian $\mathcal{L}$ of the form
$$\mathcal{L} = T-V = \frac{m}{2}\left(\dot{r}^2+r^2\dot{\phi}^2\right) + \frac{k}{r},\tag{1}$$
where $T$ is the kinetic ...
- 377
4
votes
1
answer
392
views
Confusion about imposing constraint in the action
I'm totally confused by one thing. I know that I probably shouldn't be confused about that, but at the moment I don't quite know what fails in the following:
Suppose we have a particle of unit mass ...
- 909
15
votes
3
answers
6k
views
Do we need inertial frames in Lagrangian mechanics?
Do Euler-Lagrange equations hold only for inertial systems? If yes, where is the point in the variational derivation from Hamilton's principle where we made that restriction?
My question arose because ...
- 12.4k
10
votes
2
answers
3k
views
How can you solve this "paradox"? Central potential
A mass of point performs an effectively 1-dimensional motion in the radial coordinate. If we use the conservation of angular momentum, the centrifugal potential should be added to the original one.
...
- 521
7
votes
2
answers
5k
views
Centrifugal Force and Polar Coordinates
In Classical Mechanics, both Goldstein and Taylor (authors of different books with the same title) talk about the centrifugal force term when solving the Euler-Lagrange equation for the two body ...
- 115