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:
$$m\ddot{r} = F + ...
- 17
3
votes
1
answer
1k
views
Why is total kinetic energy always equal to the sum of rotational and translational kinetic energies?
My derivation is as follows.
The total KE, $T_r$ for a rigid object purely rotating about an axis with angular velocity $\bf{ω}$ and with the $i$th particle rotating with velocity $ \textbf{v}_{(rot)...
user86425
4
votes
4
answers
542
views
Is the numerical value of the Lagrangian conserved, when moving between inertial reference frames?
I am doing a course on Lagrangian mechanics and the instructor mentioned that the numerical value of the Lagrangian is conserved when I shift between two inertial reference frames, even though their ...
- 593
1
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0
answers
378
views
Rewriting the Lagrangian in terms of the constant(s) of motion doesn't work. Why? (spherical pendulum) [duplicate]
I am trying to solve for the equations of motion to simulate a spherical pendulum. I decided to use the spherical coordinates. The Lagrange equation is,
$$
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 ...
