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Results tagged with classical-mechanics
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user 256066
Classical mechanics discusses the behaviour of macroscopic bodies under the influence of forces (without necessarily specifying the origin of these forces). If it's possible, USE MORE SPECIFIC TAGS like [newtonian-mechanics], [lagrangian-formalism], and [hamiltonian-formalism].
0
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2
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Why is this matrix symmetric?
Goldstein 3rd Ed, pg 339
In large classes of problems, it happens that $L_{2}$ is a quadratic function of the generalized velocities and $L_{1}$ is a linear function of the same variables with the fo …
- 1,330
2
votes
3
answers
1k
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Can beats be produced by two waves moving in opposite direction?
I've always seen beats to be produced when two waves are said to be moving in the same direction with different frequency.
Can beats be produced by addition of waves moving in opposite direction ?
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1
answer
243
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Why Polhode is a circle in a symmetric body
Goldstein
In the special case of a symmetrical body, the inertia ellipsoid is an ellipsoid of revolution, so that the polhode on the ellipsoid is clearly a circle about the symmetry axis. The herpol …
- 1,330
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3
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Non-uniqueness of the Lagrangian
Goldstein, 3rd ed
$$
\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}_{j}}\right)-\frac{\partial L}{\partial q_{j}}=0\tag{1.57}
$$
expressions referred to as "Lagrange's equations."
Note that fo …
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0
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1
answer
205
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Infinitesimal rotation vector [duplicate]
Chapter 4.8, Goldstein 3rd ed classical mechanics under infinitesimal rotations
(p.166~167)
($\mathbf{B}$ is an orthogonal matrix and $\mathbf{r}$ is a position vector and $d\boldsymbol{\Omega}$ is t …
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vote
0
answers
65
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"Inertia tensor possess a type of revolution about Centre of mass" ? Can't understand this s...
Goldstein pg 194, 3rd Ed.
$$
T_{\text {rotation }}=\frac{1}{2} I_{\alpha \beta} \omega_{\alpha} \omega_{\beta}
$$
where
$$
I_{\alpha \beta}=m_{i}\left(\delta_{\alpha \beta} r_{i}^{2}-r_{i \alpha} r_{ …
- 1,330
2
votes
2
answers
167
views
A doubt in a Wikipedia article discussing Bertrand's theorem
Wikipedia while deriving Bertrands theorem writes after some steps:
...For the orbits to be closed, $β$ must be a rational number. What's more, it must be the same rational number for all radii, sinc …
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2
votes
2
answers
108
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Help in understanding this derivation of Lagrange Equations in Non-Holonomic case
Whittaker, Analytical dynamics pg 215
I don't understand how we get the final equations relating $Q_r$ with $\lambda$ given the conditions above?
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2
votes
1
answer
133
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A doubt in a Wikipedia article discussing Bertrand's Theorem in Central force motion
Wikipedia on Bertrand's theorem, when discussing the deviations from a circular orbit says:
...The next step is to consider the equation for $u$ under small perturbations ${\displaystyle \eta \equiv …
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1
vote
1
answer
513
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Virtual displacement for a block sliding down a wedge
A block slides on a frictionless wedge which rests on a smooth horizontal plane. There are two constraints in this system. One that the wedge can only move horizontally and another that the block mus …
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0
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3
answers
238
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Using Lagrange equation
In classical mechanics we have a concept of generalized coordinates.
Say my generalized coordinates are $(x,y)$.
My doubt is, Is it legal to write the position vector in any vector basis say polar ba …
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0
votes
0
answers
54
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A thought experiment to prove that Newtonian gravity is incomplete [duplicate]
A particle is at rest in one frame having mass $m$. It'll attract another mass proportional to its mass ( newtons law) .
We jump into another frame moving close to speed of light. In this frame it's m …
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0
votes
1
answer
20
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Why does normal force decrease if a body rotates on and around another smooth body
We've an system of a sphere (Radius $R$) and a point particle of mass $m$.
This particle is at rest on a smooth sphere, we've then $N=mg$
Now I poke it and it begins to revolve on and around it due to …
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1
vote
1
answer
64
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How is kinetic energy $T$ given by $T=\dfrac{1}{2}\sum_{i}p_{i}\dot{q_{i}}$ in Hamiltonian a...
Im going through a website teaching Hamiltonian mechanics and I know the below
$$-\dot{p}_{i}=\dfrac{\partial H}{\partial q_{i}} \tag{14.3.12}$$
$$\dot{q}_{i}=\dfrac{\partial H}{\partial p_{i}} \tag{1 …
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0
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1
answer
475
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Transformation equations and generalized coordinates
Paraphrasing Goldstein
If we've $N$ number of particles and there exist holonomic constraints, expressed in $k$ equations, then we may use these equations to eliminate $k$ of the $3 N$ coordinates, a …
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