[tag:classical-mechanics] entails the study of the trajectory of bodies under the influence of forces. More specific subtopics are: [tag:newtonian-mechanics], [tag:lagrangian-mechanics], [tag:hamiltonian-mechanics] for point particles and [tag:fluid-dynamics], [tag:statistical-mechanics] and ...
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Why are only derivatives to the first order relevant?
Why is the Lagrangian a function of the position and velocity (possibly also of time) and why are dependences on higher order derivatives (acceleration, ...) excluded?
Is there a good reason for this ...
16
votes
7answers
2k views
Classical mechanics without coordinates book
I am a math grad student who would like to learn some classical mechanics. The caveat is I am not to interested in the standard coordinate approach. I can't help but think of the fields that arise in ...
6
votes
8answers
4k views
Which Mechanics book is the best for beginner in math major?
I'm a bachelor student majoring in math, and pretty interested in physics. I would like a book to study for classical mechanics, that will prepare me to work through Goldstein's Classical Mechanics. ...
6
votes
9answers
2k views
Book about classical mechanics
I am looking for a book about "advanced" classical mechanics. By advanced I mean a book considering directly Lagrangian and Hamiltonian formulation, and also providing a firm basis in the geometrical ...
20
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8answers
2k views
Why doesn't the bike fall if going with a high speed?
Why does the bike fall when its speed is very low or close to zero and is balanced when going with a high speed?
11
votes
2answers
1k views
Deriving the Lagrangian for a free particle
I'm a newbie in physics. Sorry, if the following questions are dumb. I began reading "Mechanics" by Landau and Lifshitz recently and hit a few roadblocks right away.
Proving that a free particle ...
8
votes
5answers
491 views
Is it possible to recover Classical Mechanics from Schrödinger's equation?
Let me explain in details. Let $\Psi=\Psi(x,t)$ be the wave function of a particle moving in a unidimensional space. Is there a way of writing $\Psi(x,t)$ so that $|\Psi(x,t)|^2$ represents the ...
7
votes
3answers
2k views
When is the Hamiltonian of a system not equal to its total energy?
I thought the Hamiltonian was always equal to the total energy of a system but have read that this isn't always true. Is there an example of this and does the Hamiltonian have a physical ...
4
votes
5answers
471 views
Classical limit of quantum mechanics
I have heard that one can recover classical mechanics from quantum mechanics in the limit the $\hbar$ goes to zero. How can this be done? (Ideally, I would love to see something like: as $\hbar$ ...
37
votes
9answers
3k views
Is Angular Momentum truly fundamental?
This may seem like a slightly trite question, but it is one that has long intrigued me.
Since I formally learned classical (Newtonian) mechanics, it has often struck me that angular momentum (and ...
11
votes
2answers
1k views
Is kinetic energy a relative quantity? Will it make inconsistent equations when applying it to the conservation of energy equations?
If the velocity is a relative quantity, will it make inconsistent equations when applying it to the conservation of energy equations?
For example:
In the train moving at $V$ relative to ground, ...
3
votes
2answers
453 views
Classical Limit of the Feynman Path Integral
I understand that in the limit that h_bar goes to zero, the Feynman path integral is dominated by the classical path, and then using the stationary phase approximation we can derive an approximation ...
6
votes
6answers
810 views
Is rotational motion relative to space?
Let's assume that there is nothing in the universe except Earth. If the Earth rotates on its axis as it does, then would we experience the effects of rotational motion like centrifugal force and ...
7
votes
1answer
550 views
Brachistochrone Problem for Inhomogeneous Potential
This recent question about holes dug through the Earth led me to wonder: if I wanted to dig out a tube from the north pole to the equator and build a water slide in it, which shape would be the ...
2
votes
1answer
206 views
What variables does the action $S$ depend on?
Action is defined as,
$$S ~=~ \int L(q, q', t) dt,$$
but my question is what variables does $S$ depend on?
Is $S = S(q, t)$ or $S = S(q, q', t)$ where $q' := \frac{dq}{dt}$?
In ...
1
vote
0answers
557 views
Do symmetries increase the number of conserved quantities? [closed]
Let us consider a classical mechanical system of N particles in a constant external field. We have 3N coordinates and 3N velocities, so totally 6N unknown variables. We have 6N ordinary differential ...
12
votes
6answers
2k views
Why the Principle of Least Action?
I'll be generous and say it might be reasonable to assume that nature would tend to minimize, or maybe even maximize, the integral over time of $T-V$. Okay, fine. You write down the action ...
26
votes
16answers
2k views
Why does one experience a short pull in the wrong direction when a vehicle stops?
When you're in a train and it slows down, you experience the push forward from the deceleration which is no surprise since the force one experiences results from good old $F=m a$. However, the moment ...
14
votes
4answers
455 views
Is the principle of least action a boundary value or initial condition problem?
Here is a question that's been bothering me since I was a sophomore in university, and should have probably asked before graduating:
In analytic (Lagrangian) mechanics, the derivation of the ...
10
votes
4answers
982 views
Newton's Bucket
This thought experiment is originally due to Sir Isaac Newton. We have a sphere of water floating freely in an opaque box in intergalactic space, held together by surface tension and not rotating with ...
1
vote
1answer
296 views
How does Newton's 2nd law correspond to GR in the weak field limit?
I can only perform the demonstration from the much simpler $E = mc^2$.
Take as given the Einstein field equation:
$G_{\mu\nu} = 8 \pi \, T_{\mu\nu}$
... can it be proved that Newton's formulation ...
6
votes
2answers
977 views
A pendulum clock problem
Below is a picture of a simple pendulum clock. Suppose that the bob (a rigid disk) on the end of the pendulum can spin without friction about its geometrical axis and is spinning at an angular ...
5
votes
4answers
726 views
Why can't any term which is added to the Lagrangian be written as a total derivative (or divergence)?
All right, I know there must be an elementary proof of this, but I am not sure why I never came across it before.
Adding a total time derivative to the Lagrangian (or a 4D divergence of some 4 ...
2
votes
2answers
124 views
Instability of a thrown tennis racquet
Someone once mentioned to me that it's impossible to throw a tennis racquet (or similarly shaped object) into the air, perpendicularly to the string plane, in such a way that it won't turn.
What is ...
19
votes
1answer
985 views
Why does this object periodically turn itself?
See this video about 30 sec in. http://www.youtube.com/watch?v=dL6Pt1O_gSE
Is this a real effect? Why does it seem to turn periodically?
Can it be explained by classical mechanics alone?
Is there a ...
14
votes
1answer
542 views
Phase space volume and relativity
Much of statistical mechanics is derived from Liouville's theorem, which can be stated as "the phase space volume occupied by an ensemble of isolated systems is conserved over time." (I'm mostly ...
5
votes
3answers
441 views
How does one quantize the phase-space semiclassically?
Often, when people give talks about semiclassical theories they are very shady about how quantization actually works.
Usually they start with talking about a partition of $\hbar$-cells then end up ...
22
votes
2answers
2k views
Why can't we feel the Earth turning?
The Earth turns with a very high velocity, around its own axis and around the Sun.
So why can't we feel that it's turning, but we can still feel earthquake.
19
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12answers
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Will a ball slide down a lumpy hill over the same path it rolls down the hill?
Suppose I have a lumpy hill. In a first experiment, the hill is frictionless and I let a ball slide down, starting from rest. I watch the path it takes (the time-independent trail it follows).
...
8
votes
1answer
281 views
Why does a cuboid spin stably around two axes but not the third?
Let $C$ be a cuboid (rectangular parallelepiped) with edges of lengths $a < b < c$.
Consider an axis that passes through the centers of two opposite faces of $C$. There are three such axes, ...
4
votes
3answers
566 views
Why is the symplectic manifold version of Hamiltonian mechanics used in Newtonian mechanics?
Books such as Mathematical methods of classical mechanics describe an approach to classical (Newtonian/Galilean) mechanics where Hamiltonian mechanics turn into a theory of symplectic forms on ...
4
votes
4answers
500 views
What does it mean, when one says that system has N constants of motion?
For example for an isolated system the energy $E$ is conserved. But then any function of energy, (like $E^2,\sin E,\frac{ln|E|}{E^{42}}$ e.t.c.)
is conserved too. Therefore one can make up infinitely ...
4
votes
3answers
644 views
Why does the cart move? [duplicate]
A while ago someone proposed the following thought experiment to me:
A horse attached to a cart is resting on a horizontal road. If the horse attempts to move by pulling the cart, according to the ...
3
votes
2answers
407 views
Differences between classical, analytical, rational and theoretical mechanics
Can you explain me what are the differences between the four following subjects?
analytical mechanics
rational mechanics
classical mechanics
theoretical mechanics
3
votes
3answers
712 views
Define Pressure at A point. Why is it a Scalar?
I have a final exam tomorrow for fluid mechanics and I was just looking over the practice exam questions. They do not provide solutions. But pretty much I have to define pressure at a point and also ...
2
votes
1answer
2k views
Analyzing the motion of a ball rolling without slipping inside a hemispherical bowl
Consider a solid ball of radius $r$ and mass $m$ rolling without slipping in a hemispherical bowl of radius $R$ (simple back and forth motion). Now, I assume the oscillations are small and so the ...
7
votes
4answers
797 views
How far does a trampoline vertically deform based on the mass of the object?
If a baseball is dropped on a trampoline, the point under the object will move a certain distance downward before starting to travel upward again. If a bowling ball is dropped, it will deform further ...
6
votes
2answers
901 views
Are there examples in classical mechanics where D'Alembert's principle fails?
D'Alembert's principle suggests that the work done by the internal forces for a virtual displacement of a mechanical system in harmony with the constraints is zero.
This is obviously true for the ...
3
votes
1answer
330 views
Coriolis force in free fall
Does the coriolis force has any measurable effect in free fall from large heights?
Take for example the sky diving experiment by F. Baumgartner who started from a height of about 40 km above New ...
3
votes
1answer
304 views
A Question about Virtual Work related to Newton's Third Law
In describing D'Alembert's principle, the lecture note I was provided with states that the total force $\mathbb F_l$ acting on a particle can be taken as,
$$\mathbb F_l=F_l+\sum_mf_{ml}+C_l,$$
...
3
votes
2answers
487 views
Classical car collision
I have a very confusing discussion with a friend of mine.
2 cars ($car_a$ and $car_b$) of the same mass $m$ are on a collision course. Both cars travel at $50_\frac{km}{h}$ towards each other.
They ...
2
votes
2answers
158 views
Foucault pendulum
The equations of motions for a Foucault pendulum are given by:
$$\ddot{x} = 2\omega \sin\lambda \dot{y} - \frac{g}{L}x,$$
$$\ddot{y} = -2\omega \sin\lambda \dot{x} - \frac{g}{L}y.$$
What are the ...
2
votes
1answer
157 views
Find the Hamiltonian given $\dot p$ and $\dot q$
I have these equations:
$$\dot p=ap+bq,$$
$$\dot q=cp+dq,$$
and I have to find the conditions such as the equations are canonical. Then, I have to find the Hamiltonian $H$.
To answer to the first ...
2
votes
0answers
109 views
Stiffness tensor
Let's have a stiffness tensor:
$$
a^{ijkl}: a^{ijkl} = a^{jikl} = a^{klij} = a^{ijlk}.
$$
It has a 21 independent components for an anisotropic body.
How does body symmetry (cubic, hexagonal ...
2
votes
4answers
292 views
Why are coordinates and velocities sufficient to completely determine the state and determine the subsequent motion of a mechanical system?
I am a Physics undergraduate, so provide references with your responses.
Landau & Lifshitz write in page one of their mechanics textbook:
If all the co-ordinates and velocities are ...
2
votes
2answers
197 views
How do I show that there exists variational/action principle for a given classical system?
We see variational principles coming into play in different places such as Classical Mechanics (Hamilton's principle which gives rise to the Euler-Lagrange equations), Optics (in the form of Fermat's ...
11
votes
3answers
807 views
Significance of the second focus in elliptical orbits
1.In classical mechanics, using Newton's laws, the ellipticity of orbits is derived. It is also said that the center of mass is at one of the foci.
2.Each body will orbit the center of the mass of ...
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votes
4answers
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Are water waves (i.e. on the surface of the ocean) longitudinal or transverse?
I'm convinced that water waves for example:
are a combination of longitudinal and transverse. Any references or proofs of this or otherwise?
7
votes
3answers
452 views
Boundary layer theory in fluids learning resources
I'm trying to understand boundary layer theory in fluids. All I've found are dimensional arguments, order of magnitude arguments, etc... What I'm looking for is more mathematically sound arguments. ...
4
votes
1answer
512 views
invariance of lagrangian in Noether's theorem
Noether's theorem needs the lagrangian to be invariant.
However, given a lagrangian $L$, we know that the lagrangians $\alpha L$ (where $\alpha$ is any constant) and $L + \frac{df}{dt}$ (where $f$ is ...
