All Questions
10,535 questions
205
votes
15
answers
59k
views
What's the point of Hamiltonian mechanics?
I've just finished a Classical Mechanics course, and looking back on it some things are not quite clear. In the first half we covered the Lagrangian formalism, which I thought was pretty cool. I ...
- 28.6k
168
votes
11
answers
18k
views
What makes a theory "Quantum"?
Say you cook up a model about a physical system. Such a model consists of, say, a system of differential equations. What criterion decides whether the model is classical or quantum-mechanical?
None ...
154
votes
9
answers
19k
views
Calculus of variations -- how does it make sense to vary the position and the velocity independently?
In the calculus of variations, particularly Lagrangian mechanics, people often say we vary the position and the velocity independently. But velocity is the derivative of position, so how can you treat ...
- 2,235
133
votes
10
answers
44k
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 ...
- 8,698
133
votes
8
answers
12k
views
Does a particle exert force on itself?
We all have elaborative discussion in physics about classical mechanics as well as interaction of particles through forces and certain laws which all particles obey.
I want to ask, does a particle ...
- 3,395
121
votes
5
answers
20k
views
Toilet paper dilemma
There are two ways to orient the toilet paper: "over" (left image), "under" (right image).
Each has it's pros and cons. For some reason, it's always easier to tear off the paper ...
- 3,494
118
votes
11
answers
16k
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 ...
- 7,458
111
votes
15
answers
16k
views
Why quantum mechanics?
Imagine you're teaching a first course on quantum mechanics in which your students are well-versed in classical mechanics, but have never seen any quantum before. How would you motivate the subject ...
- 8,698
107
votes
4
answers
11k
views
Why does nature favour the Laplacian?
The three-dimensional Laplacian can be defined as $$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}.$$ Expressed in spherical coordinates, it ...
- 1,357
97
votes
4
answers
33k
views
Physical meaning of Legendre transformation
I would like to know the physical meaning of the Legendre transformation, if there is any? I've used it in thermodynamics and classical mechanics and it seemed only a change of coordinates?
- 1,247
88
votes
4
answers
21k
views
Why is a $5-60 mph$ time slower than a $0-60 mph$ time for some automobiles?
This doesn't make a lot of sense to me, from a physics 101 point of view. I've read a few blog entries on why this is, but none of them explain it well or are convincing. "something-something launch ...
- 773
88
votes
8
answers
11k
views
Could a "living planet" alter its own trajectory only by changing its shape?
In Stanislaw Lem's novel Solaris the planet is able to correct its own trajectory by some unspecified means. Assuming its momentum and angular momentum is conserved (it doesn't eject or absorb any ...
- 3,109
77
votes
7
answers
78k
views
What is the difference between Newtonian and Lagrangian mechanics in a nutshell?
What is Lagrangian mechanics, and what's the difference compared to Newtonian mechanics? I'm a mathematician/computer scientist, not a physicist, so I'm kind of looking for something like the ...
- 951
74
votes
6
answers
56k
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 ...
- 1,820
73
votes
2
answers
15k
views
Why does dry spaghetti break into three pieces as opposed to only two?
You can try it with your own uncooked spaghetti if you want; it almost always breaks into three when you snap it. I am asking for a good physical theory on why this is along with evidence to back it ...
- 1,137
72
votes
5
answers
12k
views
Why does a system try to minimize its total energy?
Why does a system like to minimize its total energy? For example, the total energy of a $H_2$ molecule is smaller than the that of two two isolated hydrogen atoms and that is why two $H$ atoms try to ...
- 27.2k
68
votes
6
answers
48k
views
Laplace operator's interpretation
What is your interpretation of Laplace operator? When evaluating Laplacian of some scalar field at a given point one can get a value. What does this value tell us about the field or it's behaviour in ...
- 3,217
67
votes
15
answers
22k
views
Why is ascending some stairs more exhausting than descending?
I have been asked this question by school kids, colleagues and family (usually less formally):
When ascending a flight of stairs, you exchange mechanical work to attain potential Energy ($W_\text{...
- 831
65
votes
4
answers
14k
views
Lie derivative vs. covariant derivative in the context of Killing vectors
Let me start by saying that I understand the definitions of the Lie and covariant derivatives, and their fundamental differences (at least I think I do). However, when learning about Killing vectors I ...
- 28.6k
63
votes
8
answers
6k
views
Ball hits curve of same curvature [closed]
I was doing some physics problems for homework and, while procrastinating, I came up with a theoretical scenario that I couldn't figure out the result of.
The following is from a side view and in a ...
- 628
61
votes
2
answers
98k
views
Difference between $\Delta$, $d$ and $\delta$
I have read the thread regarding 'the difference between the operators $\delta$ and $d$', but it does not answer my question.
I am confused about the notation for change in Physics. In Mathematics, $\...
- 899
60
votes
3
answers
28k
views
What is the difference between implicit, explicit, and total time dependence, e.g. $\frac{\partial \rho}{\partial t}$ and $\frac{d \rho} {dt}$?
What is the difference between implicit, explicit, and total time dependence, e.g. $\frac{\partial \rho}{\partial t}$ and $\frac{d \rho} {dt}$?
I know one is a partial derivative and the other is a ...
60
votes
6
answers
19k
views
What symmetry causes the Runge-Lenz vector to be conserved?
Noether's theorem relates symmetries to conserved quantities. For a central potential $V \propto \frac{1}{r}$, the Laplace-Runge-Lenz vector is conserved. What is the symmetry associated with the ...
- 5,765
59
votes
7
answers
12k
views
Why does Taylor’s series “work”?
I am an undergraduate Physics student completing my first year shortly. The following question is based on the physical systems I’ve encountered so far. (We mostly did Newtonian mechanics.)
In all of ...
- 1,999
58
votes
15
answers
13k
views
When a balloon pops and lets a brick fall, where does the energy come from?
Let's say a scientist attaches a 1 kg brick to a large helium inflated balloon, lets the balloon go, and then it reaches an altitude of 10 000 meters before it pops, dropping the brick.
The brick ...
- 1,308
58
votes
6
answers
12k
views
Tree-level QFT and classical fields/particles
It is well known that scattering cross-sections computed at tree level correspond to cross-sections in the classical theory. For example the tree-level cross-section for electron-electron scattering ...
Squark
57
votes
7
answers
10k
views
Why isn't the Euler-Lagrange equation trivial?
The Euler-Lagrange equation gives the equations of motion of a system with Lagrangian $L$. Let $q^\alpha$ represent the generalized coordinates of a configuration manifold, $t$ represent time. The ...
- 1,883
52
votes
4
answers
12k
views
What's the real fundamental definition of energy?
Some physical quantities like position, velocity, momentum and force, have precise definition even on basic textbooks, however energy is a little confusing for me. My point here is: using our ...
- 37.4k
51
votes
3
answers
38k
views
What is the meaning of the third derivative printed on this T-shirt?
Don't be a $\frac{d^3x}{dt^3}$
What does it all mean?
50
votes
5
answers
5k
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 Euler-...
- 1,370
48
votes
8
answers
15k
views
Classical mechanics without coordinates book
I am a graduate student in mathematics who would like to learn some classical mechanics. However, there is one caveat: I am not interested in the standard coordinate approach. I can't help but think ...
47
votes
4
answers
16k
views
What is the physical meaning of the connection and the curvature tensor?
Regarding general relativity:
What is the physical meaning of the Christoffel symbol ($\Gamma^i_{\ jk}$)?
What are the (preferably physical) differences between the Riemann curvature tensor ($R^i_{\ ...
- 13.7k
45
votes
11
answers
11k
views
How do we measure time?
I'm having a little trouble trying to put to words my problem and I apologize in advance for any causation of trouble in trying to interpret it.
We define periodic events as those events that occur ...
- 575
45
votes
5
answers
4k
views
Why are we sure that integrals of motion don't exist in a chaotic system?
The stadium billiard is known to be a chaotic system. This means that the only integral of motion (quantity which is conserved along any trajectory of motion) is the energy $E=(p_x^2+p_y^2)/2m$.
Why ...
- 2,393
45
votes
13
answers
5k
views
Mechanics around a rail tank wagon
Some time ago I came across a problem which might be of interest to the physics.se, I think. The problem sounds like a homework problem, but I think it is not trivial (i am still thinking about it):
...
- 2,717
44
votes
1
answer
8k
views
Understanding Poisson brackets
In quantum mechanics, when two observables commute, it implies that the two can be measured simultaneously without perturbing each other's measurement results. Or in other words, the uncertainty in ...
- 4,810
42
votes
3
answers
4k
views
Partial derivative notation in thermodynamics
Most thermodynamics textbooks introduce a notation for partial derivatives that seems redundant to students who have already studied multivariable calculus. Moreover, the authors do not dwell on the ...
- 1,634
42
votes
4
answers
7k
views
What is momentum really?
The Wikipedia article on momentum defines momentum as in classical mechanics:
… momentum is the product of the mass and velocity of an object.
However, an electromagnetic field has momentum, which ...
- 1,546
42
votes
7
answers
11k
views
Is there a proof from the first principle that the Lagrangian $L = T - V$?
Is there a proof from the first principle that for the Lagrangian $L$,
$$L = T\text{(kinetic energy)} - V\text{(potential energy)}$$
in classical mechanics? Assume that Cartesian coordinates are used. ...
- 771
42
votes
3
answers
2k
views
Is 13 really the answer for the "Devil's problem" in physics (a rolling tube with a rod)?
Recently I chewed the fat with a physics student and got intrigued by him mentioning "the Devil's problem," which he described as a simply worded mechanics problem that is extremely ...
- 1,649
40
votes
8
answers
6k
views
How is Liouville's theorem compatible with the Second Law of Thermodynamics?
The second law says that entropy can only increase, and entropy is proportional to phase space volume. But Liouville's theorem says that phase space volume is constant.
Taken naively, this seems to ...
- 105k
39
votes
5
answers
47k
views
Why is the covariant derivative of the metric tensor zero?
I've consulted several books for the explanation of why
$$\nabla _{\mu}g_{\alpha \beta} = 0,$$
and hence derive the relation between metric tensor and affine connection $\Gamma ^{\sigma}_{\mu \beta}...
- 929
39
votes
3
answers
6k
views
Are the Hamiltonian and Lagrangian always convex functions?
The Hamiltonian and Lagrangian are related by a Legendre transform:
$$
H(\mathbf{q}, \mathbf{p}, t) = \sum_i \dot q_i p_i - \mathcal{L}(\mathbf{q}, \mathbf{\dot q}, t).
$$
For this to be a Legendre ...
- 34.9k
39
votes
3
answers
7k
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 ...
- 34.9k
38
votes
13
answers
13k
views
If water is nearly as incompressible as ground, why don't divers get injured when they plunge into it?
I have read that water (or any other liquid) cannot be compressed like gases and it is nearly as elastic as solid. So why isn’t the impact of diving into water equivalent to that of diving on hard ...
- 515
38
votes
7
answers
7k
views
Does spin really have no classical analogue?
It is often stated that the property of spin is purely quantum mechanical and that there is no classical analog. To my mind, I would assume that this means that the classical $\hbar\rightarrow 0$ ...
- 2,494
38
votes
5
answers
16k
views
What are washers for? [closed]
When you attach a bolt to something using a nut, it is clear what the roles of the nut and bold are.
The more you tighten the bolt the more secure your fastening. However, you are often also told to ...
- 1,151
38
votes
5
answers
9k
views
Equivalence between Hamiltonian and Lagrangian Mechanics
I'm reading a proof about Lagrangian => Hamiltonian and one part of it just doesn't make sense to me.
The Lagrangian is written $L(q, \dot q, t)$, and is convex in $\dot q$, and then the ...
- 559
37
votes
5
answers
6k
views
Why is a leading digit not counted as a significant figure if it is a 1?
Reading the book Schaum's Outline of Engineering Mechanics: Statics I came across something that makes no sense to me considering the subject of significant figures:
I have searched and saw that ...
- 788
37
votes
6
answers
71k
views
What are holonomic and non-holonomic constraints?
I was reading Herbert Goldstein's Classical Mechanics. Its first chapter explains holonomic and non-holonomic constraints, but I still don’t understand the underlying concept. Can anyone explain it to ...
- 381