What is Classical mechanics: Definition and 1000 Discussions
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical mechanics, if the present state is known, it is possible to predict how it will move in the future (determinism), and how it has moved in the past (reversibility).
The earliest development of classical mechanics is often referred to as Newtonian mechanics. It consists of the physical concepts based on foundational works of Sir Isaac Newton, and the mathematical methods invented by Gottfried Wilhelm Leibniz, Joseph-Louis Lagrange, Leonhard Euler, and other contemporaries, in the 17th century to describe the motion of bodies under the influence of a system of forces. Later, more abstract methods were developed, leading to the reformulations of classical mechanics known as Lagrangian mechanics and Hamiltonian mechanics. These advances, made predominantly in the 18th and 19th centuries, extend substantially beyond earlier works, particularly through their use of analytical mechanics. They are, with some modification, also used in all areas of modern physics.
Classical mechanics provides extremely accurate results when studying large objects that are not extremely massive and speeds not approaching the speed of light. When the objects being examined have about the size of an atom diameter, it becomes necessary to introduce the other major sub-field of mechanics: quantum mechanics. To describe velocities that are not small compared to the speed of light, special relativity is needed. In cases where objects become extremely massive, general relativity becomes applicable. However, a number of modern sources do include relativistic mechanics in classical physics, which in their view represents classical mechanics in its most developed and accurate form.
Hello,
I often get confused when it comes to some physical concepts in classical mechanics. That's mostly because I like to ask a ton of questions and because that I dig myself into a hole that I can't come out of. So, I'm wondering if anyone knows a really good classical mechanics book that...
I express the total kinetic energy of the body, via König theorem, as
$$T=\frac{1}{2}mv_p^2+\frac{1}{2}mI{\omega}^2$$
where $$v_p=(v_x,v_y)=(\dot{r}\cos\varphi-r\dot{\varphi}\sin\varphi-\frac{l}{2}(\dot\varphi-\dot\psi)\sin(\varphi-\psi),\dot r \sin\varphi+r\dot\varphi...
For the past few months, I've been on a look out for the best physics community on the Internet and I've just come across this one. My primary goal is to gain as much knowledge as possible in the area of classical mechanics and electromagnetism in a year. I'm fairly new to magnetism, but I can't...
(a). Getting the potential is straight forward, just take the negative derivative of F(x) and then enforce the condition that the function should approach 0 at x = infinity.
##- \int -a e^{-bx} (1 - e^{-bx})## = ##\int a e^{-bx} - a e^(-2bx)## = ## \frac{a}{b} (\frac{1}{2}e^{-2bx} - e^{-bx}) +...
So I’m a high school student and I am planing to participate in higher-category physics competitions in my country. However, I think that my theoretical understanding of the basics isn’t clear enough yet - by basics I mean classical / Newtonian mechanics. I am the type of person that learns by...
My set-up is the following: i have an iron bolt suspended on a string next to an electromagnet, of which I steadily increase the voltage and thereby the magnetic field. Supposing the force is linear with the magnetic field and dependent on the distance between the bolt and magnet. The exact...
Suppose somehow an object is moving upwards with a speed ##v##, at this point I start applying a force ##F## that is equal to its weight, so the net force on the object is zero. So it will continue moving upwards with its initial speed. Suppose after the height difference is ##h##, I stop...
I'm using rigid body dynamics/kinematics in robotics stuff but I don't have a background in mechanics, I'm interested in understanding the kinematics of frame transformations for rigid bodies.
Suppose we have two reference frames fixed on a rigid body, F_1 and F_2 and a transformation T which...
(This is a homework assignment that my sister (younger than me) didn't manage to solve, but I am not sure about the attempt I thought of either. Especially in the last point. So I ask you to correct where I am wrong).
I solved ##a)## with the following:
1. Since ##B## must be stationary, I...
[Rewriting this as per the suggestions. Thanks once again.]
I won't be using the Lagrangian because it was never explicitly stated that I have to so I'll just use conservation of energy.
$$ T = \frac{1}{2}mv^2 = \frac{1}{2}m(R\dot{\theta})^2 = \frac{1}{2}mR^2\dot{\theta}^2 $$
$$ V = mgy =...
I use ##l-1## lagrangian coordinates ##\alpha_1,...,\alpha_{l-1}## . ##\alpha_i## is the angle between ##OP_{i-1}## and ##OP_{i}##.
As the length of a chord between two rays with angle ##\alpha## is ##d=2Rsin(\alpha/2)##, I write the potential energy of the system as...
Hi, in the Hamiltonian formulation of classical mechanics, the phase space is a symplectic manifold. Namely there is a closed non-degenerate 2-form ##\omega## that assign a symplectic structure to the ##2m## even dimensional manifold (the phase space).
As explained here Darboux's theorem since...
We start with something like
If we suppose the wire is the green line, we are to figure out what the green line looks like to the right of the red bead.
What I first thought of was simply
$$\vec{r}'(t)=x'(t)\hat{i}+y'(t)\hat{j}=v_0\hat{i}+gt\hat{j}\tag{1}$$...
My answer to the question is: build a two-stage rocket. Or a ##k##-stage rocket. Then I thought: what happens if we try to make ##k=\infty##?
To cut to the chase, my question is how to calculate the infinite series
$$\lim\limits_{k\to\infty} \sum\limits_{i=1}^k \ln{\left (...
I have found the Hamiltonian to be ##H = L - 6 (q_1)^2## using the method below:
1. Find momenta using δL/δ\dot{q_i}
2. Apply Hamiltonian equation: H = sum over i (p_i \dot{q_i}) - L 3(q_1)^2. Simplifying result by combining terms
4. Comparing the given Lagrangian to the resulting Hamiltonian I...
I've started reading the Principia and have been trying to follow along with the examples. Unfortunately, I got stuck almost immediately. This example is from 'Axioms, or laws of motion', Law III, Corollary II. It is based on the following picture (everything in red is my addition):
The text...
For part (d), there is the formula a = v^2/r I can use. Note that Mg = mv^2/r, we have two unknowns, v and r. I can solve this if conservation of angular momentum is true, i.e. mvr = constant. I am not convinced I can use this however, because is increasing M torque?
My idea is that it is an...
I've heard that classical mechanics and electromagnetics are not applicable at small sizes in particle physics.
1) At what size and energy levels are they no longer considered to be applicable at all?
2) What range of size and energy levels could be considered a "transition" area where both...
We know the wave function:
$$ \frac {\partial^2\psi}{\partial t^2}=\frac {\partial^2\psi}{\partial x^2}v^2,$$
where the function ##\psi(x,t)=A\ e^{i(kx-\omega t)}## satisfies the wave function and is used to describe plane waves, which can be written as:
$$ \psi(x,t)=A\ [\cos(kx-\omega...
vA = 3u/4 and vB = u/4, and 1/8 KE is lost. I can't get to these answers however: for the first part, I got to u = vA + 3vB using conservation of momentum, and the fact that particle B is at an angle, hence I would think its momentum should be 10mvBsin(arcsin(3/5)). Doing the same for A with...
What is the correct statement of Varignon's theorem?
On the net I find some discrepancies between the various statements: in some cases the vectors of the system referred to by the theorem must be applied at the same point or such that their lines of action pass through the same point, in other...
Can somebody explain why the kinetic term for the fluctuations was already diagonal and why to normalize it, the sqrt(m) is added? Any why here Z_ij = delta_ij?
Quite confused about understanding this paragraph, can anybody explain it more easily?
I have read some text about defining the cross product. It can be defined by both a x b = epsilon_(ijk) a^j b^k e-hat^i and a x b = epsilon^(ijk) a_i b_j e-hat^k
why the a and b have opposite indice positions with the epsilon? How to understand that physically?
Consider the above system, where both the wedge and the mass can move without friction. We want to get the equations of motion for the both of them using Lagrangian formalism, where the constraints in the solution sheet are given as:
$$y_2=0$$
$$\tan \alpha=\frac {y_1}{x_1-x_2}$$
However...
Assuming that a universe operates according to Aristotle's mechanics, that is to say, the second law of Aristotle in this universe
F=mV (V is the velocity of the object's motion)True. (Note that 'm' here does not have a dimension of mass.)
In order to obtain a logically consistent Aristotle's...
For the central force ##F=-\nabla U(r_r)## where ##\vec r_r=\vec r_1-\vec r_2##, and ##\vec r_1## and ##\vec r_2## denote the positions of the masses, we get the following kinetic energy using the definition of center of mass ##\vec r_{cm}= \frac{m_1\vec r_1+m_2\vec r_2}{m_1+m_2}##:
$$T= \frac...
One of the constraints is given as ##r=R##, which is very obvious. The second constraint is however given as
$$\phi - \frac {2\pi} h z=0$$
where ##h## is the increase of ##z## in one turn of the helix. Physically, I can't see where this constraint comes from and how ##\phi=\frac {2\pi}h z##.
Suppose we have mass ##m## attached to spring with constant ##k##, and some coefficient of kinetic friction ##\mu## between the mass and the surface. Its displaced from equilibrium by some distance ##x## at ## t = 0 ##. I've come up with the following ODE to described ##x(t)## using the...
Consider the following setup
where the bead can glide along the rod without friction, and the rod rotates with a constant angular velocity ##\omega##, and we want to find the constraint force using Lagrange multipliers.
I chose the generalized coordinates ##q=\{r,\varphi\}## and the...
I'm just getting started on Lagrangian mechanics and what I can't understand is, how did Lagrange discover the Lagrangian? Did he just randomly decide to see what would happen if we calculate KE - PE or T - V and then discovered that the quantity is actually mathematically and physically...
My question is about the general relationship between the constraint functions and the constraint forces, but I found it easier to explain my problem over the example of a double pendulum:
Consider a double pendulum with the generalized coordinates ##q=\{l_1,\theta_1,l_2,\theta_2\}##,:
The...
For a rigid body with three principal axis with distinct moments of inertia, would the principal axis with the intermediate moment of inertia still be unstable in ideal conditions, e.g. no gravity, no friction etc.? From the mathematical derivation I deduce that it should be unstable, since we...
I learned some computer science basics from the book SICP ( Structure and Interpretation of Computer Programs, Authors: Gerald Jay Sussman, Hal Abelson, Julie Sussman ) and I've witnessed a book about mechanics from the same author called Structure and Interpretation of Classical Mechanics...
In classical mechanics, it seems like solving force equations are a question of finding a solvable system of equations that accounts for all existing forces and masses in question. Therefore, I'm curious if this can be mixed with reinforcement learning to create a game and reward function...
The invariance of this volume element is shown by writing the infinitesimal volume elements $$d\eta$$ and $$d\rho$$
$$d\eta=dq_1.....dq_ndp_1......dp_n$$
$$d\rho=dQ_1.......dQ_ndP_1....dP_n$$
and we know that both of them are related to each other by the absolute value of the determinant of...
The homework statement isn't exactly as is mentioned above. The actual problem statement is as follows:
This is problem 3.8 from John R. Taylor's Classical Mechanics; however, my question is not related to the main problem itself but one particular aspect of it. Now, in the same textbook (John...
Goldstein's Classical Mechanics makes the claim (pages 382 to 383) that given coordinates ##q,p##, Hamiltonian ##H##, and new coordinates ##Q(q,p),P(q,p)##, there exists a transformed Hamiltonian ##K## such that ##\dot Q_i = \frac{\partial K}{ \partial P_i}## and ##\dot P_i = -\frac{\partial...
I am attempting problem number 2.38 from John R. Taylor's Classical Mechanics and I am not getting the correct answer. My procedure is as follows:
Equation of motion (taking up as the positive direction):
$$m\dot{v}=-mg-cv^2$$
Now to find ##v_\mathrm{ter}##, the terminal velocity, we consider...
I am self studying Goldstein first chapter "A survey on the elementary principles", so far I have been enjoying it, sometimes he skips some lines while deriving a principle or so, therefore sometimes I get to PSE or Physics Forums to know the things I lack understanding in.
What are your...
I am intending to join an undergrad course in physics(actually it is an integrated masters course equivalent to bs+ms) in 1-1.5 months. The thing is, in order to take a dive into more advanced stuff during my course, I am currently studying some of the stuff that will be taught in the first...
We have 2 objects, m1 and m[SUPlB]2[/SUB]
Friction is present between the two objects but not between m1 and the floor. A force is exerted on the bottom object which causes it to accelerate parallel to the floor. The thing I'm wondering for while now is, how do I prove that the acceleration of...
From a classical mechanics perspective I understand the force interactions leading to the phenomenon, but from a matter perspective, what is a "positive" or "negative ly" charged water stream?
Is this referring to the spontaneousH(+) + OH(- )formations?
I just calculated the Lagrangian of a particle of mass ##m## in a radially symmetric potential ##V(r)##. I thought it would be a good idea to switch to spherical coordinates for that matter. What I get is
$$
L = \frac{1}{2} m \left( \dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \dot{\varphi}^2...
I have been trying to understand this proof from the book 'Introduction to classical mechanics' by David Morin. This proof comes up in the first chapter of statics and is a proof for the definition of torque.
I don't understand why the assumption taken in the beginning of the proof is...
I tried writing this out but I think there is a bug or something as its not always displaying the latex, so sorry for the image.
I have gone through various sources and it seems that the reason for u being small varies. Sometimes it is needed because of the taylor expansion, this time (below) is...
Necessary condition for a curve to provide a weak extremum.
Let ##x(t)## be the extremum curve.
Let ##x=x(t,u) = x(t) + u\eta(t)## be the curve with variation in the neighbourhood of ##(\varepsilon,\varepsilon')##.
Let $$I(u) = \int^b_aL(t,x(t,u),\dot{x}(t,u))dt = \int^b_aL(t,x(t) +...