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.

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  1. G

    Finding slip-off angle for mass off of sphere?

    [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 =...
  2. l4teLearner

    Potential of particles moving on a circle attracted by elastic force

    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...
  3. cianfa72

    A Hamiltonian formulation of classical mechanics as symplectic manifold

    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...
  4. Z

    Bead sliding along wire with constant horizontal velocity -- Shape of wire?

    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 $$\vec{r}'(t)=x'(t)\hat{i}+y'(t)\hat{j}=v_0\hat{i}+gt\hat{j}\tag{1}$$...
  5. Z

    How to calculate velocity of infinite-stage rocket?

    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 (...
  6. astroholly

    Deriving the Hamiltonian of a system given the Lagrangian

    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...
  7. G

    I Question about an example in Newton's Principia

    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...
  8. L

    Confused on whether this counts as an external torque

    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...
  9. jmheer

    I When do classical mechanics and electromagnetics stop working?

    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...
  10. deuteron

    I Physical Meaning of the Imaginary Part of a Wave Function

    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...
  11. areverseay

    Classical mech. - inelastic collision

    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...
  12. Hak

    I What is the correct statement of Varignon's theorem?

    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...
  13. G

    A Lagrangian: kinetic matrix Z_ij and mass matrix k_ij

    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?
  14. G

    A Upper indices and lower indices in Einstein notation

    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?
  15. deuteron

    Mass on a wedge, both can move

    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...
  16. lhrhzm

    I An interesting question about another view of baisc mechanics'laws

    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...
  17. deuteron

    Solving two body central force motion using Lagrangian

    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...
  18. deuteron

    Bead moving down a Helical Wire subject to Constraints

    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##.
  19. E

    I A Continuous Solution for Mass/Spring w/ Friction

    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...
  20. deuteron

    Constraint force using Lagrangian Multipliers

    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...
  21. Feynstein100

    I Alternatives to the Lagrangian?

    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...
  22. deuteron

    I Constraint Forces and Lagrange Multipliers

    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...
  23. deuteron

    I Intuition Behind Intermediate Axis Theorem in an Ideal Setting

    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...
  24. C

    Classical How is the book "Structure and Interpretation of Classical Mechanics"?

    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...
  25. giodude

    I Using Linear Algebra to discover unknown Forces

    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...
  26. Maumas

    Invariance of a volume element in phase space, What does it mean?

    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...
  27. F

    Why is the thrust equation same under gravitational force?

    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...
  28. Lagrange fanboy

    I Proof that canonical transformation implies symplectic condition

    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...
  29. Anubhav singh

    Question about the Product of Inertia for a Rolling Disk

    I think product of inertia always zero because rolling motion of disk is fully balanced
  30. F

    Vertical projectile motion with quadratic drag (sign convention)

    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...
  31. rajsekharnath

    What are your thoughts on self-studying Goldstein?

    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...
  32. rajsekharnath

    Classical Source recommendation on Differential Geometry

    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...
  33. M

    Billiards with rectangular prisms

    I plan to add on to this as I have time and ability. Apologies for the weird formatting.
  34. al4n

    B Need help understanding some concepts about friction

    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...
  35. B

    What is the source of the Kelvin Water Dropper effect?

    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?
  36. PhysicsRock

    I Why isn't the Lagrangian invariant under ##\theta## rotations?

    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...
  37. Shreya

    The Definition of Torque - a proof

    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...
  38. Reuben_Leib

    I Help with Euler Lagrange equations: neighboring curves of the extremum

    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...
  39. Reuben_Leib

    I Why does ##u## need to be small to represent the Taylor expansion

    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) +...
  40. R

    Coordinates of a point on a rotating wheel

    My issue is in deriving the coordinates of a point on a wheel that rotates without slipping. In Morin's solution he says that: My attempt at rederiving his equation: I do not understand how the triangle on the bottom with sides indicated in green is the same as the triangle on top that is...
  41. Slimy0233

    I Heating a tungsten filament to find out the maximum brightness emitted

    I was taking notes from a lecture on Quantum Physics and during the introduction, they gave an example of what led to the discovery of Quantum Physics: The electric bulb example where the brightness and colour of light depended on the temperature of the filament(see...
  42. Peter Morgan

    I The collapse of a quantum state as a joint probability construction

    The titular paper can be found here, https://doi.org/10.1088/1751-8121/ac6f2f, and on arXiv as https://arxiv.org/abs/2101.10931 (which is paginated differently, but the text and equation and section numbers are the same). Please see the abstract, but in part this 24 page paper argues that we...
  43. C

    Morin classical mechanics differential equation problem

    I was reading the oscillations chapter which was talking about how to solve linear differential equations. He was talking about how to solve the second order differential below, where a is a constant: In the textbook, he solved it using the method of substitution i.e guessing the solution...
  44. V

    Satellite mechanics: linear and rotational momentum

    [This is a continuation of OP's thread here: https://www.physicsforums.com/threads/satellite-mechanics-linear-and-rotational-momentum.1046963/ ] satellite mechanics: linear and rotational momentum I'm trying to better understand classical mechanics, and came up with a question: Say we have a...
  45. Lotto

    Can two objects moving parallel relative to a ground meet?

    I think that yes but how to explain it when someone standing on a ground sees them moving paralell? If I move properly, I can see two cars moving paralell ralative to the ground crashing, while someone on the ground do not see them crashing. Is it consistent?
  46. Lotto

    What is the smallest value of angular displacement of the raft?

    What is meant by "the smallest value of angular displacement of the raft from its original position during one cycle"? I understand that I am supposed to solve this problem using torques of the crane and and of the boxes, but I am totally confused by that "smallest angular displacement". If it...
  47. V

    I Satellite mechanics: linear and rotational momentum

    satellite mechanics: linear and rotational momentum I'm trying to better understand classical mechanics, and came up with a question: Say we have a squared satellite weighting 100kg, 1 meter on each side. it has a thruster on it's side, shown in picture thruster quickly ejects 100g of propellant...
  48. A

    Classical A replacement for Mcauley's Classical mechanics

    Mcauley's "Classical mechanics: transformations, flows, integrable and chaotic dynamics" has a very interesting table of contents, and it has a philosophy of approaching Hamiltonian flows and chaos without using the formalism of modern differential geometry. Unfortunately, after reading the...
  49. S

    Distance travelled by a car considering only air friction?

    TL;DR Summary: Distance traveled by a car considering only air friction? How much distance would a 3-ton car travel if its initial speed was 17 km/h and we only take into account air's friction? (Assume that the car has an airfoil-like shape, so that the resistance against the air is very low)...
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