Classical mechanics Definition and 1000 Threads

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. Manish_529

    How can "work" done by a force be negative?

    If the work done by a force is the consequence of the dot product of the force and the displacement of the particle, then it should be a scalar quantity which is true as well. But then why does question of the work done being positive or negative (depending on the angle between the force vector...
  2. Clockclocle

    Relation between the velocity vector and the acceleration vector of an object

    A uniform circular motion of a point always yield an equation u=cos(wt)i +sin(wt)j of position vector. Which we deduce the acceleration and velocity vector with constant magnitude and they are perpendicular at each instant. Can I use the information of them at one instant to predict the position...
  3. imperiumx

    Books to read before Arnold's Classical Mechanics

    TL;DR Summary: What book should i read before arnolds classical mechanics? I have good math background but little physics background. A little while ago, i was in a summer camp for teens who are exceptional at math. Around that time i started getting into physics. A person from there...
  4. Manish_529

    Tension in a rope hanging between 2 trees

    I know that I can draw an FBD and apply Newton's 2nd law to find the relevant equations. But my question here is why is the mg vector or the weight of the entire rope same at every point on it I mean to say that if the mass of the entire rope is say M then how can a small point on the rope also...
  5. R

    I What's the interpretation of the third term F(t)q?

    I think that time derivative of the energy of the oscillator is F times the derivative of q, which means it's the power of the external force. So it's like it is suggesting that F·q is the work done by external force, which makes no sense at all. As far as I concerned, the input energy has no...
  6. Heisenberg7

    Advice on Resources -- A really good classical mechanics book?

    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...
  7. l4teLearner

    Equivalence of Euler-Lagrange equations and Cardinal Equations for a rigid planar system

    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...
  8. Heisenberg7

    Found the Ideal Physics Community for Olympiad Prep?

    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...
  9. G

    Path of a free particle over a sphere

  10. D

    For what values of total energy is the resulting motion periodic?

    (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}) +...
  11. A

    Classical The best introductory mechanics textbook

    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...
  12. Sidsid

    Experiment measuring the distance dependence of a suspended metal bolt from a magnet

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  13. D

    B Conceptually understanding change in potential energy with 0 net work

    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...
  14. D

    I Frame Transformation in rigid bodies

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  15. HighPhy

    Two bodies connected with an elastic thread moving with friction

    (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...
  16. 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 =...
  17. l4teLearner

    Potential of particles moving on a circle attracted by elastic force

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  18. cianfa72

    A Hamiltonian formulation of classical mechanics as symplectic manifold

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  19. Z

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

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  20. Z

    How to calculate velocity of infinite-stage rocket?

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  21. astroholly

    Deriving the Hamiltonian of a system given the Lagrangian

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  22. G

    I Question about an example in Newton's Principia

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  23. L

    Confused on whether this counts as an external torque

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  24. jmheer

    I When do classical mechanics and electromagnetics stop working?

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  25. deuteron

    I Physical Meaning of the Imaginary Part of a Wave Function

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  26. areverseay

    Classical mech. - inelastic collision

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  27. 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...
  28. 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?
  29. 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?
  30. deuteron

    Why Doesn't Constant Center of Mass Velocity Reduce Degrees of Freedom?

    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...
  31. 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...
  32. 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...
  33. 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##.
  34. 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...
  35. 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...
  36. 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...
  37. 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...
  38. 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...
  39. 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...
  40. 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...
  41. 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...
  42. 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...
  43. 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...
  44. 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
  45. 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...
  46. 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...
  47. 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...
  48. M

    How Can Rectangular Prisms Transform Billiards?

    I plan to add on to this as I have time and ability. Apologies for the weird formatting.
  49. 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...
  50. 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?
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