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. Classical Physics by Prof. V. Balakrishnan (NPTEL):- Module 1, Lecture 5: Autonomous Dynamical Systems (Part 1)

    Classical Physics by Prof. V. Balakrishnan (NPTEL):- Module 1, Lecture 5: Autonomous Dynamical Systems (Part 1)

    All copyright reserved to NPTEL, Govt. Of India and Prof. Balakrishnan.
  2. Classical Physics by Prof. V. Balakrishnan (NPTEL):- Module 1, Lecture 6: Autonomous Dynamical Systems (Part 2)

    Classical Physics by Prof. V. Balakrishnan (NPTEL):- Module 1, Lecture 6: Autonomous Dynamical Systems (Part 2)

    All copyright reserved to NPTEL, Govt. Of India and Prof. Balakrishnan.
  3. Classical Physics by Prof. V. Balakrishnan (NPTEL):- Module 1, Lecture 7: Lagrangian Formalism

    Classical Physics by Prof. V. Balakrishnan (NPTEL):- Module 1, Lecture 7: Lagrangian Formalism

    All copyright reserved to NPTEL, Govt. Of India and Prof. Balakrishnan.
  4. Classical Physics by Prof. V. Balakrishnan (NPTEL):- Module 1, Lecture 8: Summary of Classical Electromagnetism

    Classical Physics by Prof. V. Balakrishnan (NPTEL):- Module 1, Lecture 8: Summary of Classical Electromagnetism

    All copyright reserved to NPTEL, Govt. Of India and Prof. Balakrishnan.
  5. Classical Physics by Prof. V. Balakrishnan (NPTEL):- Module 1, Lecture 9: Charged particle in an electric field

    Classical Physics by Prof. V. Balakrishnan (NPTEL):- Module 1, Lecture 9: Charged particle in an electric field

    All copyright reserved to NPTEL, Govt. Of India and Prof. Balakrishnan.
  6. Classical Physics by Prof. V. Balakrishnan (NPTEL):- Module 1, Lecture 10: Hamiltonian Dynamics (Part 1)

    Classical Physics by Prof. V. Balakrishnan (NPTEL):- Module 1, Lecture 10: Hamiltonian Dynamics (Part 1)

    All copyright reserved to NPTEL, Govt. Of India and Prof. Balakrishnan.
  7. Toby_phys

    Using Noether's Theorem to get conserved quantities

    Homework Statement N point particles of mass mα, α = 1,...,N move in their mutual gravitational field. Write down the Lagrangian for this system. Use Noether’s theorem to derive six constants of motion for the system, none of which is the energy Homework Equations Noethers Theorem: If a...
  8. D

    Lagrangian of two equal masses attached by a spring

    (note: I'm going to represent the lagrangian as simply L because I don't know how to do script L in latex.) Homework Statement Two particles of equal masses m are confined to move along the x-axis and are connected by a spring with potential energy ##U = \frac{1}/{2}kx^2## (here x is the...
  9. Amitayas Banerjee

    Kinetic energy as seen from a different frame

    Homework Statement A mass m is initially at rest. A constant force $F$ (directed to the right) acts on it over a distance d. The increase in kinetic energy is therefore $Fd$. Consider the situation from the point of view of someone moving to the left at speed $V$ . Show explicitly that this...
  10. ZapperZ

    "Classical Physics Is Wrong" Fallacy - Comments

    Greg Bernhardt submitted a new PF Insights post "Classical Physics Is Wrong" Fallacy Continue reading the Original PF Insights Post.
  11. D

    Energy analysis of a particle moving in a shrinking circle

    Homework Statement A particle of mass m is moving on a frictionless horizontal table and is attached to a massless string, whose other end passes through a hole in the table, where I am holding it. Initially the particle is moving in a circle of radius ##r_0## with angular velocity ##w_0##, but...
  12. Ben Geoffrey

    I Taylor Series Expansion of Quadratic Derivatives: Goldstein Ch. 6, Pg. 240

    Can anyone tell me how if the derivative of n(n') is quadratic the second term in the taylor series expansion given below vanishes. This doubt is from the book Classical Mechanics by Goldstein Chapter 6 page 240 3rd edition. I have attached a screenshot below
  13. qttv

    A Classical Mechanics: Continuous or Discrete universe

    Good morning. The question of the "continuous" or "discrete" nature of the universe is the subject of diatribe among the greatest physicists in the world. I would like to discuss the same topic, but asking a question about the aspect of continuum in classical mechanics. The use of mathematical...
  14. Gene Naden

    Classical Undergrad Classical Mechanics with Hamiltonian formulation

    I am looking for an undergraduate textbook on Classical Mechanics that includes Hamiltonian and Lagrangian formulations. One reason for this is that I am interested in quantization and second quantization. It should include treatment of harmonics oscillators. Thanks!
  15. Manolisjam

    3 balls in a moving mechanics problem

    Homework Statement Consider 2 balls A,B on the same line . and they are connected to a third one G with a rope L. AG, AB. now the system monves in the effect of the mass of G and its projection to the line AB is in the middle. No friction. mass of A=mass of B=m and mass of G=2m .FInd the time...
  16. Manolisjam

    Classical Mechanics Problem with balls

    Homework Statement Consider 2 balls A,B on the same line . and they are connected to a third one G with a rope L. AG, AB. now the system monves in the effect of the mass of G and its projection to the line AB is in the middle. No friction. mass of A=mass of B=m and mass of G=2m .FInd the time...
  17. Hamza Qayyum

    I Force applied to a sphere off center of mass

    I'm trying to model a sphere having force applied at position P in the following diagram: I know that this applied force will have an x and y component; the y component will propel it upwards, but what I am confused about is the x component of the force. I know that the x component will propel...
  18. R

    Volume of smaller pyramid is less than the volume of the whole.

    I have a test coming up next week and while doing some practice questions I found one I can't wrap my head around. The question is: A pyramid (assume uniform density) is divided in two parts by a horizontal plane through its center of mass. How do the masses of the two parts compare ? There are...
  19. H

    Find the forces on a table that is lifted in a tilted position

    Homework Statement Homework EquationsThe Attempt at a Solution what's the Fa and Fb? my logic is Fa = Wa - N = mg - ##\frac{N}{\cos \theta}## so is Fb... so the ratio is 1. But I know it's wrong. What will make the difference between Fa and Fb? can anyone give a clue please?
  20. R

    Centre of mass of binary system calculation

    Homework Statement Given a graph (see below) containing the velocities of two stars with respect to the sun, I am asked to calculate the velocity of the centre of mass of a binary system. I am not given the mass of either star, nor the shape of the orbit nor the velocity of the centre of mass...
  21. F

    Set up the Lagrangian for a CO2 molecule

    Homework Statement The carbon dioxide molecule can be considered a linear molecule with a central carbon atom, bound to two oxygen atoms with a pair of identical springs in opposing directions. Study the longitudinal motion of the molecule. If three coordinates are used, one of the normal...
  22. D

    A rocket needs to hover stationary over the ground

    Homework Statement A rocket (initial mass ##m_o##, constant exhaust velocity ##v_{ex}## needs to use its engines to hover stationary, just above the ground. a) If it can afford to burn no more than a mass ##\lambda m_o## of its fuel (##\lambda \lt##), for how long can it hover? b) If ##v_{ex} =...
  23. Ben Geoffrey

    I Definite Integrals and the Principle of Least Action: Exploring the Connection

    This is with regard to my doubt in the derivation of the principle of least of action in Goldstein Is there any theorem in math about definite integrals like this ∫a+cb+df(x)dx = f(a)c-f(b)d The relevant portion of the derivation is given in the image.
  24. F

    Velocity needed to escape gravitational potential

    I am solving a problem where I need to decide if an asteroids velocity is high enough to escape the planets gravitational pull. The way I did it was use conservation of energy and angular momentum to find an expression for the radial velocity and show that it remains positive as r tends to...
  25. V

    Minimum friction to roll an imperfect sphere on an incline

    Homework Statement I have a sphere, it's imperfectly spherical, I put it on an incline and apparently it needs a minimum friction to start rolling or moving. Homework Equations I = 2/5 MR^2 torque = I*alpha = R x F The Attempt at a Solution [/B] before anything, I don't get why there's a...
  26. Ben Geoffrey

    I Free Vibrations of a linear triatomic molecule

    This question is regarding equation 6.59 in Classical Mechanics Goldstein 3rd edition. I don't understand how he reaches that step. He says he does by inverting 6.41. Is he taking the transpose matrix composed of the eigen vectors by meaning inversion ? I thought I could upload the screenshot...
  27. Zack K

    Mass of a picture in static equilibrium

    Homework Statement An artist must push with a minimum of 75N at an angle of 45° to a picture to hold it in equilibrium. The coefficient of friction between the wall and the picture frame is 0.30. What is the mass of the picture? Homework Equations ΣF=0 Ffr=μFN F=ma The Attempt at a Solution I...
  28. A

    Classical Textbook for Classical Mechanics

    I am self studying Classical Mechanics and I am a bit confused about this. After reading Kleppner Kolenkow Mechanics can I read Symon's Mechanics or do I need to read anything in between like Fowels or Marion&Thornton? I already studied the Maths portion upto Multivariable Calculus,Real...
  29. R

    Classical Books with exercises on Classical Mechanics (Introductory)

    I need a book with exercises, to try to excel my next classical mechanics, I really to get a good grade. Thank you very much!
  30. R

    Studying Bad grade in classical mechanics

    I’m felling demotivated because I had a bad grade, I don’t know how I got such kind of grade. However, I’m looking to get a very good grade in the next test, that will count 100%. I’m in the first year of physics, what advices do you have on how to study(CM), which exercises? What books should...
  31. V

    Classical mechanics: motion and acceleration

    Homework Statement Referring to the attached diagram, the object ABCD is kept in an isolated system where it is under no external influence of any kind. A· An instantaneous force F acts along the edge BC of the block ABCD, where M is its mass, with its center of mass located at O...
  32. A

    Will a sphere rotate on a frictionless inclined surface?

    Well, my physics teacher taught us about rotation the other day and I came across a scenario where a sphere and a ring roll down a friction-less inclined plane from a point of absolute rest. I found it counter-intuitive as I started to think about why would they start rolling in the first place...
  33. K

    Find the angular speed of the smaller gear

    1.Why does the angular speed of small gear depend only on larger gear only? 2.Why does the length of linkage connecting two gears does not have any influence on the angular speed of smaller gear? The first question can be answered by looking at slack and tightening of chain caused by rotation...
  34. U

    Hamiltonian and Lagrangian in classical mechanics

    Is the following logic correct?: If you have an hamiltonian, that has time has a variable explicitly, and you get the lagrangian,L, from it, and then you get an equivalent L', since L has the total time derivate of a function, both lagrangians will lead to the same equations euler-lagrange...
  35. U

    Classical Goldstein 3ed Classical mechanics solutions

    is there a way for me see EVERY solution from goldstein's book? i already have some solutions, but not every one
  36. chrononaut 114

    Past Exam Advanced Dynamics Question; Help, Please

    Homework Statement The Attempt at a SolutionSo I first tried by saying consider a time t in which mass m is directly above the origin O. I.e., mass m at the Cartesian coordinate (0, 4l/3). I wrote a = a(t) as the extension function of the spring, which has 0 natural length. So, I applied the...
  37. B

    How Does Goldstein Derive the Kinetic Energy Equation in Classical Mechanics?

    how is md^2r/dt^2 . dr/dt = d/dt (1/2 m (dr/dt)^2 ) Thank You
  38. V

    Two blocks of mass m1, m2 are attached to a spring

    Homework Statement Now these two blocks are pulled separately by x1 and x2 distance. Find the frequency of shm Homework Equations a = -w^2 x The Attempt at a Solution Now after m1 being pulled, it moves with acceleration a1=Force of spring/m1. acceleration of block 2 = a2=/force of...
  39. W

    Writing Hamiltonian: Classical Mechanics

    Homework Statement I'm having some issues understanding a number of concepts in this section here. I attached the corresponding figure at the end of the post for reference. Issue 1) 1st of all, I understand that a Hamiltonian can be written as such $$H = T_2 - T_0 + U$$ whereby ##T_2##...
  40. VSayantan

    Elastic Collision between a Ball and a Stick

    Homework Statement [/B] A mass ##m## travels in a straight line with velocity ##v_0## perpendicular to a uniform stick of mass ##m## and length ##l##, which is initially at rest.. The distance from the center of mass of the stick to the path of the traveling mass is ##h##. Now the traveling...
  41. mcaay

    Lagrange Multipliers in Classical Mechanics - exercise 1

    Homework Statement The skier is skiing without friction down the mountain, being all the time in a specified plane. The skier's altitude y(x) is described as a certain defined function of parameter x, which stands for the horizontal distance of the skier from the initial position. The skier is...
  42. Luxucs

    Normal force as a function of time (oscillator)

    Homework Statement A spring with spring constant k is attached to a box of mass M in which is placed a small body of mass m. The system is displaced a distance A from equilibrium and released from rest. Find the normal force between the box and the small mass as a function of time. For what...
  43. Clara Chung

    Classical mechanics question (pendulum)

    Homework Statement Homework EquationsThe Attempt at a Solution I have done part a, I have no idea on part b, here is my attempt,
  44. V

    Potential energy of a ball on top of a vertical rod

    Homework Statement The problem is to take a small ball of mass m on a uniform rod of mass m which is hung on a hinge vertically downward, they're asking to find the velocity to be imparted for it undergo a complete rotation. I have the kinetic energy bit worked out but I'm getting the wrong...
  45. Krunchyman

    Teach me about classical mechanics please

    I have no prior experience in physics, but I am a math undergrad so I know calculus, differential equations, linear algebra and stuff like that. So I'm ready to move from the conceptual "bedtime story physics" to stuff with real mathematical rigor. So, what is classical mechanics all about...
  46. S

    Classical Source of Difficult Newtonian Mechanics Problems?

    I'm looking for a source of harder mechanics problems, preferably with solutions. I'm supposed to use the Young and Freedman book, but I find it rather teadious and superficial; most problems don't match the difficulty and depth of the ones on the exam. I had my mid-term tests recently, and got...
  47. Anupama

    Linearity of time evolution in classical mechanics

    I came to know that time evolution in classical mechanics is highly non linear. Is there any case that it become linear?
  48. Wrichik Basu

    B Balancing a bicycle that is moving fast, rather than one that is moving slow

    Why is it easier to balance a bicycle that is moving fast, rather than one that is moving slow?
  49. astrocytosis

    Ball rolls without slipping in an accelerating car

    Homework Statement A bowling ball sits on the smooth floor of a subway car. If the car has a horizontal acceleration a, what is the acceleration of the ball? Assume that the ball rolls without slipping. Homework Equations torque = R x F = Iα aball= Rα Isphere = (2/5) MR2 The Attempt at a...
  50. V

    If a bead is placed on a rod and rotated...

    Homework Statement A bead is placed on a rod with coefficient of friction m, it's accelerated from rest with angular velocity a, find the time it takes to start moving. Neglect gravity Homework Equations Force of friction =Nm The Attempt at a Solution They said there's no gravity, so i don't...
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