What is Classical dynamics: Definition and 48 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.
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...
Hello,
I understand the equation that describes the angular accelerator when 3 reference frames are involved is:
Now I would like to ask what is the resulting equation when accounting more than 3 reference frames, i.e. when n-reference frames are involved.
Thank you.
I am attaching the solution given, but I am not convinced with the approach. I am also asking for some suggestions and hints if I am wrong or is there any other way around.
The thing I do not understand is the arguments from geometry they have made. How they can conclude the extension of spring...
This question is from Collection of Problems in Classical Mechanics by Kotkin & Serbo, here, the answer is given as the following:
However, the graph of ##-Ax^4## looks like:
so shouldn't the trajectory be just ##x(t)=0##?
Consider the above setup. Here, to get the kinetic energy of the body, the moment of inertia with respect to the ##y-##axis has to be calculated. This can be done in two ways:
1. The moment of inertia of the rotation around the center of mass is ##\Theta_s##, then the kinetic energy is...
While deriving the Lagrange equations from d'Alembert's principle, we get from $$\displaystyle\sum_i(m\ddot x_i-F_i)\delta x_i=0\tag{1}$$ to $$\displaystyle\sum_k (\frac {\partial\mathcal L}{\partial\ q_k}-(\frac d {dt}\frac {\partial\mathcal L}{\partial\dot q_k}))\delta q_k=0\tag{2}$$
However...
One of the first things Landau does in his Mechanics book is give an argument as to why the Lagrangian of a free particle must be our conventional kinetic energy. Heuristically, he justifies it, but leaves out the details, perhaps being too obvious. They aren't obvious to me. While in free space...
Assume that, in a binary system, one (and only one) of the two stars has a non-zero quadrupole moment. Then the other star feels the usual gravity force $F_g$ plus an additional force $F_q$ coming from the quadrupole potential. On the other hand, the first star feels only the usual gravity force...
How can I find the maximum bending moment and maximum deflection for a spring?
It would be very helpful if you could explain the specific procedure and formula in an easy-to-understand manner.
that's all, thank you very much.
Goldstein 3 ed, pg 171, under" rate of change of a vector " :
The author derives the relationship between the change of a vector in a stationary and rotating coordinate system.
In the process he uses this assumption :>It is no loss of generality to take the space and body axes as...
If we change the orientation of a coordinate system as shown above, (the standard eluer angles , ##x_1y_1z_1## the initial configuration and ##x_by _b z_b## the final one), then the formula for the coordinates of a vector in the new system is given by
##x'=Ax##
where...
Goldstein 2nd ed.
In its Appendix is given the derivation of Bertrands Theorem.Here ##x=u-u_0## is the deviation from circularity and ##J(u)=-\frac{m}{l^{2}} \frac{d}{d u} V\left(\frac{1}{u}\right)=-\frac{m}{l^{2} u^{2}} f\left(\frac{1}{u}\right)##
If the R.H.S of A-10 was zero, the solution...
We have Rayleigh's dissipation function, defined as
##
\mathcal{F}=\frac{1}{2} \sum_{i}\left(k_{x} v_{i x}^{2}+k_{y} v_{i j}^{2}+k_{z} v_{i z}^{2}\right)
##
Also we have transformation equations to generalized coordinates as
##\begin{aligned} \mathbf{r}_{1} &=\mathbf{r}_{1}\left(q_{1}, q_{2}...
I am struggling to understand shocks in a one dimensional lattice with a linear spring connecting the masses. Say I have a one dimensional lattice with a linear spring constant, k and lattice spacing a. If the particles in the lattice has mass, m then my speed of sound c is a*sqrt(k/m). That is...
Hello, PF members. This is my first post here. I got my undergrad result for final semester few days ago and to my surprise it showed that I have failed in 'classical dynamics'. This is not at all possible as I clearly remember it being easy and quite simple. What bothers me more is that I had...
Homework Statement
This is the problem 8.62(in screenshot) from Morin's textbook of Classical mechanics. I solved it using conservation of momentum in y direction. However in solution manual,he neglects the momentum in y direction by calling stick frictionless. What is this frictionless stick...
Homework Statement
While solving equation of rocket motion with Newton's law in 1-d,I pondered to apply Lagrangian method on this. However, I didn't get correct result. Because I can eliminate last 2nd equation using last equation and get some other equation which is certainly not rockets...
Mentor note: Moved from non-homework forum to here hence no template.
So I was able to solve part 1.A of the first problem by hand, the phase portrait is a sideways parabola. However, I want to also show on this on mathematica. I want to solve the equation first and then plot the phase...
Homework Statement
F=-kx+kx3/α2 where k and α are constants and k > 0. Determine U(x) and discuss the motion. What happens when E=kα2/4?
Homework Equations
F=ma=mv2d/dx
U=-∫Fdx
The Attempt at a Solution
The first part is easy.
U(x) = kx2/2-kx4/4α2
Now I'm looking for what happens when E=kα2/4...
Homework Statement
so I was going over my notes on classical mechanics and just started to review rotation matrices which is the first topic the book starts with. On page 3, I've uploaded the page here
The rotation matrix associated with 1.2a and 1.2b is
\begin{pmatrix}
\cos\theta &...
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...
I don't understand the motion of an accelerating wedge and a block. I'd really appreciate if you make me understand the motion in both an inertial and a non inertial reference frame.
Here's a figure I have made, a0 is the acceleration on the wedge with respect to an inertial frame, towards right.
Today I was doing some reading and I came across this topic. If we have a stationary hydrogen atom with a single electron in orbit around the nucleus and want to calculate the kinetic energy of the electron we would take the following approach.
1) Using Newton's second law:
F = ma ⇒ FE = mac...
This book should introduce me to Lagrangian and Hamiltonian Mechanics and slowly teach me how to do problems. I know about Goldstein's Classical Mechanics, but don't know how do I approach the book.
Homework Statement
I am stuck over a classical mechanics problem. I tried to solve it, but after having solved the first point, I got stuck on the second one. Here is the problem:
>A mechanical structure is composed by 4 rigid thin bars of length $\ell = 8\ m$, mass $m = 5\ kg$ each one. Those...
I'm working on some classical mechanics and just got a question stated:
Is the Hamiltonian for this system conserved? Is it the total energy?
In my problem it was indeed the total energy and it was conserved but it got me thinking, isn't the Hamiltonian always the total energy of a system...
I know how to implement Lagrangian mechanics at a mathematical level and also know that it follows the approach of calculus of variations (i.e. optimisation of functionals, finding their stationary values etc.), however, I'm unsure whether I've grasped the physical intuition behind the...
Homework Statement
I have a 2-DOF system, whereby I have one body that is grounded by a spring (body A), and a second body (body B) attached to the first by a spring and a viscous damper. For body A, I know the velocity and amplitude (before body B is added). I think I also have the stiffness...
Homework Statement
How do I go about finding the most general form of the canonical transformation of the form
Q = f(q) + g(p)
P = c[f(q) + h(p)]
where f,g and h are differential functions and c is a constant not equal to zero. Where (Q,P) and (q,p) represent the generalised cordinates and...
Hey guys I'm a sophomore in college currently taking physics 2(intro E&M), Multivariable calculus, and Differential Equations. I was hoping some of you guys could recommend some good books for intro mechanics and E&M. I'm currently using University Physics by Young in my E&M class, and I used...
Homework Statement
The statement of the question is:A chain of uniform linear mass density ##\rho##, length ##b## and mass ##M## hands as shown in the figure below. At time t=0, the ends A and B are adjacent, but end B is released. Find the tension in the chain at point A after end B has...
From "Greenwood Donald T. - Classical Dynamics", Chapter 1, Section 1-4 (virtual work), Example 1-4:
https://books.google.it/books?id=x7rj83I98yMC&lpg=PP1&hl=it&pg=PA26#v=onepage&q&f=false
1) There are 3 mass points of the same mass m moving on a plane (even if the text doesn't specify this)...
Homework Statement
A mass m1 is attached to a fixed spring on a horizontal surface and attached across a pulley (ignore the pulley mass) to another freely hanging m2. Write the Lagrangian in terms of a single parameter. Find the equation of motion and determine the frequency of oscillation...
This is an image of Classical Dynamics of Particles & Systems, chapter 1
In deriving the equations for the rotation of a coordinate system
I understand the equations 1.2a & 1.2b b, but why is the projection of x2 on the x'1 equal to ab +bc
and why is the vector de equal to the vector Of?
I...
Author: Jorge José and Eugene Saletan
Title: Classical Dynamics: A Contemporary Approach
Amazon Link: https://www.amazon.com/dp/0521636361/?tag=pfamazon01-20
Author: Stephen T. Thornton (Author), Jerry B. Marion (Author)
Title: Classical Dynamics of Particles and Systems
Amazon Link: https://www.amazon.com/dp/0534408966/?tag=pfamazon01-20
Prerequisities: Calculus, Ordinary and Partial Differential Equations, Introductory Physics
Level...
My friend recommend this book to me.
Actually, I don't have enough time to read Goldstein. But this is book is not so thicker as Goldstein's.
May I use this book as a substitution?
Hi, I'm doing this Classical Dynamics section II question which can be found here (http://www.maths.cam.ac.uk/undergrad/pastpapers/2008/Part_2/list_II.pdf ) on page 27.
I have done most of the question but am unsure about the last part. Specifically using Hamilton's equations to show there's...
I'm designing a satellite( flying in formation).
It looks like 3 couples of daughter-satellites(180degree apart) orbiting around the mother satellite in 3 orthogonal planes to measure flyby anomally effect. I want to measure the effect in 3-separate directions. But I'm afraid that the orbiting...
Homework Statement
The speed of a particle of mass m varies with the distance x as v(x) = (alpha)*x-n.
Assume v(x=0) = 0 at t = 0.
(a) Find the force F(x) responsible.
(b) Determine x(t) and
(c) F(t)Homework Equations
Likely:
F = maThe Attempt at a Solution
I obtain
a(x) = -n(alpha)x-(n+1)
So...
Moderation note: In reference to http://farside.ph.utexas.edu/teaching/336k/lectures.pdf
Lagrangian(L) and Hamiltonian(H),
Dear Greg I am studying the L and H.
If kinetic energy(K) and potential(U) are given it seems that L=K-U.
Hamilton defines (p_i, dot q_i being components of momentum...
book is "Classical Dynamics of Particles and Systems"
hello again all.
im just preparing for my first semester at a real university. I transferred from community college.
i will be taking mechanics and the book is "Classical Dynamics of Particles and Systems" by Thornton.
I was wondering if...
Hello,
Does anyone have marion and thornton's classical dynamics book? I have a possible error that I wanted to point out. Actually, goldstein's classical mechanics book would work also since I found the same "error" in there as well.
A complete set of lecture notes for an upper-division classical dynamics course. The course concentrates on those aspects of classical dynamics which can be studied analytically. Topics covered include oscillations, Keplerian orbits, two-body scattering, rotating frames of reference, rotation of...
Hi all
I am wondering about recent developments of classical dynamics. It seems most physicists are now devoted to quantum world. Is there any effort to broaden the scopes of classical world?
Thanks
[SOLVED] Analytical Classical Dynamics: Newton's Laws
Homework Statement
Consider a system of N mutually interacting point objects. Let the ith object have mass mi and position vector ri. Suppose that the jth object exerts a central force fij on the ith. In addition, let the ith object be...
The question is as follows: The height of a hill (meters) is given by [z=(2xy)-(3x^2)-(4y^2)-(18x)+(28y)+12], where x is the distance east, y is the distance north of the origin. a). where is the top of the hil (x,y,z) and how high is it (z=?)? b). How steep is the hill at x=y=1, that is...