1
vote
0answers
55 views

Restrained double pendulum

The equations of motion of a double pendulum are well-known. Usually you'd have the them expressed in the rotations $\theta_1(t)$ and $\theta_2(t)$. There are two degrees of freedom. Now consider the ...
3
votes
2answers
41 views

Internal potential energy and relative distance of the particle

Today, I read a line in Goldstein Classical mechanics and got confused about one line. To satisfy the strong law of action and reaction, $V_{ij}$ can be a function only of the distance between ...
0
votes
2answers
87 views

Derivation of Lagrangian?

I know that the Lagrangian $L$ is defined to be $T-V$, i.e. the difference between kinetic energy and potential energy. Also the Action $S$ is defined to be $\int Ldx$ and from this we can derive ...
0
votes
1answer
56 views

Lagrangian for a moving spring device

How can I write the proper Lagrangian for such as system as the one shown in picture? Am confused about what is the suitable way to designate the coordinate.
0
votes
1answer
64 views

Atwood machine with spring

I'm just beginning to learn about Lagrangian mechanics, and I am asked to find the kinetic energy of this Atwood machine (See figure). I am told, that the kinetic energy should be: ...
2
votes
1answer
92 views

Lagrangian approach to spinning thread reel

I am trying to better understand Lagrangian dynamics and am struggling to complete the following question: A reel of thread of mass $m$ and radius $r$ is allowed to unwind under gravity, the upper ...
5
votes
1answer
174 views

Why isn't $F = \frac{\partial \mathcal{L}}{\partial q}$?

If momentum is, $$p = \frac{\partial \mathcal{L}}{\partial \dot{q}}$$ and force is, $$ F = \frac{dp}{dt}$$ and by Euler-Langrange equations, $$ \frac{d}{dt}\frac{\partial \mathcal{L}}{\partial ...
0
votes
1answer
81 views

Block on cart, equation of motion

Consider a rigid block of $b \times h$ having mass $m$ on cart (as depicted below). The cart is given an acceleration $a$, this leads to overturning of the block. The angle of rotation is indicated by ...
0
votes
0answers
37 views

Spring Force in a Dynamic Equation

I am working on dynamic simulation of a bipedal robot, I have dynamic equations of a preliminary structure. But Now I have to add some springs to the dynamics. I am having a problem of how to account ...
0
votes
1answer
109 views

Can't understand the principle of least action [closed]

I tried many hours to understand the principle of least action, and those hours become days... and I still didn't understand that principle/ and how it relates to Newtonian mechanics? Could someone ...
1
vote
1answer
68 views

Why does a particle fall in a straight line?

In Lagrangian Mechanics we choose the path of least action. Given a uniform gravitational field, and a particle of finite mass; and fixing two points the start & end-point we consider all paths ...
2
votes
1answer
302 views

Equations of motion for a pendulum in 3D?

I am trying to solve for the equations of motion to simulate a pendulum. I decided to use the spherical coordinates. The Lagrange equation is: where L = length of the rope ϕ= angle of the ...
1
vote
5answers
211 views

Motivation for form of Lagrangian

This question (in lagrangian mechanics) might be silly, but why is the Lagrangian L defined as: $L = T - V$? I understand that the total mechanical energy of an isolated system is conserved, and that ...
5
votes
2answers
315 views

How can you solve this “paradox”? Central potential

A mass of point performs an effectively 1-dimensional motion in the radial coordinate. If we use the conservation of angular momentum, the centrifugal potential should be added to the original one. ...
2
votes
1answer
446 views

Deriving D'Alembert's Principle

The wiki article states that D'Alembert's Principle cannot derived from Newton's Laws alone and must stated as a postulate. Can someone explain why this is? It seems to me a rather obvious principle.
2
votes
0answers
40 views

Acceleration of 2 bodies tied with a string [closed]

Find the acceleration of the block of mass M shown in the figure . The co-efficient of friction between the 2 blocks is μ1 and that between the bigger block and ground is μ2. Could someone help ...
8
votes
4answers
390 views

D'Alembert's Principle: Necesssity of virtual displacements

Why is the D'Alembert's Principle $$\sum_{i} ( {F}_{i} - m_i \bf{a}_i )\cdot \delta \bf r_i = 0$$ stated in terms of "virtual" displacements instead of actual displacements? Why is it so necessary ...
1
vote
1answer
93 views

Landau Lifshitz energy for uniform rotation

Landau Lifshitz claim in their Mechanics book (39.11) that for a uniform rotation we have $ E = \frac{mv^2}{2} - \frac{m}{2} (\omega \times r)^2 + U,$ where the rotation is given by $v' = v + \omega ...
2
votes
2answers
971 views

Expression of kinetic energy in polar coordinates

Expression for kinetic energy in Cartesian coordinate: Expression for kinetic energy in polar coordinate (applying the transformation of coordinates): Why can't we express it in the following ...
3
votes
2answers
478 views

Small oscillations of the double pendulum

From the Lagrangian I've got the following equations of motion for the double pendulum in 2D. (The masses are different but the lengths of the two pendula are equal.) Let $m_2$ be the lowest-hanging ...
1
vote
0answers
871 views

Equations of motion for a pendulum and spring system

The question is available here: I've modeled the building as a rod on a torsional spring (with a pendulum hanging from the top). $\phi$ is the angle from the centre for the pendulum and $\theta$ ...
3
votes
1answer
239 views

D'Alembert's principle

Actually I have some troubles to understand what this principle is all about, so I want to use the simple pendulum in order to get the idea. Since I have read a few passages that dealt with this ...
2
votes
1answer
169 views

Definition of Kinetic energy

In class we had that $ T= \frac{1}{2}T_{ij}v_iv_j$ where we used the Einstein summation convention. Hitherto we only discussed examples where the kinetic energy was dependent of the square of one ...
3
votes
2answers
160 views

Bertrand's theorem

I found in Goldstein's Classical Mechanics that the condition for closed orbits is given by $\frac{d^2 V_{eff}}{dr^2}>0$.(bertrand's theorem). Can somebody explain to me, how this inequality is ...
-1
votes
1answer
151 views

Solving differential Equation for the Two-Body Problem

So, I'm following the derivation in D. Morin, Introduction to Classical Mechanics, of the equations for a two-body system. I understand all of it, aside from this one step. When he's talking about ...
0
votes
0answers
81 views

Stability of trajectory of disc which moves along a straight curve

Let's have a disc which moves along a straight curve on a plane in a uniform gravitational field. There need to discover the stability of it's trajectory. I represented the possible deviation of the ...
9
votes
3answers
408 views

No closed orbits for a Newtonian gravitational field in 4 spatial dimensions

We are supposed to show that orbits in 4D are not closed. Therefore I derived a Lagrangian in hyperspherical coordinates $$L=\frac{m}{2}(\dot{r}^2+\sin^2(\gamma)(\sin^2(\theta)r^2 \dot{\phi}^2+r^2 ...
0
votes
2answers
224 views

A small oscillations of a rod on the cylinder

Let's have the next case. A rod (with mass $m$, length $L$ and a momentum of inertia $I$) at the initial time is located on a cylinder (with radius $R$) surface so that it's (rod's) center of mass ...
4
votes
3answers
206 views

Virtual differentials approach to Euler-Lagrange equation - necessary?

I'm currently teaching myself intermediate mechanics & am really struggling with the d'Alembert-based virtual differentials derivation for the Euler-Lagrange equation. The whole notion of, and ...
0
votes
2answers
319 views

Lagrange-Euler equations for a bead moving on a ring

A bead with mass $m$ is free to glide on a ring that rotates about an axis with constant angular velocity. Form the Lagrange-Euler equations for the movement of the bead. Solution: Let us ...
2
votes
1answer
258 views

Euler-Lagrange Equation

A particle moving towards the origin has initial conditions $x(t=0) = 1$ and $\dot{x}(t=0)=0$ If the Lagrangian is L:=$\frac{m}{2}\dot{x}^2 -\frac{m}{2}ln|x|$ This should satisfy Euler Lagrange ...
1
vote
1answer
251 views

What's the motivation behind the action principle? [closed]

What's the motivation behind the action principle? Why does the action principle lead to Newtonian law? If Newton's law of motion is more fundamental so why doesn't one derive Lagrangians and ...
4
votes
3answers
2k views

What exactly are Hamiltonian Mechanics (and Lagrangian mechanics)

What exactly are Hamiltonian Mechanics (and Lagrangian mechanics)? I want to self-study QM, and I've heard from most people that Hamiltonian mechanics is a prereq. So I wikipedia'd it and the entry ...
0
votes
0answers
139 views

Describing the movement of the object in a particular situation in Lagrangian way

Suppose there is a object M, (sliding motion) moving by the initial speed $v$ and the initial location $x_0$. Otherwise noted, friction is assumed to be nonexistent. It then meets a circular mold ...
1
vote
1answer
295 views

Questions regarding solving the Brachistochrone problem using Lagrangian

brachistochrone problem: Suppose that there is a rollercoaster. There is point 1 ($0,0$) and point 2 ($x_2, y_2)$. Point 1 is at the higher place when compared to the point 2, so the rollercoaster ...
6
votes
2answers
286 views

What are the reasons for leaving the dissipative energy term out of the Hamiltonian when writing the Lyapunov function?

I have a problem with one of my study questions for an oral exam: The Hamiltonian of a nonlinear mechanical system, i.e. the sum of the kinetic and potential energies, is often used as a Lyapunov ...
4
votes
1answer
403 views

Lagrangian dynamics with initial conditions: motion of free particle

I am very new to Lagrangian dynamics so I am trying to get my head around the practical usage. So far on here all I could find were proofs and they did not make much sense to me, especially when time ...
5
votes
1answer
152 views

Elementary derivation of the motion equations for an inverted pendulum on a cart

Consider a cart of mass $M$ constrained to move on the horizontal axis. A massless rod is attached to the midpoint of the cart, having a mass $m$ on its endpoint. See wikipedia for a picture and for a ...
2
votes
1answer
234 views

Do Lagrangian points actually maintain a fixed distance?

I was reading on up Lagrangian points and the restricted three-body problem. From what I was able to tell, the Lagrangian points are 5 points in a two-body system such that a third body would be ...
1
vote
1answer
438 views

Relation between between linear momentum and translational kinetic energy

The momentum $m v$ of a particle is formally the same as the derivative its translational kinetic energy $\frac{1}{2} m v^2$ with respect to $v$. Similarly the angular momentum $I \omega$ is the ...
5
votes
2answers
431 views

Is it circular reasoning to derive Newton's laws from action minimization?

Usually, a typical example of the use of the action principle that I've read a lot is the derivation of Newton's equation (generalized to coordinate $q(t)$). However, in the classical mechanics ...
4
votes
1answer
469 views

A Question about Virtual Work related to Newton's Third Law

In describing D'Alembert's principle, the lecture note I was provided with states that the total force $\mathbb F_l$ acting on a particle can be taken as, $$\mathbb F_l=F_l+\sum_mf_{ml}+C_l,$$ ...
6
votes
1answer
651 views

When is the principle of virtual work valid?

The principle of virtual work says that forces of constraint don't do net work under virtual displacements that are consistent with constraints. Goldstein says something I don't understand. He says ...
3
votes
1answer
215 views

Showing constraint is nonholonomic

One example of a nonholonomic constraint is a disk rolling around in the cartesian plane that is constrained to not be slipping. These leads to the constraint $dx - a \sin\theta d\phi = 0$ and $dy - ...
1
vote
1answer
156 views

Is the number of independent constants of a system equal to the number of degree of freedom of it?

Maybe the question is not very clear myself since I am not a physics major.But can you help me make this question clearer and then give me some comments on it? I got that this holds in gravitional ...
11
votes
3answers
2k views

Galilean invariance of Lagrangian for non-relativistic free point particle?

In QFT, the Lagrangian density is explicitly constructed to be Lorentz-invariant from the beginning. However the Lagrangian $$L = \frac{1}{2} mv^2$$ for a non-relativistic free point particle is ...
17
votes
7answers
9k views

What is the difference between Newtonian and Lagrangian mechanics in a nutshell?

What is Lagrangian mechanics, and what's the difference compared to Newtonian mechanics? I'm a mathematician/computer scientist, not a physicist, so I'm kind of looking for something like the ...
11
votes
2answers
2k views

Are there examples in classical mechanics where D'Alembert's principle fails?

D'Alembert's principle suggests that the work done by the internal forces for a virtual displacement of a mechanical system in harmony with the constraints is zero. This is obviously true for the ...
2
votes
3answers
347 views

What is the definition of momentum when a mass distribution $\rho(r,t)$ is given?

This question is Edited after recieving comments. What is the definition of momentum when a mass distribution $\rho(r,t)$ is given? Assuming a particle as a point mass we know the definition of ...