5
votes
1answer
139 views

Why are D'Alembert's Principle and the Principle of Least Action Related?

Why do we get the same differential equations from both principles? Surely there is a fundamental connection between them? When written out, the two seem to have nothing in common. $$\sum _i ( ...
0
votes
0answers
47 views

Question about an example of non-integrable constraints

The example is a thin disk rotating of an inclined plane. The disk can roll not only down the plane, but also "sideways". Let $(x,y)$ be the position of the CM, where the $y$ axis is down the slope ...
1
vote
0answers
71 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 ...
0
votes
0answers
50 views

Classical rod-wall-floor system

I have a homogeneous thin rod $AB$. $A$ can slide along $z$-axis, $B$ along $xy$ plane. There is no friction. We have any initial conditions. Now the question is to write down the equation of motion ...
8
votes
4answers
507 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
3answers
731 views

Frequency of small oscillation of particle under gravity constrained to move in curve $y=ax^4$

How to find the frequency of small oscillation of a particle under gravity that moves along curve $y = a x^4$ where $y$ is vertical height and $(a>0)$ is constant? I tried comparing $V(x) = \frac ...
3
votes
1answer
268 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 ...
1
vote
1answer
200 views

Constrained motion in a parabolic tube [closed]

A smooth parabolic tube is placed with vertex downwards in a vertical plane. A particle slides down the tube from rest under the influence of gravity. Prove that in any position, the reaction of ...
6
votes
1answer
746 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
235 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 - ...
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 ...