Explain the proof in goldstein

In summary, when analyzing the motion of a small bead attached to a wire rotating along a fixed axis, Lagrange's equation can be used with the generalized coordinate being the distance of the particle along the wire. In this case, the generalized force Q is zero due to the constraint of moving along the wire and the absence of non-conservative forces.
  • #1
pardesi
339
0
Question:
Analyze the motion of a small bead attached to a wire which is rotating along a fixed axis?

Proof(Using Lagrangian formulation):
Clearly here the generalized coordinate is the distance of the particle along the wire.
so we have the formulae
[tex]\frac{d \frac{\delta T}{\delta r}}{dt} - \frac{\delta T}{\delta r}=Q[/tex]
where [tex]Q[/tex] is the generalized force acting on the object ...
goldstein claims that is 0 here i don't get that how?
 
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  • #2
First, correcting the typos, Lagrange's equation is

[tex]\frac{d}{dt} \frac{\partial L}{\partial \dot{r}} - \frac{\partial L}{\partial r}=Q[/tex]

There is no potential so L=T in this example as you have stated. In the method described on p. 26, the generalized force Q is zero because the constraint of moving along the wire is built into the generalized coordinates instead, and there are no non-conservative forces (i.e., friction).
 
  • #3


The proof in Goldstein's analysis of the motion of a small bead attached to a rotating wire is based on the Lagrangian formulation. This approach uses the concept of generalized coordinates, which in this case is the distance of the particle along the wire. The equation used in this proof is the Lagrange's equation, which states that the time derivative of the partial derivative of the kinetic energy with respect to the generalized coordinate minus the partial derivative of the kinetic energy with respect to the generalized coordinate equals the generalized force acting on the object.

Goldstein claims that the generalized force in this case is equal to 0. This can be understood by considering the system as a whole, where the wire and the rotating axis are also part of the system. In this system, there is no external force acting on the bead, as the wire and the axis are providing the necessary centripetal force for the bead to rotate along with them. Therefore, the generalized force acting on the bead is indeed 0.

This proof in Goldstein's analysis shows that the motion of the bead is completely determined by the geometry and dynamics of the rotating wire and axis, and there is no need to consider any external forces. This is a fundamental concept in classical mechanics and highlights the power of the Lagrangian formulation in analyzing complex systems.
 

What is the proof in Goldstein?

The proof in Goldstein refers to the mathematical proof of the variational principle in classical mechanics, which was first presented by Austrian physicist and philosopher, Ernst Mach, in his book "The Science of Mechanics". This proof is also known as the Goldstein's theorem or the principle of least action.

Why is the proof in Goldstein important?

The proof in Goldstein is important because it provides a fundamental principle in classical mechanics that explains the motion of particles and systems. It is a powerful tool for solving complex problems and has been used extensively in various fields such as physics, engineering, and economics.

What is the variational principle?

The variational principle states that the actual path followed by a system between two points in time is the one that minimizes the action, which is the integral of the Lagrangian over time. In simpler terms, it means that the actual path of a system is the one that takes the least amount of time to travel between two points.

How does the proof in Goldstein relate to classical mechanics?

The proof in Goldstein is closely related to classical mechanics as it provides a mathematical basis for the fundamental laws of motion, such as Newton's laws and the principle of conservation of energy. It also helps to explain the behavior of systems in terms of the least action principle.

Are there any limitations to the proof in Goldstein?

Yes, there are some limitations to the proof in Goldstein. It assumes that the system under consideration is conservative, meaning that there are no external forces acting on it. It also assumes that the system can be described using a finite number of generalized coordinates. These limitations make it unsuitable for describing certain types of systems, such as those with non-conservative forces or those with an infinite number of degrees of freedom.

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