# Mechanics Goldstein doubt

• pardesi
In summary: . not at equilibrium the principle of virtual work still holds meaning there must be some force on each particle acting on it

#### pardesi

in the "derivation" of 'principle of vitual work' Goldstein started with this
he wrote the force F acting on each particle as the sum of $$F_{a}$$ (Force applied) and $$F_{i}$$(Constraint force)
ok i have some three queries
1)What are constraint forces?
2)Are all internal forces constraint forces
3)He said the cond. of equilibriia of a system is equivalent to
$$\sum \vec{F_{a,i}} .d \vec{r_{i}} =0$$
that is the virtual work is $$0$$ i don't get how that happens(because virtual work being 0 doesn't imply force on each particle of the system is $$0$$ or is it just by definition?

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The constraint forces are the forces which restrict the particle to move in some way. One example is the normal force, when two surfaces touch. For a rigid body, if you look at the forces on a single constituent particle, then the internal forces are indeed constraint forces.

What Goldstein says is that, if $F_i$ is the total force on a particle, and for a virtual displacement $\delta r_i$, the product $F_i . \delta r_i$ is 0 since each particle is at equilibrium, and so the sum over all particles $\sum_i \bf{F}_i . \delta \bf{r}_i=0$.

If you write $F_i$ as the sum of an applied force $F_i^a$ and a constraint force $f_i$, and only consider systems where $\sum_i f_i.\delta r_i$ is assumed to be 0, the condition for equilibrium is now, $\sum F_i^a . \delta r_i=0$

This assumption holds for many cases. For example, in rigid bodies where the internal forces are constraint forces, then because of Newton's third law, the work due to the constraint forces is 0. Similarly, when the normal force from a surface is the constraint force, the work done is again 0, because the constraint force will always be perpendicular to the virtual displacement.

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i was clear about the rigid body part but does the notion of constarint force hold good anywhere else?
also about the principle of virtual work
it's clear that if the body is in equilibrium with constarint foprces doing no work then we have
$$\sum \vec{F_{i,a}}.\vec{dr_{i}}=0$$ but $$\sum \vec{F_{i,a}}.\vec{dr_{i}}=0$$ does not imply that the body has to be at equilibrium

pardesi said:
i was clear about the rigid body part but does the notion of constarint force hold good anywhere else?
also about the principle of virtual work
it's clear that if the body is in equilibrium with constarint foprces doing no work then we have
$$\sum \vec{F_{i,a}}.\vec{dr_{i}}=0$$ but $$\sum \vec{F_{i,a}}.\vec{dr_{i}}=0$$ does not imply that the body has to be at equilibrium

For systems with ideal constraint forces, where $\sum f_i.\delta r_i$ is assumed to be 0, then if $\sum_i \bf{F}_i^a . \delta \bf{r}_i=0$ for any arbitrary displacement $\delta r_i$, it implies the system is in equilibrium. Just reverse the sequence of arguments.

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oh ok i think i got the point ...the thing is for any disp

## 1. What is the Goldstein doubt in mechanics?

The Goldstein doubt in mechanics is a question raised by physicist Herbert Goldstein in his book Classical Mechanics. It concerns the validity of the Lagrangian and Hamiltonian formalism in classical mechanics, and whether they are truly fundamental or merely mathematical constructs.

## 2. What is the difference between Lagrangian and Hamiltonian mechanics?

Lagrangian mechanics is a formulation of classical mechanics that uses generalized coordinates and the principle of least action to derive the equations of motion. Hamiltonian mechanics is a related formulation that uses canonical coordinates and the Hamiltonian function to derive the equations of motion. Both approaches are mathematically equivalent, but differ in their choice of variables and how they are derived.

## 3. How does the Goldstein doubt impact classical mechanics?

The Goldstein doubt raises questions about the true nature of classical mechanics and whether the Lagrangian and Hamiltonian formalism are fundamental or merely convenient mathematical tools. It has led to further research and debate about the foundations of classical mechanics and the relationship between mathematics and physical reality.

## 4. What is the significance of the Goldstein doubt in modern physics?

The Goldstein doubt has sparked discussions and investigations into the foundations of classical mechanics, and has also influenced the development of quantum mechanics. It highlights the importance of understanding the philosophical and conceptual underpinnings of physics, and how they can impact our understanding of the physical world.

## 5. Is there a consensus on the validity of Lagrangian and Hamiltonian mechanics?

While there is no definitive answer to the Goldstein doubt, the majority of physicists and mathematicians view the Lagrangian and Hamiltonian formalism as valid and useful approaches to classical mechanics. However, there are still ongoing debates and research on the fundamental nature of these formalisms and their relationship to physical reality.