## 物理代写|分析力学代写Analytical Mechanics代考|PHY225

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## 物理代写|分析力学代写Analytical Mechanics代考|Virtual Work

The importance of introducing the notion of virtual displacement stems from the following observation: if the surface to which the particle is confined is ideally smooth, the contact force of the surface on the particle, which is the constraint force, has no tangential component and therefore is normal to the surface. Thus, the work done by the constraint force as the particle undergoes a virtual displacement is zero even if the surface is in mótion, differently from thé work donne during a reeal displaccement, which does nôt necessarily vanish. In most physically interesting cases, the total virtual work of the constraint forces is zero, as the next examples attest.

Example 1.10 Two particles joined by a rigid rod move in space. Let $\mathbf{f}_1$ and $\mathbf{f}_2$ be the constraint forces on the particles. By Newton’s third law $\mathbf{f}_1=-\mathbf{f}_2$ with both $\mathbf{f}_1$ and $\mathbf{f}_2$ parallel to the line connecting the particles. The virtual work done by the constraint forces is
$$\delta W_v=\mathbf{f}_1 \cdot \delta \mathbf{r}_1+\mathbf{f}_2 \cdot \delta \mathbf{r}_2=\mathbf{f}_2 \cdot\left(\delta \mathbf{r}_2-\delta \mathbf{r}_1\right) .$$
Setting $\mathbf{r}=\mathbf{r}_2-\mathbf{r}_1$, the constraint equation takes the form (1.38), namely $r^2-l^2=0$. In terms of the variable $\mathbf{r}$, the situation is equivalent to the one discussed in Example 1.9. Taking $f(\mathbf{r}, t)=r^2-l^2$, Eq. (1.51) reduces to $\mathbf{r} \cdot \delta \mathbf{r}=0$. Since $\mathbf{f}_2$ and $\mathbf{r}$ are collinear, there exists a scalar $\lambda$ such that $\mathbf{f}_2=\lambda \mathbf{r}$, hence $\delta W_v=\lambda \mathbf{r} \cdot \delta \mathbf{r}=0$. Inasmuch as a rigid body consists of a vast number of particles whose mutual distances are invariable, one concludes that the total virtual work done by the forces responsible for the body’s rigidity is zero.

Example 1.11 A rigid body rolls without slipping on a fixed surface. As a rule, in order to prevent slipping, a friction force between the fixed surface and the surface of the body is needed, that is, the surfaces in contact must be rough. Upon rolling without slipping, the body’s particles at each instant are rotating about an axis that contains the body’s point of contact with the surface. Thus, the friction force acts on a point of the body whose velocity at each instant is zero, because it is on the instantaneous axis of rotation. Virtual displacements are such that the body does not slip on the surface, that is, $\delta \mathbf{r}=0$ at the point of contact between the body and the fixed surface. Therefore, the virtual work done by the constraint force is $\delta W_v=\mathbf{f} \cdot \delta \mathbf{r}=0$ because $\delta \mathbf{r}=0$, even though $\mathbf{f} \neq 0$.

## 物理代写|分析力学代写Analytical Mechanics代考|Principle of Virtual Work

Newton’s formulation of mechanics is characterised by the set of differential equations
$$m_i \ddot{\mathbf{r}}_i=\mathbf{F}_i, \quad i=1, \ldots, N,$$
where $\mathbf{F}_i$ is the total or resultant force on the $i$ th particle, supposedly a known function of positions, velocities and time. This system of differential equations determines a unique solution for the $\mathbf{r}_i(t)$ once the positions and velocities are specified at an initial instant. ${ }^{10}$
In the presence of constraints, it is patently clear how inconvenient the Newtonian formulation is. First of all, it usually requires the use of more coordinates than are necessary to specify the configuration of the system. When the constraints are holonomic, for instance, the positions $\mathbf{r}_1, \ldots, \mathbf{r}_N$ are not mutually independent, making the Newtonian approach uneconomical by demanding the employment of redundant variables. Furthermore, the total force on the $i$ th particle can be decomposed as
$$\mathbf{F}_i=\mathbf{F}_i^{(a)}+\mathbf{f}_i,$$
where $\mathbf{F}_i^{(a)}$ is the applied force and $\mathbf{f}_i$ is the constraint force. In the case of the double pendulum in Example 1.4, $\mathbf{F}_1^{(a)}$ and $\mathbf{F}_2^{(a)}$ are the weights of the particles, whereas $\mathbf{f}_1$ and $\mathbf{f}_2$ are determined by the tensions on the rods or strings. The difficulty here lies in that one does not a priori know how the constraint forces depend on the positions and velocities. What one knows, in fact, are the effects produced by the constraint forces. One may also argue that the applied forces are the true causes of the motion, the constraint forces merely serving to ensure the preservation of the geometric or kinematic restrictions in the course of time. No less important is the fact that Newton’s laws – the second law together with the strong version of the third law – turn out to be incapable of correctly describing the motion of certain constrained systems (Stadler, 1982; Casey, 2014).

For all these reasons, it is highly desirable to obtain a formulation of classical mechanics as parsimonious as possible, namely involving only the applied forces and employing only independent coordinates. We shall soon see that this goal is achieved by the Lagrangian formalism when all constraints are holonomic. As an intermediate step towards Lagrange’s formulation, we shall discuss d’Alembert’s principle, which is a method of writing down the equations of motion in terms of the applied forces alone, the derivation of which explores the fact that the virtual work of the constraint forces is zero.

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