- #1
SheikYerbouti
- 19
- 0
I understand that accepting Hamilton's principle will yield identical results as accepting Newton's laws. However, simply accepting that the integral of the difference between kinetic and potential energies is an extrema seems not intuitively obvious. The textbook that I used for my classical mechanics class (Fowles) states that Lagrange developed his mechanics through the use of the principle of virtual work. Elsewhere, I have read that the acceptance of the principle of virtual work is equivalent to the acceptance of Newton's first law. Since the whole concept of virtual displacements is somewhat counter-intuitive (at least initially), I am trying to understand its derivation from the Newton's first law. From the definition of virtual work we have:
[tex] \delta W = \sum_{i=1}^m \vec{F_i} \centerdot \delta \vec{r_i} [/tex]
Where [itex]\vec{r_i}[/itex] is a function of n generalized coordinates and time. Since [itex] \delta t = 0[/itex], it follows from the properties of virtual displacement that [tex]\delta \vec{r_i} = \sum_{j=1}^n \frac{\partial \vec{r_i}}{\partial q_j} \delta q_j [/tex]
After substituting this into the expression for virtual work and doing some rearrangement we find that
[tex] \delta W = \sum_{j=1}^n (\sum_{i=1}^m \vec{F_i} \centerdot \frac{\partial \vec{r_i}}{\partial q_j}) \delta q_j [/tex]
We define the term in parentheses to be the generalized force [itex] Q_j [/itex]. The principle states that the virtual work is zero for a static system only when the generalized forces are all zero, and this readily clear here. However, I do not see how this follows from Newton's first law, which was only initially applied the forces [itex] \vec{F_i} [/itex]. Since the virtual displacements in the first sum are those caused only by the corresponding force and are completely arbitrary, I don't see how this relationship between the forces must carry on to the generalized forces, which are essentially a sum of dot products of arbitrary vectors. I feel like this is where my mistake is; the virtual differential operator is not present in the partial derivative of the position with respect to a generalized coordinate. Does this mean that [itex]\frac{\partial \vec{r_1}}{\partial q_j} = ... = \frac{\partial \vec{r_m}}{\partial q_j}? [/itex] Sorry for the lengthy post, I would greatly appreciate some clarification on this topic and/ or how to derive the principle of virtual work and Hamilton's variational principle from Newton's laws. (Simply showing that they yield equivalent results does not give me the deep, theoretical understanding that I would like to have.)
[tex] \delta W = \sum_{i=1}^m \vec{F_i} \centerdot \delta \vec{r_i} [/tex]
Where [itex]\vec{r_i}[/itex] is a function of n generalized coordinates and time. Since [itex] \delta t = 0[/itex], it follows from the properties of virtual displacement that [tex]\delta \vec{r_i} = \sum_{j=1}^n \frac{\partial \vec{r_i}}{\partial q_j} \delta q_j [/tex]
After substituting this into the expression for virtual work and doing some rearrangement we find that
[tex] \delta W = \sum_{j=1}^n (\sum_{i=1}^m \vec{F_i} \centerdot \frac{\partial \vec{r_i}}{\partial q_j}) \delta q_j [/tex]
We define the term in parentheses to be the generalized force [itex] Q_j [/itex]. The principle states that the virtual work is zero for a static system only when the generalized forces are all zero, and this readily clear here. However, I do not see how this follows from Newton's first law, which was only initially applied the forces [itex] \vec{F_i} [/itex]. Since the virtual displacements in the first sum are those caused only by the corresponding force and are completely arbitrary, I don't see how this relationship between the forces must carry on to the generalized forces, which are essentially a sum of dot products of arbitrary vectors. I feel like this is where my mistake is; the virtual differential operator is not present in the partial derivative of the position with respect to a generalized coordinate. Does this mean that [itex]\frac{\partial \vec{r_1}}{\partial q_j} = ... = \frac{\partial \vec{r_m}}{\partial q_j}? [/itex] Sorry for the lengthy post, I would greatly appreciate some clarification on this topic and/ or how to derive the principle of virtual work and Hamilton's variational principle from Newton's laws. (Simply showing that they yield equivalent results does not give me the deep, theoretical understanding that I would like to have.)