- #1
Helmholtz
- 19
- 0
I am following along with Goldstien's Classical Mechanics Book and I am on page 11. The text is breaking down the total potential energy of a system into two parts: the external conservative forces and the internal conservative forces. My question pertains to the internal forces.
Writing the sum of the internal forces as: $$\sum_{i,j} \int_1^2 \vec F_{ji} \cdot d \vec s_i.$$
He then uses the fact that the interacting force is conservative to rewrite in a potential $$\vec F_{ji} = - \vec \nabla_i V_{ji}.$$
Now we have: $$\sum_{i,j} \int_1^2 \vec F_{ji} \cdot d \vec s_i=-\sum_{i,j} \int_1^2 \vec \nabla_i V_{ji} \cdot d \vec s_i.$$
Still a bit shaky on formally why, but I recognize the integral works out to simplify to: $$\sum_{i,j} \int_1^2 \vec F_{ji} \cdot d \vec s_i = -\sum_{i,j} V_{ji} \Big |_1^2.$$
However, there is at least one mistake here, as I should have gotten a factor of 1/2 out front. Which I understand the need for because you are double counting potential. However, I am failing to see where it comes out of from the math. From my understanding you do want to "double count" the forces to get the correct total work, but double counting the potential get you twice the work you should have. Any insight would be appreciated.
Writing the sum of the internal forces as: $$\sum_{i,j} \int_1^2 \vec F_{ji} \cdot d \vec s_i.$$
He then uses the fact that the interacting force is conservative to rewrite in a potential $$\vec F_{ji} = - \vec \nabla_i V_{ji}.$$
Now we have: $$\sum_{i,j} \int_1^2 \vec F_{ji} \cdot d \vec s_i=-\sum_{i,j} \int_1^2 \vec \nabla_i V_{ji} \cdot d \vec s_i.$$
Still a bit shaky on formally why, but I recognize the integral works out to simplify to: $$\sum_{i,j} \int_1^2 \vec F_{ji} \cdot d \vec s_i = -\sum_{i,j} V_{ji} \Big |_1^2.$$
However, there is at least one mistake here, as I should have gotten a factor of 1/2 out front. Which I understand the need for because you are double counting potential. However, I am failing to see where it comes out of from the math. From my understanding you do want to "double count" the forces to get the correct total work, but double counting the potential get you twice the work you should have. Any insight would be appreciated.