Monday 13 November 2017

newtonian mechanics - Internal potential energy and relative distance of the particle


Today, I read a line in Goldstein Classical mechanics and got confused about one line.



To satisfy the strong law of action and reaction, $V_{ij}$ can be a function only of the distance between the particles: $V_{ij} = V_{ij}(|{\bf r}_i-{\bf r}_j|)$.



What confuses me is that I can't see the logic between these two statements. Obviously, I understand strong law of action and reaction and Internal energy. But why the strong law of action and reaction leads to internal energy only depending on relative distances?


I prefer to receive mathematical proof (not thorough, but provide a direction so that I can know where I'm going); yet, intuitive illustration is also welcome.



Answer



The quote is taken from just above eq. (1.32) in Ref. 1:




[...] If the internal forces are also conservative, then the mutual forces between the $i$th and $j$th particles, ${\bf F}_{ij}$ and ${\bf F}_{ji}$, can be obtained from a potential function $V_{ij}$. To satisfy the strong law of action and reaction, $V_{ij}$ can be a function only of the distance between the particles: $$V_{ij} ~=~ V_{ij}(|{\bf r}_i-{\bf r}_j|). \tag{1.32}$$



The structure of internal forces among $N$ point particles can be quite rich in general, see e.g. this Phys.SE post. However the first sentence in the quote makes it clear that Ref. 1 is additionally assuming:




  1. that the internal forces on one particle is a sum of forces from the other particles. Thus it is enough to study the internal force ${\bf F}_{ij}$ from the $i$th particle on the $j$th particle.




  2. that ${\bf F}_{ij}({\bf r}_i,{\bf r}_j)$ only depends on the two positions ${\bf r}_i$ and ${\bf r}_j$ of the $i$th and $j$th particles, respectively.





  3. that ${\bf F}_{ij}({\bf r}_i,{\bf r}_j)$ is a conservative force, meaning that there exists a potential $V_{ij} = V_{ij}({\bf r}_i,{\bf r}_j)$ such that $$ {\bf F}_{ij}~=~-{\bf \nabla}_{j} V_{ij}. $$




  4. that the potential $$V_{ij}({\bf r}_i,{\bf r}_j)~=~V_{ji}({\bf r}_j,{\bf r}_i)$$ is symmetric.




The weak form of Newton's 3rd law then implies that $$ {\bf 0}~=-{\bf F}_{ij}-{\bf F}_{ji}~=~ {\bf \nabla}_{j} V_{ij}+{\bf \nabla}_{i} V_{ji}~=~ ({\bf \nabla}_{i}+ {\bf \nabla}_{j}) V_{ij},$$ which in turn implies that the potential $V_{ij} = V_{ij}({\bf r}_{ij})$ only depends on the difference ${\bf r}_{ij}:={\bf r}_j-{\bf r}_i$ in positions.


Finally the strong form of Newton's 3rd law implies that $${\bf r}_{ij}~\parallel~ {\bf F}_{ij}~=~-{\bf \nabla}_{j} V_{ij}({\bf r}_{ij}),$$ which in turn implies that the potential $V_{ij} = V_{ij}(|{\bf r}_{ij}|)$ only depends on the distance $|{\bf r}_{ij}|$. [The last point follows from the fact that a scalar function $V:\mathbb{R}^3\to \mathbb{R}$, with the property that the gradient $${\bf \nabla} V({\bf r})~\parallel~ {\bf r}$$ is parallel to the position vector ${\bf r}$, can only depend on the length $|{\bf r}|$. Can you see why? Hint: Decompose the gradient in spherical coordinates.]



References:



  1. Herbert Goldstein, Classical Mechanics, Chapter 1.


No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...