I'm reading up on classical mechanics. In my book there is an example of a simple classical polymer model, which consists of N point particles that are connected by nearest neighbor harmonic interactions. The Hamiltonian of this system is:

$$H = \sum\limits_{i=1}^N \frac{\vec{p}_i^2}{2m} + \frac{1}{2}\sum\limits_{i=1}^{N-1} m \omega^2(|\vec{r}_i - \vec{r}_{i+1}| - b_i)^2, $$

where $p$ stands for momentum, $r$ for position and $m$ for the mass of the respective particle in the chain. Now comes the part that I don't understand: $b_i$ is the equilibrium bond length. I don't understand why one subtracts the bond length from the distance between two neighboring particles in the chain in the above formula. What is the physical meaning of doing this?

I have included the picture below for visualization of the problem. In my case, $k_1 = k_2 = k_3 = k_4 = k$ and also all masses are equal to m. In my understanding, the $b_i$ (equilibrium bond length) would be the black spring (spiral line between two circles) in the picture below.

polymer chain model


Each point of mass has a "default" position where the springs are relaxed, and they don't exert force on the masses. That's when the springs have the length $b$, thus, the distance between $r_i$ and $r_{i+1}$ is $b$. This in turn means


The right term is then zero, because there is no potential energy in the system.

It's easier to understand if you define $d_i=|r_i-r_{i+1}|$, then the above formula reads $(d_i-b_i)^2$, which means that the potential energy is zero if the actual spring length is equal to the equilibrium spring length.

Furthermore, I would use $r_{i+1}-r_i$ instead of $r_i-r_{i+1}$, because you can get rid of the absolute value $||$ if you define $r_{i+1}>r_i$ (which is the standard case).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.