- #1

l4teLearner

- 7

- 3

- Homework Statement
- I would like to derive the potential energy for ##l## equal point particles ##P_1, P_2, . . . , P_l## ##(l > 2)## on a circle of radius ##R## and centre ##O##. The costraint is smooth and the particle ##P_i## is attracted to ##P_{i−1}, P_{i+1}## with an elastic force (##P_0= P_l##).

- Relevant Equations
- Assuming the elastic potential energy between each couple of adjacent particles as ##1/2kd^2## where ##d## is the distance between the particles.

I use ##l-1## lagrangian coordinates ##\alpha_1,...,\alpha_{l-1}## . ##\alpha_i## is the angle between ##OP_{i-1}## and ##OP_{i}##.

As the length of a chord between two rays with angle ##\alpha## is ##d=2Rsin(\alpha/2)##, I write the potential energy of the system as

$$V=2kR^2\{\sum_{1}^{l-1}sin^2(\frac{\alpha_i}{2})+sin^2(\frac{2\pi-\sum_{1}^{l-1}\alpha_i}{2})\}$$

I am not fully convinced of this solution because when I evaluate the equilibrium position of 3 particles (placed at the vertices of an equilateral triangle) and 4 particles (forming a square) I find they are not stable ( hessian of V is not positive definite around the equilibrium position).

Any hint?

thanks

As the length of a chord between two rays with angle ##\alpha## is ##d=2Rsin(\alpha/2)##, I write the potential energy of the system as

$$V=2kR^2\{\sum_{1}^{l-1}sin^2(\frac{\alpha_i}{2})+sin^2(\frac{2\pi-\sum_{1}^{l-1}\alpha_i}{2})\}$$

I am not fully convinced of this solution because when I evaluate the equilibrium position of 3 particles (placed at the vertices of an equilateral triangle) and 4 particles (forming a square) I find they are not stable ( hessian of V is not positive definite around the equilibrium position).

Any hint?

thanks