Recent content by l4teLearner

I think we can prove that ##M## is positive definite because it corresponds to the following quadratic form: $$\sum_{1}^{l}x_i^2+\sum_{1\leq i<j \leq l}x_i x_j $$ It can be proved by contradiction that ##\sum_{1}^{l}x_i^2+\sum_{1\leq i<j \leq l}x_i x_j > 0## if some ##x_i \neq 0##. In fact...

So my current status is the following. I am evaluating the Hessian ##H## of ##V## to look at the eigenvalues for the equilibrium position ##\alpha=\alpha_i=\frac{2\pi}{l}## for each ##i##. $$\frac{\partial^2 V}{\partial^2 \alpha_i}=2kR^2\{...

Thanks a ton for your help. Do I have to mark this as solved, or close it, or something? thanks

Thank you! I think you plotted my formula for ##V## when varying just one of the ##\alpha_i## right? Thinking about this, I think that the reason behind this, at least with ##l=3##, is that the constraint reaction component projected on the spring is stronger when the spring is extended, so the...

Here is the complete statement (I will omit the book it comes from as I dont't know the policy of this forum): Consider ##l## equal point particles ##P_1, P_2, . . . , P_l(l > 2)## on a circle of radius #R$# and centre ##O##. All particles move without friction and the point ##P_i## is...

thanks BvU for your reply. Actually the way in which I wrote the formula was a bit ambigous, I will re-edit the post with ##\frac{1}{2}kd^2##. I am expecting stability because of 1) for both cases ##l=3## and ##l=4## if I displace one point mass by ##\epsilon## with respect to its equilibrium...

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...

Hi everyone, IT professional from Italy, I have always been fascinated by maths and fundamental physics. Here to learn more about nature. Cheers