It can be shown that for n-many masses the system of masses and springs can be written in time form M x + K x = 0 ( equation 1)

where x's are vectors and the first x has two dots over it.

[K]_ij = [K]_ji => K^T = K

where K is a matrix of spring constants and M is a matrix of masses

show that equation 1 is a conservative system.

Hint: d/dt(x^T * Mx) = 2x^TMx

where the x^T 's have a dot above them and the lone x has two dots

x with a dot above it = dx/dt where x is a vector

can someone please go through this step by step? I missed this lecture in class and the teacher said he went over this and I can't figure out how to do it.