Solving dynamic problem for many variables

As shown in the Wikipedia for Lagrangian mechanics, it's another way to describe and solve mechanical problems.

In resume, once we know the energy E of the system, we can use the calcul of variations to calculate the motion equations from the derivatives of E.

The lagrangian operator is given by

And so, if we have the variables x0, x1, ..., xn, we will have n+1 equations given by setting L_xi = 0.

Now we are going to use an example such that the energy by:

Where M and K are symmetric matrix.

Applying the Lagrange operator we get

Where W is a symmetric matrix gotten by W = V - V^T

In this case, where we have a linear matrix equation, we can use the same [methods for linear equations].

So, our matrix interaction become