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
For simple problems, where we don't have any external load or any constraint, we have only L_xi = 0 .
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 is the mass matrix and K is the rigidity matrix, both are symmetric matrices.
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. For this, we have that the lagrangian is equal to the external force.
So, our matrix interaction become (with M different of mass matrix)
Another term appears when we allow constraint functions* to exist in our system. We will call these equations as g and our objective is that always g = 0.
For example, if we have two variables x and y, which describe the cartesian coordinates of a circle of radius a. So, a constraint equation is g(x, y) = x^2 + y^2 - a^2
When we add the constraint equations, our matrix change of the system changes.
- Put the new matrix, after adding Lagrange multiplicators