Dynamics and and Dynamical Systems. Appendix B

B. Lagrange equations

This appendix shows the derivation of Lagrange equations from Newton's second law. Let us consider a system formed by rigid bodies whose center of mass position vectors are: , , …, . Thus, to determine the configuration of the whole system at a given instant, it is necessary to know the values of coordinates, which could be a combination of distances and angles.

The system is called holonomic, if some of those are related, making it possible to reduce the number of independent coordinates to generalized coordinates ( ):

Each position vector can depend on some of the generalized coordinates and on time:

and the velocity vector of the th body is then

Therefore, depends also on time, the generalized coordinates, and the generalized velocities :

and the partial derivatives of are then

(B.1)

The acceleration vector of the th rigid body is:

(B.2)

If at a given instant the value of each coordinate is changed to , where the virtual displacements are consistent with the constraints, each position vector will undergo a displacement:

(B.3)

Taking the dot product of each side of this equation times the sides of equation B.2, we obtain

(B.4)

since the derivative of the product is,

According to equations B.1, the derivative and the term inside the parenthesis on the right-hand side of the equation are the partial derivatives of , with respect to and . Therefore, the following result is obtained

Equation B.4 can then be written as,

(B.5)

Furthermore, notice that the partial derivatives of with respect to the generalized coordinates and velocities are:

substituting these two expressions in equation B.5 and multiplying both sides of the equation by the mass of the th body we get,

where is the kinetic energy of the th body. According to Newton's second law, is the total force acting on the th body. Using the expression B.3 and summing over all the th body, we obtain

which leads to Lagrange equations:

(B.6)

where is the total kinetic energy of the system and the generalized force has been defined as

(B.7)