Lagrange’s equations are ordinary differential equations that the path must

satisfy. They can be used to test if a proposed path is a realizable path of the

system. However, we can also use them to develop a path, starting with initial

conditions.

Assume that the state of a system is given by the tuple . If we are

given a prescription for computing the acceleration , then

and we have as a consequence

and so on.

So the higher-derivative components of the local tuple are given by functions . Each of these functions depends on lower-derivative components of the local tuple. All we need to deduce the path from the state is a function that gives the next-higher derivative component of the local description from the state. We use the Lagrange equations to find this function.

— Structure and Interpretation of Classical Mechanics

.

Eq. (1.113):

.

Eq. (1.114):

.

The Lagrange equation:

.

.

where is a structure that can be represented by a symmetric square matrix, so we can compute its inverse.

— Me@2022-06-30 11:33:27 AM

.

.

2022.06.30 Thursday (c) All rights reserved by ACHK