Our goal is to solve the equation of motion for the high-order acceleration , a given higher-order Lagrangian .
Calculations are done using SageManifolds.
Vectors (partial derivatives)
is an unknown function that is the analogue of acceleration for a higher-order equation of motion.
The Equation of Lagrange can be defined with the aid of the following auxillary fields
Number of terms in this expression:
We want to solve this equation (eq1 = 0) for in terms of .
The equation can also be written in the following form:
But we would prefer to write it as a polynomial in , , and powers of . If we write instead of , and above then we get new coefficients and terms in powers of .
Terms 2 order in arise from the Liebniz rule applied to the differential operator on the term in . For example
The necessary coefficient can be read from the above.
After removing the term above, the remaining terms are at most first-order in .
Similarly the first-order terms arise from applied to the term in .