Structure and Interpretation of Classical Mechanics
Ex 1.10 Higher-derivative Lagrangians
Derive Lagrange’s equations for Lagrangians that depend on accelerations. In particular, show that the Lagrange equations for Lagrangians of the form with terms are
In general, these equations, first derived by Poisson, will involve the fourth derivative of . Note that the derivation is completely analogous to the derivation of the Lagrange equations without accelerations; it is just longer. What restrictions must we place on the variations so that the critical path satisfies a differential equation?
Varying the action
Chain rule of functional variation
Since variation commutes with integration,
By the chain rule of functional variation:
If is path-independent,
But is path-independent?
The is path-dependent. Its input is a path , not just , the value of at the time . However, itself is a path-independent function, because its input is not a path , but a quadruple of values .
Since is path-independent,
Since and ,
Here is a trick for integration by parts:
As long as the boundary term ,
So if and ,
Since and ,
By the principle of stationary action, . So
Since this is true for any function that satisfies and ,
The notation of the path function is , not .
The notation means that takes a path as input. And then returns a path-independent function , which takes time as input, returns a value .
The other notation makes no sense, because takes a path , not a value , as input.
— Me@2020-11-11 05:37:13 PM
2020.11.11 Wednesday (c) All rights reserved by ACHK