Exercise 1.10: Higher-derivative Lagrangians
The only reason that the Lagrangians we've been considering don't take any local tuple components beyond velocity is that the physics of our universe seems concerned with updating velocities and nothing beyond. Newton's second law gives us an update rule for the velocities in a system, and we picked the Lagrangian so that Lagrange's equations would match Newton's second law.
But the formula for action works just as well if the Lagrangian takes many, or infinite, derivatives of the original coordinates. This exercise asks us to:
Derive Lagrange's equations for Lagrangians that depend on accelerations. In particular, show that the Lagrange equations for Lagrangians of the form inline_formula not implemented with inline_formula not implemented terms are
formula not implemented
In other words, find the constraint that has to be true of the Lagrangian for the action to be stationary along the supplied path.
This derivation follows the derivation of the Lagrange equations from the book, starting on page 28. Begin with the equation for action with an acceleration argument to inline_formula not implemented:
formula not implementedapply the variation operator, inline_formula not implemented:
formula not implementedExpand the right side out using the chain rule for variations, equation \eqref{eq:var-chain}:
formula not implementedFrom equations 1.20 and 1.21 in the book, we know that
formula not implementedExpand the chain rule up to the inline_formula not implementedth derivative of the coordinate:
formula not implementedOur goal now is to find some quantity inside the integral that doesn't depend on inline_formula not implemented. Setting that quantity to inline_formula not implemented will give us the Lagrange equations. Focus on the inline_formula not implemented term:
formula not implementedIntegrate by parts:
formula not implementedThe first of the two terms disappears, since, by definition, inline_formula not implemented, leaving us with:
formula not implementedAnd reducing the full equation to:
formula not implementedThe original Lagrange equations are peeking at us, multiplied by inline_formula not implemented.
Can we get another term into that form and move it in with the original Lagrange equation terms? Take the next term and integrate by parts twice:
formula not implementedThe second of the two definite evaluations disappears, since, as before, inline_formula not implemented. The first of the definite evaluations suggests that we need a new constraint to achieve higher-derivative Lagrange equations.
If we require inline_formula not implemented, then the first definite evaluation disappears as well. So, for a Lagrangian that considers acceleration, we have to impose the constraint that the path's endpoint velocities can't vary. The path-wiggle's endpoints can't be in motion.
The term collapses to:
formula not implementedFold this back in to the full equation:
formula not implementedFor a Lagrangian of the form inline_formula not implemented, the remaining terms disappear, leaving us with
formula not implementedThe goal was to find conditions under which the action is stationary, ie, inline_formula not implemented. For arbitrary inline_formula not implemented with fixed endpoints, this can only occur if the non-inline_formula not implemented factor inside the integral is 0:
formula not implementedThis is the result we were seeking.
Higher dimensions
To keep going, we have to integrate by parts once more for each new term of the local tuple that the Lagrangian depends on. For each new term we gain a new constraint:
formula not implementedAnd a new term in the ever-higher-dimensional Lagrange's equations:
formula not implementedThe fully general Lagrange's equations are, for a Lagrangian that depends on the local tuple up to the inline_formula not implementedth derivative of inline_formula not implemented:
formula not implementedConstrained by, for all inline_formula not implemented from 0 to inline_formula not implemented:
formula not implementedEquivalently, the constraint is that all derivatives of inline_formula not implemented from inline_formula not implemented to inline_formula not implemented must remain constant at inline_formula not implemented and inline_formula not implemented.
Exercise 1.13 implements a procedure that generates the residual required by these higher-dimensional Lagrange equations in Scheme.