Direct Energy, 2018a

11 CALCULUS OF VARIATIONS 245
Part II
Theoretical Ideas
11 Calculus of Variations
11.1 Introduction
The previous chapters surveyed various energy conversion devices. The purpose
of Chapters 11 and 12 is to establish a general framework to describe
any energy conversion process. By placing energy conversion processes in
a larger framework, we may be able to see relationships between processes
or identify additional energy conversion processes to study. Establishing
this framework requires some abstraction and hence some mathematics. In
the next section, we dene the Principle of Least Action and the idea of
calculus of variations. In the followingsections, we apply these ideas to two
example energy conversion systems: a mass spring system and a capacitor
inductor system.
An advantage of using calculus of variations over other techniques is that
the analysis is based on energy, which is a scalar, instead of the potential,
which may be a scalar or vector. Workingwith a scalar quantity like energy
instead of a vector can make the mathematics quite a bit more manageable.
11.2 Lagrangian and Hamiltonian
Consider a process which converts energy from one form to another. We
are interested in how some quantity evolves duringthe energy conversion
process, and we call this quantity the generalized path, y(t). For simplicity,
we consider only the case where this path has one independent variable t
and one dependent variable y. In this chapter, t represents time, but it
can also represent position or another independent variable. These ideas
generalize directly to situations with multiple independent and dependent
variables [163] [164], but the multiple variable problem requires more involved
mathematics. The units of generalized path depend on the energy
conversion process under consideration. In the mass springexample of Sec.
11.5, it represents position of a mass. In the capacitor inductor example of
Sec. 11.6, it represents the charge built up on the plates of the capacitor.
Aside from the energy conversion process under consideration, assume that
no other energy conversion processes occur, even though this situation is
unlikely. The system goes from having all energy in the rst form to having
all energy in the second form following the path y(t).