Calculus of variations, more than any other branch of mathematics, is intimately connected with the physical world in which we live. Nature favors extremum principles, and calculus of variations provides the mathematical framework in which to express such principles. As a result, many of the laws of physics find their most natural mathematical expression in variational form. In Chapter 1, we considered the cases of Fermat's principle of optics and minimization of total energy to determine the shape of a liquid drop on a solid surface. The objective of Part II is to provide a brief introduction to a variety of physical phenomena from a unified variational point of view. The emphasis is on illustrating the wide range of applications of the calculus of variations, and the reader is referred to dedicated texts for more complete treatments of each topic. The centerpiece of these seemingly disparate subjects is Hamilton's principle, which provides a compact form of the dynamical equations of motion – its traditional area of application – and the governing equations for many other physical phenomena as illustrated throughout this and subsequent chapters. Much of the historical development of the calculus of variations is centered around its application to dynamical systems; therefore, a number of the important principles and historical figures intimately connected with the calculus of variations will be highlighted in this chapter.

It is hard to imagine a time when our sight was not bombarded with images of far off people and places, when one had to actually be in the presence of someone to see their face without it being filtered through the hands of an artist (or while sitting on a Seat of Seeing). Images allow us to render the very small, the very large, and the very distant; they provide a visual record of time and place; they move us and entertain us; they replace perception with reality. Images are acquired by Earth-orbiting satellites, interplanetary probes, subterranean robots, surveillance cameras in unmanned aircraft and on nearly every street corner, and our mobile phones. They are created using optical principles and magnetic fields, and images display what is “seen” in every corner of the electromagnetic spectrum.

Before proceeding to the applications of variational methods that will occupy our attention throughout the remainder of the text, it is worthwhile to pause and consider how the solution of variational problems is accomplished when a closed form solution is not possible. We have already encountered several such scenarios, particularly when dealing with functionals with multiple independent variables that produce partial differential equations, and we will encounter numerous more situations in the balance of the text.

If an Euler equation cannot be solved in closed form, we may obtain approximate solutions using the following techniques:

1. Solve the differential Euler equation using numerical methods, such as finite difference methods, spectral methods, or finite-element methods, where the latter two are based on the Galerkin (or other method of weighted residual) approach.

2. Solve the integral variational form approximately using the Rayleigh-Ritz method or finite-element methods based on the Rayleigh-Ritz method.

As described in Chapter 5, classical (Newtonian) mechanics is centered around Hamilton's principle and the associated Newton's second law, particularly as applied to statics and dynamics of nondeformable (discrete) and deformable (continuous) bodies. Such systems are typically of O(1) size operating at O(1) speeds. Modern physics generally refers to branches of physics that have arisen since Einstein's relativity theory in the early part of the twentieth century, including special relativity, general relativity, and quantum mechanics, for example. Modern physics addresses physical phenomena that occur at very small sizes, such as atoms, or very large sizes, such as galaxies, and/or very high speeds close to the speed of light. We focus on the theory of relativity and quantum mechanics, both of which reduce to the classical mechanics limit that has occupied our considerations thus far. Relativity reduces to classical mechanics when υ ≪ c, where c is the speed of light in a vacuum, and υ is a characteristic velocity of the system; quantum mechanics reduces to classical mechanics when length scales dominant within the system are much larger than Planck's universal constant ħ.

Electromagnetic wave radiation is remarkable for two reasons: 1) it spans a vast spectrum of wavelengths providing for a wide range of physics and applications, and 2) it does not require a medium within which to travel. While all electromagnetic waves travel at the speed of light if traveling through a vacuum, they span ten orders of magnitude in wavelength, essentially corresponding to the full range of sizes that exist in the universe. Amazingly, the governing equations of electromagnetic waves, which do not require a medium, and physical waves, which do, are essentially the same. Rather than a guitar string or gas molecules, however, electromagnetic waves travel via massless photons. While the photons do not have mass, they do carry energy; it is this energy that determines their wavelength. A photon's energy is directly proportional to its frequency (and inversely proportional to the wavelength).

The above excerpt cryptically recounts the legend of Dido, the Queen of Carthage, from the ninth century bc. After being exiled from Tyre in Lebanon, Dido purportedly sailed to the shores of North Africa (now Tunisia) and requested that the local inhabitants give her and her party the land that could be enclosed by the hide of an ox. Not thinking that an oxhide could encompass a large portion of land, they granted her wish. She then proceeded to have the hide cut into narrow strips and extended end-to-end to form a semicircle bounding the shoreline and encompassing a nearby hill. This area became known as Carthage.

Certain branches of mathematics have arisen out of consideration of the theoretical consequences of known mathematical theorems. More often than not, however, new branches of mathematics have been developed to provide the means to address certain types of practical problems that initially are often very specific. For example, how did Dido know to arrange the oxhide in a circular shape in order to enclose the largest possible area with a given perimeter length?

Our final application of optimization theory is in grid generation, which is an important topic in many computational fields in which we desire to obtain numerical solutions of partial differential equations with complex physics and/or on domains with complex shapes. When choosing a numerical grid on which to solve differential equations, it is necessary that the grid both faithfully represent the geometry of the domain and be sufficiently refined in order to maintain the numerical truncation errors at an acceptable level. Traditionally, this has been done in a rather ad hoc manner by choosing a grid that “looks good” in some qualitative sense. Calculus of variations allows us to optimize the choice of a grid in a formal manner by explicitly enforcing certain criteria on the generation of the grid, thereby providing a more intuitive and mathematically formal basis for grid generation. Variational grid generation is generally formulated to produce the “best” grid in a least squares sense, eliminating the trial and error necessary in other grid generation approaches, such as algebraic and elliptic grid generation.

Fluid mechanics is one of the fields that is contributing significantly to the current resurgence of interest in variational methods. Historically, calculus of variations has not been taught or utilized as a core mathematical tool in the arsenal of the fluid mechanics practitioner; it is a rare fluid mechanics textbook that even mentions variational calculus. However, recent research developments are revolutionizing the field by reframing certain fluid mechanics phenomena within a variational framework. This is particularly the case in the areas of flow control and hydrodynamic stability.

Traditionally, flow control has been implemented in an ad hoc manner involving a significant amount of trial and error within an empirical (experimental or numerical) framework. Beginning in the 1990s, flow control is increasingly being framed within the context of optimal control theory. As will be seen in Chapter 10, the solution to optimal control formulations, particularly for large problems involving many degrees of freedom, is very computationally intensive. It is only recently that the computational resources have become available that are capable of solving realistic fluid mechanics scenarios within such an optimal control framework.

Similarly, significant advances are being made in our understanding of stability of fluid flows through the application of transient growth analysis that seeks the “optimal,” or most unstable, initial perturbation (disturbance) that results in the greatest growth of the instability.

The focus of Chapters 4 through 9 has been to develop physical variational principles and their associated differential equations of mathematical physics. The number and variety of physical phenomena captured in Hamilton's principle are truly remarkable and are a testament to the fundamental nature of the law of conservation of energy on which it is based. Whereas our emphasis has been to elucidate the laws of physics using calculus of variations, we now turn our attention to optimization and control of these systems. Interestingly, the same variational methods used to formulate the governing Euler equations provide the framework for performing their optimization and control. The governing Euler-Lagrange equations of motion obtained in the previous chapters become the differential constraints for control of such dynamical, fluid mechanical, and electromagnetic systems.

Classical mechanics encompasses those areas of physics that originated prior to the development of relativistic mechanics at the beginning of the twentieth century. Its primary focus is on the application of Newtonian mechanics to macroscopic systems. Classical mechanics provides the basis for many of the important fields in engineering, including solid mechanics, fluid mechanics, transport phenomena, and dynamics. These fields are employed broadly in the design of almost all devices that make modern life possible, including being sure that your mobile phone can withstand a fall on a hard surface, placing and maintaining communication satellites in their orbits, and both terrestrial and extraterrestrial transportation systems.