The Flavor of Analysis

Mathematics, as a discipline that organizes concepts based on quantity and rigorous logic, has continued to develop since ancient times. Since the 17th century, beginning with mechanics, mathematical thinking has been earnestly applied to understand natural phenomena. Within this context, a new field of mathematics called “analysis” was born. In studying natural phenomena, the concept of a “function” emerges as a means to express phenomena that change from moment to moment in mathematical formulas. For example, to investigate the motion of a planet, which often appears to “wander,” we first assume there is a rule that gives the planet's position at each moment in time. We represent this rule as 𝑓 and the planet's position at time 𝑡 as 𝑓(𝑡).

The rule 𝑓 is called a function. At this point, it is unclear what the function 𝑓 specifically is, but it is presumed that 𝑓 is not determined randomly but follows some “law” of the natural world. Discovering these “laws” is an important role of physics.

As a law that the function 𝑓 describing planetary motion should follow, there is, for example, the “law of motion,” which states that “acceleration is proportional to force.” The rate of change of position is velocity, and the rate of change of velocity is acceleration. By requiring the function 𝑓 to satisfy the condition that “acceleration is proportional to force,” we obtain a certain equation concerning 𝑓 (an ordinary differential equation). If we can find the solution 𝑓 to this equation and further understand the properties of 𝑓 in detail, we can then understand the motion of the planet. A seemingly different law from the law of motion is the “principle of least action,” which states that “the motion that actually occurs is the one described by the function 𝑓 that minimizes the ‘cost.’” In other words, among the infinite number of candidate functions (paths) representing the planet's motion, some candidates take a “detour,” resulting in a high “cost,” while others proceed “efficiently,” resulting in a low “cost.” In this situation, if we can find the function 𝑓 (path) that minimizes the “cost,” we can understand the planet's motion. Clarifying what this “cost” specifically is is also an important role of physics. Although it is a somewhat rough expression, it can be shown that the law of motion and the principle of least action are mathematically equivalent. Analysis provides the mathematical framework to conduct these discussions rigorously. One of the most fundamental concepts in analysis is “differentiation and integration.” To formulate velocity and acceleration from the position function 𝑓, we first consider the average rate of change

but to eliminate the ambiguity in how ℎ is chosen, we define the velocity 𝑓'(𝑡) as the value that 𝑣(𝑡;ℎ) approaches indefinitely as ℎ approaches 0.  

This is the concept called “differentiation.” Whether 𝑓'(𝑡) exists is non-trivial, and various matters concerning the differentiation of functions are organized along with a rigorous definition of “approaching indefinitely (limit).” Analysis also provides a well-developed theory for solving general ordinary differential equations, including those derived from the laws of motion. Specifically, for the general form of an initial value problem for an ordinary differential equation (higher-order ordinary differential equations can also be transformed into this form)

it is possible to prove the existence and uniqueness of a solution. A powerful method for demonstrating the existence of a solution to a differential equation is the “construction of approximate solutions and their convergence.” For example, the following is known as an approximation for 𝑦'(𝑡)=F(𝑡,𝑦(𝑡)): let ℎ be a positive constant, 𝑘=0,1,2…, and 𝑡𝑘=𝑘ℎ.  

This is obtained by discretizing the continuous time variable with a step size ℎ and approximating the derivative 𝑦'(𝑡) with the average rate of change over the time interval ℎ. Starting from 𝑘=0, 𝑦0, 𝑦1,𝑦2 … can be obtained immediately through arithmetic operations. For each ℎ, a piecewise linear function 𝑦ℎ is obtained by connecting adjacent points of the graph of the approximate solution { (𝑡𝑘,𝑦𝑘) | 𝑘=0,1,2,…} with line segments. Here, as ℎ approaches 0, the “kinks” in the function 𝑦ℎ become progressively finer, and in the limit, it is expected that 𝑦ℎ will “converge” to some smooth function 𝑦, and that 𝑦 will be the true solution. It is possible to prove this conjecture rigorously. To do so, it is necessary to investigate in detail the “convergence of a sequence of functions” (in the example above, the convergence of the sequence of piecewise linear functions 𝑦ℎ), which is a generalization of the concept of the convergence of a numerical sequence. “Integration” is related to the concept of the area of curved figures and is also the inverse operation of differentiation, playing a very important role in various situations. The “cost” that a function (path) possesses, as mentioned in the principle of least action, is given, for example, by an integral such as the following:  

where L is a known function and γ is the function representing the path. ℓ is also called a “functional” in the sense that it is “a function defined on the set of all possible paths γ, that is, a set of functions.” The problem of finding a function that minimizes ℓ (i.e., an “efficient” path) can be seen as a generalization of the problem of finding the minimum point of a function defined on the number line, such as 𝑦=𝑥4−4𝑥3+𝑥2+3𝑥−1. Since the domain of ℓ is a set of functions rather than the number line, the problem becomes significantly more difficult. Here too, the concept of “convergence of a sequence of functions” plays a very important role.  

In addition to the above, differential equations corresponding to various other phenomena have been derived, and analysis is rapidly developing to solve them. Furthermore, the usefulness of the idea of “cost minimization” has become apparent in various problems outside of mechanics, and a theory called the “calculus of variations” has developed within analysis. Setting up an appropriate set of functions according to the problem and clarifying the convergence of sequences of functions and other properties within that set forms the foundation of analysis. Using these frameworks to elucidate phenomena is also an important role of analysis.

I hope that through this article, you have been able to catch a glimpse of the background and role of analysis, a relatively new field of mathematics.