The spectacle of the universe becomes so much the grander, so much more beautiful, the worthier of its Author, when one knows that a small number of laws, most wisely established, suffice for all movements.Pierre Louis Maupertuis (1744)
Among the more or less general laws, the discovery of which characterize the development of physical science during the last century, the principle of Least Action is at present certainly one which, by its form and comprehensiveness, may be said to have approached most closely to the ideal aim of theoretical inquiry. Its significance, properly understood, extends, not only to mechanical processes, but also to thermal and electrodynamic problems. In all the branches of science to which it applies, it gives, not only an explanation of certain characteristics of phenomena at present encountered, but furnishes rules whereby their variations with time and space can be completely determined. It provides the answers to all questions relating to them, provided only that the necessary constants are known and the underlying external conditions appropriately chosen.Max Planck (1909)
Table of Contents
IV. Modern Formulation <— You are here
IV. Modern Formulation <— You are here
This is the fourth installment in the series on the Principle of Least Action. In the first installment, “Principle of Least Action I,” we were introduced to teleology. We learned that teleology is a reason or an explanation for something that serves as a function of its purpose, as opposed to something that serves as a function of its cause. Things are believed to have a purpose, a goal, or an end they strive towards. The end purpose is called the final cause (causa finalis). We also learned that Judaism is undoubtedly teleological to the core and that the Jewish faith is based on the belief in a purposeful G‑d, who created a purposeful world. In the second installment, we were introduced to the principle of least action. We learned that the principle of least action is a formal representation of such simple ideas as efficiency and optimization. Nature strives to emulate its Creator as best as it can. Therefore, the efficiency of nature is the manifestation of the perfection of the infinite Creator within the finite confines of the physical world. In the third installment, we reviewed the history of the development of the principle of least action. We met Heron, who was the first to introduce the concept of the shortest distance; Pierre de Fermat and Gottfried Leibniz, who proposed the “Principle of Least Time”; Johann Bernoulli, who formulated the variational principle; Pierre-Louis de Maupertuis, who introduced the principle of least action; Leonhard Euler and Joseph-Louis Lagrange, who derived Euler-Lagrange equations of motion. We saw how each of these formulations had strong teleological overtones and theological parallels. In this installment, we will learn about the modern formation of Lagrangian mechanics based on the principle of least action.
The Lagrangian formulation of classical mechanics is a reformulation of Newtonian mechanics and provides a unified framework for studying the motion of systems. In Newtonian mechanics, to predict the evolution of a particle, we must know its initial conditions—its initial position and initial velocity—and all forces acting on the particle. In the formulations using the principle of least action, we are not required to know the initial velocity or the forces acting on the particle. All we need to know are the initial and final positions of the particle and its energy.
The heart of this formulation is a Lagrangian, L, which is defined as the difference between a system’s kinetic energy (energy of motion) and its potential energy (energy due to position). Mathematically:
L = T – V,
where T is kinetic energy, and V is potential energy.
Instead of using traditional cartesian coordinates like x, y, and z, the Lagrangian formulation often uses “generalized coordinates,” denoted q. These can be angles, lengths, or any other convenient parameters that describe the system.
As mentioned before, the main idea of the Principle of Least Action is that nature “prefers” paths of motion that minimize (or, more precisely, extremize) a certain quantity called the “action.” The action is the integral (think of it as a summation) of the lagrangian over time. In more layman’s terms, it means that out of all the possible ways something could move, it will choose the path that “costs” the least in terms of this action.
The Euler-Lagrange Equation is the mathematical formulation resulting from the least action principle. It is a differential equation determining how a system evolves over time. By solving this equation, you can predict the evolution of a system. Whereas the principle of least action is an integral principle (in which that action—the integral of the lagrangian over time—is extremized) that has strong teleological overtones, Euler-Lagrange Equations of motion are differential equations that have no hint of teleology. Just as Newtonian mechanics, Euler-Lagrange Equations describe the motion through efficient causation—each state is determined by the state immediately before it. These two approaches—integral and differential—albeit each carrying a very different metaphysical baggage—are mathematically equivalent to each other.
The Lagrangian formulation of classical mechanics provides a unified, elegant, and often more straightforward way to study the dynamics of systems. Instead of dealing directly with forces, it works with energy functions and the idea that systems evolve in a way that minimizes the action.
However, the Lagrangian formulation carries strong teleological overtones. In Newtonian mechanics, we only know the particle’s initial conditions—its initial position and velocity. We also need to know all the forces acting on the particle to predict its evolution in time. The Lagrangian mechanics dispenses with forces. However, it requires knowledge not only of the initial position of the particle but its final position as well. The question is, how does a particle know its final position to figure out how to move? The reliance on the final cause is a characteristic feature of teleology.
The teleological nature of the principle of least action is hotly debated by modern philosophers primarily based on the fact that Euler-Lagrange equations, which are equivalent to the principle of least action in its variational form, dispense with the final cause (the final position of the particle) replacing it with efficient causation just as in Newtonian mechanics. But Euler-Lagrange equations are derived from the principle of least action. So, even if they do not explicitly contain information about the final position of the particle, it is implicit because it is found in the variational principle from which Euler-Lagrange equations are derived. Some philosophers claim that the explanatory arrow goes either way, and it is one’s personal choice, depending on one’s metaphysical commitments, how to look at it.
I agree, all its elegance notwithstanding, the principle of least action does not prove the existence of an intelligent Creator. In a sense, when it comes to inanimate matter, the hand of G‑d is concealed, preserving our freedom of choice—to believe or not to believe. However, when it comes to live matter, which, unlike passive inanimate matter, is actively pursuing the goals of survival and procreation, (locally) violating the second law of thermodynamics, there is no more choice. Acknowledging an intelligent Creator, who imbues live matter with goals it must labor to pursue, is no longer a matter of metaphysical commitments, it is a matter of intellectual honesty. But this is a different topic, which requires a discussion of its own.
For now, suffice it is to say that many great physicists, being fully aware of the mathematical equivalence between the variational principle of least action and Euler-Lagrange equations, nevertheless maintained the teleological character of the former. The grandfather of quantum physics, Max Plank, said this about the principle of least action:
The least-action principle introduces a completely new idea into the concept of causality: The causa efficiens, which operates from the present into the future and makes future situations appear as determined by earlier ones, is joined by the causa finalis, for which, inversely, the future—namely, a definite goal—serves as the premise from which there can be deduced the development of the processes which lead to this goal.
Not only did Planck maintain the teleological import of the least action principle, but he exposed the scientific and philosophical bias that prevents other thinkers from acknowledging this obvious truth:
The most adequate formulation of this law creates the impression in every unbiased mind that the nature is ruled by a rational purposeful will.
Another objection raised against the teleological interpretation of the principle of least action has to do with Feynman’s Path Integral formulation of quantum mechanics.
But what do we make of this strange quantity, the lagrangian, defined as the difference between kinetic and potential energy? In physics, we are used to seeing the sum of these two forms of energy, making up the system’s total energy, but not the difference. We will see later how Jewish mysticism sheds light on this puzzle.
 To understand the real significance of the Lagrangian approach, imagine you are studying the motion of a pendulum. In Newton’s formulation, you would deal directly with forces, which can get quite tricky if the pendulum is, say, swinging in two dimensions or if there is a constraint on its motion. With the Lagrangian approach, you would use an angle as your generalized coordinate and develop an equation based on energies. This can be much simpler and more intuitive in many cases.
 Max Planck, Scientific Autobiography and Other Papers, (New York, Philosophical library, 1949), , 179-80, See also Yemima Ben-Menahem, Causation in science, (Princeton University Press, 2018), p. 150.
 Ibid, p. 177.