The Principles of Least Action and Virtual Work

To seek out what Euler was referring to let us consider the basic case of a harmonic oscillator which can be seen as a body attached to one end of a spring the other end being fixed with the body moving back and forth on a frictionless table (along a straight line). The motion of the body satisfies the following differential equation expressing balance of dynamic and spring forces:

• $\ddot x(t) = - x(t) $ for $0<t<T$     (1)

where $x(t)$ is the position of the body being equal to the length of the spring, $\dot x(t)=\frac{dx}{dt}(t)$ is the velocity and $\ddot x(t)$ is the acceleration of the body at time $t$.  Here  $0$ is an initial time with $x(0)$ and $\dot x(0)$ given as initial conditions and $T$ is a final time.

The equation (1) can be solved analytically and with $x(0)=0$ and $\dot x(0)=1$ the solution is $x(t)=\sin (t)$ as a periodic harmonic oscillation. 

Following Euler, let us now formally multiply (1) with an arbitrary function $y(t)$ satisfying $y(0)=0$ and $y(T)=0$ and integrate over $[0,T]$ including integration by parts to get 

• $\int_0^T(-\dot x\dot y + xy) dt = 0$ for all $y(t)$ with $y(0)=y(T)=0$.   (2)

The equation (2) expresses stationarity of the Lagrangian 

• $L(x)=\int_0^T (-\frac{1}{2} {\dot x}^2 +\frac{1}{2}x^2 )dt$

in the sense that $L(x+\epsilon y)$ does not change from $L(x)$ for a perturbation $\epsilon y(t)$ with small $\epsilon$ i.e,  

• $\frac{d}{d\epsilon}L(x+\epsilon y)=0$ for $\epsilon =0$.   (3)

We thus see that:

• The Equation of Motion EoM (1) expresses Stationarity of the Lagrangian (2) or (3).  

The stationarity of the Lagrangian is also named the Principle of Least Action PLA where action or work is force times displacement as expressed by multiplying the force balance equation $\ddot x+x=0$ by the displacement $x$ or $y$ as in (2). The Principle of Least Action in the from (2) is also referred to as the Principle of Virtual Work PVW with $y(t)$ a virtual displacement.

The Principal of Virtual Work is the starting point for the Finite Element Method as a computational method to solve EoM in cases when analytical solution is not feasible.

We can now summarise:

• Physics is modeled by EoM expressing balance of forces. 
• Formally EoM expresses PLA/PVW.
• Computational methods build on PLA/PLW.

We understand that PVW is a formality since virtual work is a formality. 

We can naturally connect to the distinction between ontology (what is) and epistemology (what one can say). What is are the body and spring on the table with dynamic and spring forces. What we can say is PLA or PVW and from that can we construct computational methods. 

What is less natural is to view a physical system to evolve according to PLA or PVW since it is not equipped with any (brain) power to compute action and then seek least action. This is something a a human with computer can do but not the physical system itself. A physical system evolves in order to satisfy EoM but not to satisfy PLA or PVW. 

We know that light is wave phenomenon satisfying certain EoM with a connected PLA expressing quickest path which can used in computation but does not govern the real physics. 

The formalism of PLA was given a prominent role in classical physics because it was useful in computation and so the distinction from the physics of EoM became unclear. The formalism was picked up in modern physics with Lagrangians being the holy grail.  In particular, quantum mechanics was based on formalities without physics, which has led to endless discussions about physicality without resolution as made clear in previous posts. 

Hollywood vs Principle of Least Action

The fictional character of the Principle of Least Action viewed to serve a fundamental role in physics,  can be understood by comparing with making movies:

The dimension of action as energy x time comes out very naturally in movie making as actor energy x length of the scene.  However, outside Hollywood a quantity of dimension energy x time is questionable from physical point of view, since there seems to be no natural movie camera which can record and store such a quantity.   

Principle of Least Action vs Adam Smith's Invisible Hand

                                     Violation of the PLA of the capitalistic system in 1929.

The Principle of Least Action (PLA) expressing

• Stationarity of the Action (the integral in time of the Lagrangian), 

with the Lagrangian the difference between kinetic and potential energies, is cherished by physicists as a deep truth about physics: Tell me the Lagrangian and I will tell you the physics, because a dynamical system will (by reaction to local forces) evolve so as to keep the Action stationary as if led by an invisible hand steering the system towards a final cause of least action.

PLA is similar to the invisible hand of Adam Smith supposedly steering an economy towards a final cause of maximal effectivity or least action (maximal common happiness) by asking each member of the economy to seek to maximize individual profit (individual happiness). This is the essence of the capitalistic system. The idea is that a final cause of maximal effectivity can be reached without telling the members the meaning of the whole thing, just telling each one to seek to maximize his/her own individual profit (happiness).

Today the capitalistic system is shaking and nobody knows how to steer towards a final cause of maximal efficiency. So the PLA of economy seems to be rather empty of content. It may be that similarly the PLA of physics is void of real physics. In particular, the idea of a smallest quantum of action as a basis of quantum mechanics, may well be unphysical.

  

Lagrange's Biggest Mistake: Least Action Principle Not Physics!

The reader will find no figures in this work. The methods which I set forth do not require either constructions or geometrical or mechanical reasonings: but only algebraic operations, subject to a regular and uniform rule of procedure. (Preface to Mécanique Analytique)

The Principle of Least Action formulated by Lagrange in his monumental treatise Mecanique Analytique (1811) collecting 50 years work, is viewed to be the crown jewel of the Calculus of Newton and Leibniz as the mathematical basis of the scientific revolution:

• The equations of motion of a dynamical system are the same equations that express that the action as the integral over time of the difference of kinetic and potential energies, is stationary that is does not change under small variations.   

The basic idea goes back to Leibniz:

• In change of motion, the quantity of action takes on a Maximum or Minimum. 

And to Maupertis (1746):

• Whenever any action occurs in nature, the quantity of action employed in this change is the least possible.

In mathematical terms, the Principle of Least Action expresses that the trajectory $u(t)$ followed by a dynamical system over a given time interval $I$ with time coordinate $t$, is determined by the condition of stationarity of the action:

• $\frac{d}{d\epsilon}\int_I(T(u(t)+\epsilon v(t)) - V(u(t)+\epsilon v(t)))\, dt =0$,  

where $T(u(t))$ is kinetic energy and $V(u(t))$ is potential energy of $u(t)$ at time $t$, and $v(t)$ is an arbitrary perturbation of $u(t)$,  combined with an initial condition. In the basic case of a harmonic oscillator;

• $T(u(t))=\frac{1}{2}\dot u^2(t)$ with $\dot u=\frac{du}{dt}$,
• $V(u(t))=\frac{1}{2}u^2(t)$
• stationarity is expressed as force balance as Newton's 2nd law: $\ddot u (t) +u(t) = 0$.  

The Principle of Least Action is viewed as a constructive way of deriving the equations of motion expressing force balance according to Newton's 2nd law, in situations with specific choices of coordinates for which direct establishment of the equations is tricky. 

From the success in this respect the Principle of Least Action has been elevated from mathematical trick to physical principle asking Nature to arrange itself so as to keep the action stationary, as if Nature could compare the action integral for different trajectories and choose the trajectory with least action towards a teleological final cause, while in fact Nature can only respond to forces as expressed in equations of motion.

But if Nature does not have the capability of evaluating and comparing action integrals, it can be misleading to think this way. In the worst case it leads to invention of physics without real meaning, which is acknowledged by Lagrange in the Preface to Mecanique Analytique.

The ultimate example is the very foundation of quantum physics as the pillar of modern physics based on a concept of elementary (smallest) quantum of action  denoted by $h$ and named Planck's constant with dimension $force \times time$. Physicists are trained to view the elementary quantum of action to represent a "quantization" of reality expressed as follows on Wikipedia:

• In physics, a quantum (plural: quanta) is the minimum amount of any physical entity involved in an interaction. 
• Behind this, one finds the fundamental notion that a physical property may be "quantized," referred to as "the hypothesis of quantization".This means that the magnitude can take on only certain discrete values.
• A photon is a single quantum of light, and is referred to as a "light quantum".

In the quantum world light consists of a stream of discrete light quanta named photons. Although Einstein in his 1905 article on the photoelectric effect found it useful as a heuristic idea to speak about light quanta, he later changed mind:

• The quanta really are a hopeless mess. (to Pauli)
• All these fifty years of conscious brooding have brought me no nearer to the answer to the question, 'What are light quanta?' Nowadays every Tom, Dick and Harry thinks he knows it, but he is mistaken. (1954)

But nobody listened to Einstein and there we are today with an elementary quantum of action which is viewed as the basis of modern physics but has not physical reality. Schrödinger supported by Einstein said:

• There are no particles or quanta. All is waves.

Connecting to the previous post, note that to compute a solution according the Principle of Least Action typically an iterative method based on relaxation of the equations of motion is used, which has a physical meaning as response to imbalance of forces. This shows the strong connection between computational mathematics as iterative time-stepping and analog physics as motion in time subject to forces, which can be seen as a mindless evolution towards a hidden final cause, as if directed by an invisible hand of a mind understanding the final cause.