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.