The principle of least action

Abstract

We start with a review of the principle of least action. We are familiar with the concept of “total energy” for a simple mechanical system: it is the sum of kinetic and potential energies. For example, for a simple harmonic oscillator, the potential energy isV = 12 kx2, (1)where x represents the displacement of the spring from its equilibrium (relaxed) position and k, the spring constant. The kinetic energy for this system isT = 12 mx˙2, (2)where x˙ = dx dtrepresents the rate of change of position of the mass ‘m’ attached to the spring. The total energy isH = T + V = 12 mx˙2 +2 kx2. (3)Surprisingly, although the total energy is a useful quantity, it is not essential to our understanding of classical mechanics. Instead, the really important object in mechanics is the Lagrangian (Wikipedia: Lagrange) defined byL = T − V. (4)Explicitly, for the simple harmonic oscillator, this readsL(x, x˙, t) = 12 mx˙2 − 12 kx2. (5)The reason the Lagrangian is such an important object is because of the “principle of least action” which we now state. Consider a classical system at position x1 at time t1 and at position x2 at time t2. This system will always move from x1 to x2 in a manner such that the integralS =L(x, x˙, t) dt, (6)takes the least possible value1. This is referred to as the principle of least action (S being the action) and may be expressed as the following variationδS = δL(x, x˙, t) dt = 0. (7)It is amazing that all of classical mechanics (including laws like the conservation of energy) follows from this beautiful2 principle. The entire field of mathematics, called the “Calculus of variations”, is devoted to extremizing functionals as in (7). Physics is littered with principles governed by the calculus of variations - a wonderful example of the key role mathematics plays in the physical sciences (Wikipedia: Calculus_of_variations). Performing the variation in (7) produces the Euler-Lagrange equations of motion (Landau and Lifshitz, 1976)ddt( ∂L∂x˙) − ∂L ∂x = 0, (8)which for the simple harmonic oscillator impliesmx¨ = −kx, (9)which matches what we obtain by using Newton’s laws. Each of the forces of Nature is also decribed by a Lagrangian. Conveniently, three of the four forces are described by the Yang-Mills Lagrangian. Gravity, on the other hand, is described by the Einstein-Hilbert Lagrangian. In the next section, we will briefly describe these two Lagrangians and then be in a position to highlight the problem with gravity.