An Observation on Least Action Principle in Classical Mechanics Oriented on the Evaluation of Relativistic Error Concerning the Measurements of Light Propagating in a Liquid

2. The Action Functional In Different Sets Of Admissible Functions

Let us consider the action functional for a single particle moving along x-axis (1) where L(t,x,) is the Lagrangian. If x(t) is the real physical curve described by the particle (trajectory), such that x(t₁)=a, x(t₂)=b, we consider the functional (1) in the following sets of admissible functions

Here D₁(t₁,t₂) is the space of all functions defined on the interval [t₁,t₂] which are continuous and have continuous first derivatives; h(t) is an arbitrary variable function, h₀(t) and h₁(t) are arbitrary but fixed functions; α is a real variable parameter. In Figure 1 is reported an example of x₂-trajectory.

Figure 1.

Example of the trajectory x₂(t,α)

The increment of action functional corresponding to the increment h(t) of x(t) is written as (3) where δS[x;h] is the differential of the functional and ε→0 for ||h||→0. The last action principle states that in the set Γ₁ we have

We now observe that (δS⁽¹⁾[x]=0)→ (δS⁽²⁾[x]=0)→ (δS⁽³⁾[x]=0).

In fact an explicit form of the formula (4) can be derived by using the Taylor’s theorem. So we have (5) where the subscript denotes partial derivatives with respect to the corresponding arguments, and the dots denote terms of order higher than 1 relative to h and _.

In this manner the formula (4) can be written as

By considering the action functional in the set Γ₂ we have obviously

So we have

We can also prove the implication (9) by showing that (Gelfand et al., 1963)