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As we want to characterize the evolution of states over time, the appropriate setting is phase space extended with the time variable12,13. That is, the space charted by position q, momentum p and time t as can be seen in Fig.1. In the same way that we write (x^i = [ x, y, z ]) for the three dimensions of space, we write

$$begin{aligned} xi ^a = [ q, p, t] end{aligned}$$

(1)

for the three dimensions of the extended phase space.

Evolutions in the extended phase space and the divergence-free displacement field.

Under the assumption that

$$begin{aligned}&the,,system ,,undergoes ,,deterministic ,,and ,,reversible \&evolution end{aligned}$$

(DR)

we can define a displacement vector field

$$begin{aligned} begin{aligned} vec {S}&= left[ frac{dq}{dt},frac{dp}{dt},frac{dt}{dt} right] \&= S^a e_a = frac{dxi ^a}{dt} e_a. end{aligned} end{aligned}$$

(2)

that describes how states move in time. [Where possible, we will be writing the same expression in both vector calculus and component notations.] In dynamical system literature, this is referred to as the vector field of the dynamical system. The time component of the displacement vector field is constrained, as we have

$$begin{aligned} S^t=frac{dt}{dt}=1. end{aligned}$$

(3)

If assumption (DR) is valid, we expect the flow of states through a closed surface to be zero: as many states flow in as flow out of the region. Alternatively, if we assign a probability, or probability density, to each trajectory, the assumption requires that probability not to change, so integrating the probability over a closed surface must yield zero. However we see it, assumption (DR) means the field is divergence-free. [Given that this is a three-dimensional space, we can use the standard tools of vector calculus.] That is,

$$begin{aligned} nabla cdot vec {S} = partial _a S^a = 0. end{aligned}$$

(4)

Since the displacement field is divergence-free, it admits a vector potential. We have

$$begin{aligned} begin{aligned} vec {theta }&= [theta _q, theta _p, theta _t] = theta _a e^a \ vec {S}&= - nabla times vec {theta } = - epsilon ^{abc} partial _b theta _c , e_a. \ end{aligned} end{aligned}$$

(5)

The minus sign is introduced to match conventions. Mathematically, this is analogous to what is done for a magnetic field or for an incompressible fluid.

Because the displacement field must satisfy (3), without loss of generality we can set

$$begin{aligned} begin{aligned} vec {theta }&= [p, 0, -H(q,p,t)] \&= p e^q - H(q,p,t) e^t, end{aligned} end{aligned}$$

(6)

where H is a suitable function of q, p and t. The potential (vec {theta }) is closely related to the canonical one-form of symplectic geometry and the contact form of contact geometry. By applying definition (2) and expanding (5) with (6), we have

$$begin{aligned} left[ frac{dq}{dt},frac{dp}{dt},frac{dt}{dt} right] = - nabla times vec {theta } = left[ frac{partial H}{partial p},-frac{partial H}{partial q}, 1 right] , end{aligned}$$

(7)

which yields Hamiltons equations. Note that the argument works in reverse: any Hamiltonian system with one degree of freedom yields a divergence-free displacement field, and therefore satisfies (DR).

As shown in (a), the variation of the action is the flow of the displacement field (vec {S}) through the surface (Sigma) that sits between the path (gamma) and its variation (gamma '). In (b) we see that the flow is zero if the path is an actual evolution of the system, since the displacement field will be parallel to the path (gamma) and therefore tangent to the surface (Sigma).

We now turn to constructing the principle of stationary action. As illustrated in Fig.2a, take a path (gamma) with endpoints A and B, not necessarily a solution of the equations of motion. Then take a variation (gamma ') of that path and identify a surface (Sigma) between them. We can ask: what is the flow of the displacement field (vec {S}) through (Sigma)? Because (vec {S}) is divergence-free, the flow through (Sigma) will depend only on the contour, therefore the question is well posed. Using Stokes theorem, we find

$$begin{aligned} begin{aligned} - iint _{Sigma } vec {S} cdot dvec {Sigma }&= iint _{Sigma } left( nabla times vec {theta } right) cdot dvec {Sigma } \&= oint _{partial Sigma = gamma cup gamma '} vec {theta } cdot dvec {gamma } \&= int _{gamma } vec {theta } cdot dvec {gamma } - int _{gamma '} vec {theta } cdot dvec {gamma }' \&= delta int _{gamma } vec {theta } cdot dvec {gamma }. end{aligned} end{aligned}$$

(8)

Now suppose (gamma) is a solution of the equation of motion, as in Fig.2b. Then (gamma) is a field line and the flow is tangent to (Sigma) no matter what (gamma ') we picked. The converse is true: if we look for those paths for which the flow through (Sigma) is zero no matter what (gamma '), (gamma) must be everywhere tangent to (vec {S}) so we find a solution to the equation of motion. The solutions, then, are those paths and only those paths for which

$$begin{aligned} 0 =delta int _{gamma } vec {theta } cdot dvec {gamma } = - iint _{Sigma } vec {S} cdot dvec {Sigma } end{aligned}$$

(9)

We call this the principle of stationary action in Hamiltonian form.

The last step is to express the principle exclusively in terms of kinematic variables: position, time and velocity. This can be done if we assume that

$$begin{aligned} the,, kinematics ,,of ,,the ,,system ,,is ,,enough ,,to ,,reconstruct ,,its ,,dynamics. end{aligned}$$

(KE)

This means that by looking at just the trajectory in space q(t), we are able to reconstruct the state at each moment in time. Therefore we must be able to write (p=p(q,dot{q})), and therefore we can also write

$$begin{aligned} begin{aligned} delta int _{gamma } vec {theta } cdot dvec {gamma }&= delta int ^{t_2}_{t_1} vec {theta } cdot frac{dvec {gamma }}{dt} dt \&= delta int ^{t_2}_{t_1} left( p frac{dq}{dt} - H right) dt \&= delta int ^{t_2}_{t_1}L(q, dot{q}, t) dt = 0. end{aligned} end{aligned}$$

(10)

We find that a system for which (DR) and (KE) are valid can be characterized in terms of the principle of stationary action with a suitable Lagrangian. The converse is also true: if the principle of stationary action allows for a unique solution, then the conjugate momentum and the Hamiltonian are well defined and the system satisfies both (DR) and (KE).

We have thus demystified the principle of stationary action, and turned it into a geometric property: requiring the principle of stationary action is equivalent to requiring that the solutions are the field lines of a divergence-free field in phase space. We also have a clear physical meaning: the principle of stationary action is equivalent to assuming determinism/reversibility (DR) and kinematic equivalence (KE). However, we do feel that the principle expresses these requirements in a very roundabout way.

Originally posted here:

Geometric and physical interpretation of the action principle ... - Nature.com