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Now, let's talk a bit about how we find minima

and solve the classical mechanics for our cause of potential.

First, let's discuss minimization of this potential.

This is really easier,

as the potential is a fourth degree polynomial,

it has four roots,

but two of them coincide at x equals zero.

So, the point x equals zero is doubly degenerate.

It has two roots.

The remaining two roots,

are given by roots of a quadratic equation.

So, this gives us the formula for roots x bar one and x bar two as shown here.

Clearly, if we want to have a real [inaudible] roots,

the discriminant in this equation should be positive.

Also, another interesting point to note here is that

the point x equals zero is a natural boundary of the score.

This means that if a particle touches this point,

it gets stuck or absorbed there.

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The reason is that at this point,

both the drift and diffusion terms in our model vanish.

Therefore if the particle touches x equals zero, it cannot escape.

The zero level is an absorbing boundary.

It is also instructive to look the formula of the solutions x bar

one x bar two that is obtained when parameter g is very small.

This can be obtained directly from the exact solution,

and the corresponding formulas are shown here in equation 26.

It is interesting that the first root is non-perturbative in

Kappa as Kappa stands in the denominator and perturbative

in g. But the second root is non-perturbative in both Kappa and g. Now from mechanics,

we know that a total energy of a particle,

E, is made of a kinetic energy and the potential energy, your facts.

Because the total energy is conserved,

the particle on this graph can only move in regions where the sum of

its kinetic and potential energy equals E. Because kinetic energy is always non-negative,

it means that in classical mechanics,

a particle cannot go to regions where its potential energy

U exceeds its total energy E. Now,

let's discuss how particle can move from one point to another in such potential.

These can be obtained from Newton's second law of mechanics.

To remind you this law,

it says in simple terms that the mass times the

acceleration should be equal to the force applied to particle.

In our case, the particle mass M is one,

and the force is the negative of the gradient of the potential.

So, if we use all these,

we get the second Newton's law in the form of equation 27.

It is interesting that the Newtonian law that I repeat here

in equation 28 has some interesting invariances.

Let's consider these three transformations that we will call

the CPT transformation using an analogy with physics.

The C-transform also called C-parity changes the sign of Kappa.

The second transform called P-parity changes x to negative x.

The last transform, which is called T-reversal changes the direction of time.

So, you can check that the Newtonian Second Law of equation 28 for

our system is separately invariant with respect to CP transform and the T-transform.

As a result, it's also invariant with respect to joint CPT transform.

Informally speaking, the joint CPT transform describes

an anti-world or rather an anti-market where all prices are flipped,

frictions are flipped, and in addition,

the market waves backwards in time.

According to classical mechanics,

such market would work the same way as our conventional market.