thanks to all theorists as well as Mark A Simpson as well as Sarah Barua

STRING THEORY DEVELOPMENT August 12 2012

                                                   

Strictly speaking as per definitive axioms, No, because of bijective relationships between Hamilton’s Principle (Hamiltonian matrix functions) and Holonomic (or Non-holonomic/anholonomic) functions (and in that various constraints) are very much dichotomous; as per classical approach neither of these can be subset of each other, respectively!! Classical approach poses an impossibility condition!

In this same context, quantized classical approach poses iterative stochasticity (not so good as a behavioral system) and has no clue for heteroscedasticity, so, for a simplistic input-byproduct-output system, this works as a heterodyne of known and unknown parameters and as a result the output erratic; whereas in quantum approach the output for stochasticity is determinably exact and for heteroscedastic distribution it is either fully deterministic or exact-stochastic which is wonderful and desirable for a system’s behavior to have!!

Frankly speaking, this query about Hamiltonian and non-holonomic functions rose years ago because of the very fundamental literary confusion within the understanding of their implications in the real world.. It is like wave-particular entwinement of a particle-wavy nature.. I will try and make this simpler with not too much of mathematics involved..

Holonomic means ‘wholly’ integrable. Wherever there is an integrality, results being erroneous is very trivial therefore not very trustworthy. Differentiability (also, integration was invented/discovered earlier than differentiation) of a system is the most efficient approach for a perfect outcome!!

There are striking duality and differences between Hamilton’s Principle and Non-holonomic constraints:

~Some amazing duality,

1) Definitions:

● Hamilton’s Principle is an “Integration form,” in general,

Action of evolution of a path with path integration:

A= ∫_t1^t2 [L(ψ(t),ψ'(t) ,t)dt]; L(Φ)is the Lagragian function,t1 & t2 are time states.

●● Non-holonomic constraints have “Differentiation form,” in general,

Non-integrable, no total differentiation but in accordance to partial differentiation:

∑_(i=1,2,…,n)^c:1,2,…,k [a_(c,i)dψ(t)_i ]+ a_(c,t) dt=constant

; c is the number of constraints, “n” number of co-ordinates and “a” is the constraint coefficient. And so for this my personal preference and favourite is the most amazing distribution, Poissan distribution, and the Jacobian matrices!

PART 2

‎2) Type of dependency:

● Hamilton’s Principle is based on “Variational parameters”

In A= ∫_t1^t2 [L(ψ(t),ψ'(t) ,t)dt]

; ψ(t):q_1(t_1 ) ,q_2(t_2 ) ,…,q_n(t_n ) and ψ' (t):q_1(t_1' )',…,q_n(t_n' )' are static point time states and so even with negligible variation, the path evolution will change!

●● Non-holonomic constraints are “Non-variational”

Better definition,

∑_(i=1,2,…,n)^(c:1,2,…,k) [a_((c,i) ) dψ(t)_i]⊕ a_((c,t)) dt={const_1⊕const_2,const_1 ⊗const_2}

Here, one amazing and surprisingly mind-blowing mathematical magic is if I replace “a” with “א” as a coefficient and function “ψ(t)” with “λ(z),” I will get a different sets of unique prime numbers as a diagonal matrix with maximum rank of (c,n)!!

∑_(i=1,2,…,n)^(c:1,2,…,k) [א_((c,i) ) dλ(z)_i ] ⊕ א_(c,t)dz= {{prime numbers},{semi-prime numbers},{coprime numbers}}

■ Corollary: Use of quaternion q and –q will completely change the behavior of both,

A= ∫_q(t)^(-q(t)) [L(q(t), q'(t) ,t)dt] &

∑_(i=1,2,…,n)^c:1,2,…,k [a_((c,i) ) ∂^2A(q, q’)_i ]+ a_(c,qq’) ∂q∂q’= scalar function

Combining both, they specifically represent Reimannian manifolds irrespective of coordinates and orbifolds for irregular bodies!

HERE, IT CAN BE SAID THAT HAMILTON’S PRINCIPLE IS A TYPE OF “NON-HOLONOMIC LINEAR CONSTRAINT, CONDITIONALLY!!”

3) System representation:

● Hamilton’s Principle represents the system “explicitly” as in “overview/overall.”

●● Non-holonomic constraints represent the system “implicitly” with step-wise equilibrium of quasi-states.

4) Behaviour :

● Hamilton’s Principle is purely non-ergodic, which means that it final state or each and every prevailing steps of evolution of path is dependent on the initial state occurrence.

●● Non-holonomic constraints may or may not be ergodic, but, can be made ergodic!

5) Operationally:

● Hamilton’s principle is “multiplicative,” which means that the evolution of path is the product of causal states to be the next state path coordination,

Actual and total path evolution Q(t): L(ψ(t), ψ'(t) ,t) + δ(t)f(ψ(t), ψ’(t), t), where δ(t)f(ψ(t), ψ’(t), t) is the multiplier which forms the basis of Multiplier theorem.

●● Non-holonomic constraints are “additive.” Both simple addition and cocyclotomic additions are inclusive as per the definition shown.

Part 3

1) Hamiltonian systems based on generalized Hamilton’s Principle and special cases, the study scope is basically for the systems with Optimal Controls. For example, autonomous systems, robots etc. generalization of classical Lagrangian and Hamiltonian systems in which one allows position constraints only.

Non-holonomic systems are mechanical systems with constraints on their velocity that are not derivable from position constraints. For example, rolling contact (rolling of wheels without slipping), certain kinds of sliding contact (bar chain mechanism).

2) Hamiltonian systems are Variation based which comes under Vakonomics (Variational Action Kinematics). Non-holonomic systems are “Non-Variational” based systems which are derived from the Lagrange-d’Alembert principle and not from Hamilton’s principle. Lagrangian function is the core functional variation in Hamiltonian but with constraints it has dimensional limitations.

3) Energy is preserved for non-holonomic systems, momentum is not always preserved for systems with symmetry.

4) Non-holonomic systems are almost Poisson but not Poisson and so it doesn’t generally satisfy the Jacobi identity for partial differentiation of Tensors.

5) Unlike the Hamiltonian, volume being scalar, it may not be preserved in the phase space for non-holonomic constraints which leads to asymptotic stability in some cases, despite energy conservation. Recall Greene’s Theorem!!

‎6) Constraints and Degrees of Freedom: (Certainly the most widespread subject detailing can be done in this one, comparatively)

Degrees of Freedom as the name suggests, it gives us an idea about the scope of prevailingness of/for a system and with what order and degree of levels does it exist independently until the perturbation and even more!! Statistically speaking, it is the most perfect estimate for a class distribution with a valid upper and lower class and an unique (for odd) or neighbouring (for even) mean class to exist under all trivial and non-trivial conditions, all of these to be independent!

Constraints, as the name suggests, are the controlling features which accounts for the number of parameters that create dependencies. There are various types of constraints, viz,

~ Linear/non-linear constraints

~ Circular/spherical/elliptical/general conical constraints

~ Quantitative parametric constraints such as Time (variant/invariant), Mass, Energy, Momentum, Velocity, Volume, Dimensional, Force, Potential, Field, ENTROPY (most interesting one) and so on…

The rotation, translation and propagation pertaining to Specially Orthogonal, SO(3) or SO(3, R) for the fundamental general cover SO(n, R) under the Weyl grouping for Lie Algebraic Topology is the most crucial part of understanding for testing the validity. This validity is concentrated for 3D, (3d+1) and (3d-1).

The non-polynomial algebraic equation for En will be,

[|{2^(n-1) - n} x d > ± {0, ± 1 ± j, j ⌆ j | j : C / C ⊌ (∞)}] D complex dimensions..

Extracting only the canonical part then it is [{2^(n-1) - n} x d] D complex dimensions pertaining to a design in which “n” that is negated from “2^(n-1)” is the number of constraints that reduce the degree of freedom for any En topological design. “n” number of constraints can have Nilradical rings, Antipodal points, etc. For example, in E8, there are at least n=8 types of constraints and thus reducing its Degrees of Freedom to 248!

Hamilton’s Principle is so much in relation with this because these are non-polynomial dimensional gapping paths for the evolution pertaining to centralized/decentralized action, which can be non-integrable (non-holonomic with constraints).

This suggests that there is a thin leaf of congruence in coexistence of Hamiltonians and Non-holonomic functions with constraints, but, the Lagrangian dynamics is unstable for 3D and higher dimensions because it gives erratic outcomes to the evolution hence Hamiltonian being one of the types of non-holonomic constraints will have to dealt with lots of instability and a bit of premature to conclude as in!! One of the solutions is to play Lagrangian dynamics with sub-3 (3-n) dimensions like an unitary-2 matrix algebra.

A very simple but very strong illustration that cites the importance of Degrees of Freedom is in Particle Physics. Pertaining strictly for the Standard Model, Quarks enjoy 6 degrees of freedom, hence, six types of Quarks! For Bosons, it varies from 3 to 4, hence Photon, W (W+, W-) bosons and Z boson. For complete understanding of the Standard Model, there is a need to identify at least 5 different and unique types of the Higgs-Boson having equal Degrees of Freedom!!

7) Functionally:

● Hamiltonian space has ingredients such as time invariant point states and a path definition and thus it is functionally “Evolutionary” and as it is purely non-ergodic too. Causal time-invariant states define the path of evolution for the upcoming states.

●● Non-holonomic constraints can be ergodic as well as non-ergodic and with that it’s functionally both “Evolutionary” and “Involutional (not devolutionary).”

 PART 4 EXAMPLES

Some fascinating Real-World illustrations that witness the outer-rim of intersectional area between applied Hamiltonian Principle and Non-holonomic functions with constraints:-

1) Optochemical:

In the colloidal Brownian motion and when bombarded with visible light, when resultant spectrum forms specific Raman lines and Stokes/Anti-Stokes lines pertaining to Raman spectroscopy, the path consideration of not-so-random motion and these lines illustrate a classic example.

2) In Aerodynamics:

Ever wondered about the reason that why is Helicopter piloting/flying much more difficult than a fighter jet plane with Mach-4 speeds!!! The reason is Degrees of Freedom and Coupled constraints!!

How??

Helicopter as a whole can be considered as two part construction,

a) Helicopter body with tail,

b) Helicopter blades.

In which helicopter body is coupled with blades that rotate and provide an aerodynamic thrust. In a 3D space, degrees of freedom, F, for a maximum value for a freely moving body is 3n (n is number of dimensions/coordinate values). And for a coupled moving body it gets reduced as, F = 3n – r (r is the number of couplings).

Now, for a Helicopter body,

F_max (body) = 3(3) = 9 ; n = 3 i.e. for rotational, translational and propagating dimensions in a 3D space.

For helicopter blades which generally has two blades with 4 endpoints rotates in a plane, i.e. n = 2,

F_max (blades) = 3(2) = 6.

But, helicopter’s body is coupled with blades in 3:1 one fashion, i.e. 3 dimensional singular points pivoted at one single point. Therefore, r, here will be 3. So, degrees of freedom for body is,

F(body) = F_max(body) – 3 = 9 -3 = 6

Similarly, coupling for blades will be 1 because of singular ball-bearing coupling with a rotor. So, degrees of freedom for blades will be,

F(blades) = F_max(blades) – 1 = 6 - 1 = 5

Therefore, Total F_max (heli) = F_max(body) + F_max(blades) = 9 + 6 = 15

Whereas TotalF(heli) = F(body) + F(blades) = 6 + 5 = 11

Now, difference ∆F = TotalF_max – TotalF = 15 - 11 = 4 = No. of constraints.

As found out, no. of constraints that is No. of Controlling constraints for a Helicopter are 4. This 4 is significant enough for how a helicopter is controlled mechanically by 2 controlling rods each having 2 dimensional state-space, i.e. each controlling rod has translational and rotational dimension for the pilot of a helicopter to control instead of just 1 steering for a fighter plane pilot!!!

PART 5 BIOLOGY

3) In Biology:

~ The study of a group of sperms in a semen, semen consisting of billions of sperms and how among all these only one sperm of male is more than enough to fertilize the embryonic egg of a female to create a zygote!! It is not only about potency of one sperm but it is also about the path (how? and why?) that is taken by that one sperm out of many to reach the goal.

~ The study of neurons and neural networks pertaining to conscious to sub-conscious mind through P, Q, R, S and T waves generated inside brain as an electric pulse forming point-tensor quantities of surds. Specifically in that, for a long-term memory how C-Kinase plays the most critical part.

4) In Quantum Cosmological world:

There are so many examples and illustrations but I am citing only one and that is Electron Binning effect and quantum wells.

Epochal Inflation and before that beyond Planck's Time & Length, force diversification from 10^-50 seconds to 10^-43 seconds after the supposed Big Bang gives a provision to look at it as Entropy-Enthalpy system of Latency!

5) In Mechanics:

When a vehicle turns sideways on a road, suppose, towards the left side of an on-going travel then due to the effect of center of gravity and inertia, the tyre (tire) and with that whole vehicle with its center of gravity and center of momentum bends towards that left side by balancing the centrifugal force due to motion.

Whereas, as per minute study with quantum approach, that tyre actually drifts towards the right-side (when it is to turn left) at the local-point of contact which in a sense creates a negative-friction!! This is the classic example in relation to Hamiltonian and Non-holonomic constraints.

One more example is that of an optical illusion that we get to see when observing the tyre grooves in motion. It creates an illusion as if the tyre is rotating towards the upward direction from the ground-level when the motion is actually about to move forward!!

6) In Acoustics/Musical instruments:

Citing an of Flute here, while blowing air into the flute pipe which has many holes, due to vibration, hindrance and interference of acoustic waves, we get to hear specific pattern of sound. As per Indian music it is “Sa Re Ga Ma Pa Dha Ni Saa” (from low to high pitch) and as per Western music it is “Do Re Me Fa So La Ti Do” (from low to high pitch). Flute is a wonderful example of path differences and its concerned output..

When the air is blown and if the orifice that is farthest from the mouth-piece is open then the sound that will come out will be of high-pitch that is “Saa/Do”. Eventually, when orifice next to farthest will be open along with all other orifices closed, the sound will be of “Ni/Ti.” When both orifices will be kept open then the sound will be the pattern of “Ni Saa” or “Ti Do.” Similarly, for all different tones and pitches, the path for the acoustic waves to travel from the mouth-piece to the orifices play a very significant role in the pattern-making of the outcome to be, here, music!!

There are still many more fascination real world examples but I will wrap it upto this only..

CONCLUDING REMARKS

Some concluding remarks and points:

1) Hamilton’s Principle is very much accountable with/for Linear Non-holonomic constraint types.

2) The information provided by Hamilton’s Principle is not enough to fully understand Non-holonomic constraints. Thus, relating each other for sub-types will be rather incomplete to estimate!

3) Hamilton's Principle as in the Variational Principle that Hamilton's Action is Extremized, strictly speaking, no such "Nonholonomic Constrained Action Principle" exists, except in the special case that the "nonholonomic constraints" are "integrable" which in itself is self-contradictory. That is they are "disguised" holonomic constraints in non-holonomic forms which creates mathematical null-spaces which are not unique.