Tensor-Centric Warfare IV : Kähler Dynamics of Battlefields

This paper presents the complex dynamics synthesis of the combat dynamics series called tensor-centric warfare (TCW; for the first three parts of the series, see [1] [2] [3]), which includes tensor generalization of classical Lanchester-type combat equations, entropic Lie-dragging and commutators for modeling warfare uncertainty and symmetry, and various delta-strikes and missiles (both deterministic and random). The present paper gives a unique synthesis of the Red vs. Blue vectorfields into a single complex battle-vectorfield, using dynamics on Kähler manifolds as a rigorous framework for extending the TCW concept. The global Kähler dynamics framework, with its rigorous underpinning called the Kähler-Ricci flow, provides not only a new insight into the “geometry of warfare”, but also into the “physics of warfare”, in terms of Lagrangian and Hamiltonian structures of the battlefields. It also provides a convenient and efficient computational framework for entropic wargaming.


Introduction
In the series of papers called the tensor-centric warfare (TCW; see [1] [2] [3]) we have developed a tensor architecture for general Red-Blue combat dynamics.
The TCW framework starts by providing tensorial generalization of the Lanchester-type combat equations [1], and then includes entropic Lie-dragging (for modeling warfare uncertainty) and commutators (for analysis of warfare V. Ivancevic [2], as well as various kinds of delta-strikes and missiles [3].The whole TCW architecture is defined by the following pair of Red-and-Blue tensorial dynamical systems (formally, the pair of Red-Blue vectorfields): ( ) In Equations (1) the Red and Blue Hamilton-Langevin delta strikes, (weighted by the random noise; see [3] for details), read: In the Red-Blue Equations ( 1 C defined via bipartite and tripartite adjacency matrices, respectively defining Red and Blue aircraft formations (according to the aircraft-combat scenario from [4] and [1]);  Quadratic Lanchester-type terms, , so that the battlefield entropy is conserved, 0 t S ∂ = (see [2]); , for modeling warfare symmetry (see [2]); • Hamilton-Langevin delta strikes, ( ) ( ) , on both sides, including discrete striking spectra (slow-fire missiles) and continuous striking spectra (rapid-fire missiles), as well as bidirectional random strikes, Hamiltonian vectorfields, self-dissipation, opponent-caused dissipation and non-delta random forces (see [3]).
The first three models of the TCW series have been developed on the Red and Blue configuration manifolds,

Kähler Dynamics of Battlefields
The concept of Kähler dynamics, or tensor dynamics on Kähler manifolds (see Appendix 1 for a technical exposition), has been formally developed in [5], based on our previous work on self-organization entropy [6], controllable complexity [7] and autonomy of cyber-physical-cognitive systems [8].
Briefly, the Kähler dynamics is defined by the complex-valued vectorfield: which flows along the Kähler battle-manifold  defined by the complexified sum (i.e., the sum with the imaginary unit:

Geometry of Warfare: Kähler Battle-Vectorfield
Our Kähler dynamics of the battlefield is defined as a complex-valued nD vectorfield ( ) , called the Kähler battle-vectorfield, flowing along the Kähler battle-manifold  and defined in the following two steps.Firstly, the above two real-valued Red-and-Blue vectorfields (1) can be rewritten in terms of the real and imaginary components of a single complex-valued vectorfield, defined on  as: Secondly, from the split real-Red and imaginary-Blue vectorfields, ( ) where ∈ is the unique complex vector.Its time derivative, ( ) , is our main actor, the battle-vectorfield, defined as the mapping from the Kähler battle-manifold  to the complex plane  : .
, defined by the complex-valued system of tensor differential Equations ( 4), represents a dynamical game played on the Kähler battle-manifold  , in which the actors are the following complex tensors: The promised rigorous geometric underpinning of the battle-vectorfield ( ) , as:  ( ) ( ) The solutions of these four KR equations are called the KR solitons.They uniquely exist in the case of Kähler-Einstein metric: ij R g λ = , for some real constant λ .KR solitons can be threefold: shrinking (if For more technical details on the Kähler-Ricci Intelligent Control and Automation flow, see Appendix 1.2.
In summary, the proposed Red-Blue combat dynamics model: is defined on the Kähler battle-manifold  by a single battle-vectorfield: which is underpinned by the geometric Kähler-Ricci flow on  :

Physics of Warfare: Lagrangian and Hamiltonian Structures of the Red and Blue Forces
Now we give a physical interpretation of warfare, using geometric insights from the Kähler dynamics provided above (and in the Appendix 1).The Kähler battle-manifold ( ) , g  , with the fundamental complex structure defined by its , includes the Riemannian structure R g (for the Red force) and the symplectic structure S ω (for the Blue force)-or vice versa.
The Riemannian structure, Next, we recall that general forced-and-dissipative mechanics (see, e.g.[5] and the references therein) in Lagrangian form reads: and in Hamiltonian form reads: , , where new ( ) , , x p q -indices denote partial derivatives (which is common with PDEs), a F represents the covector of generalized external forces and the scalar function Φ , given by the mappings ( ) By comparing the general forced-and-dissipative mechanics with our Red and Blue vectorfields, we make the following two observations.Firstly, we can see that there are no any covectors of external forces a F in the Red and Blue (pure velocity) vectorfields, so we can reduce our matches to: ( ) Secondly, since the Red and Blue vectorfields are generalized from classical Lanchester equations (which are the 1st-order ODEs), there are no any covectors of inertial (internal) forces either.In other words, the Red and Blue vectorfields physically correspond to dynamics of highly viscous/dissipative fluids, in which inertial forces can be neglected, so we can make the second reduction as: ( )

Computational Wargame: "Entropy Battle"
The computational wargame called the Entropy Battle (see Figure 1) is currently being implemented in C# language (on.Net 4.7), using Irrlicht 3D graphics engine and Bullet 3D physics engine, and implementing the metaphysics of wargaming outlined in the next subsection.The core version of the wargame simulates the aircraft battle scenario from [4] and [1] with 30 aircraft on each Intelligent Control and Automation , moving/flying predominantly in the horizontal complex plane  .It is numerically solved in adaptive time steps using the complex-valued RKF45 (Cash-Karp) integrator, which is fast, accurate and almost symplectic.

The extended version of the Entropy Battle wargame has two levels:
Top level is the core aircraft battle, and Bottom level has two scenarios (both formally defined by a simplified version of the battle-vectorfield ( ) , a a t V z t ∂ moving in the complex plane):  Land battle between Red and Blue land vehicles, and  Sea battle between Red and Blue boats.
In both cases, the Entropy Battle wargame follows the general metaphysics of wargaming outlined as follows.

Metaphysics of Wargaming: Warfare Entropy and "Combat Signatures" in the Battlespace
• The stage for combat dynamics is the Red-Blue battlespace, which can be modeled by a dynamical concept of the phase-space.From a bird-view (or, from God's Eye), the phase-space reduces to its 2D order-parameter subspace, the Red-Blue phase-plane, which is usually used in simulations.
• The concept of the phase-space (in our case spanned by the Red and Blue There is no chaos in the 2D phase plane (theorem), but the entropy growth is still observable, since, e.g., the Kolmogorov-Sinai entropy is a sum of all Lyapunov exponents (both positive-chaotic and negativenonchaotic).
• We can assume that, at least in a short time interval, the Red-Blue combat dynamics in the battlespace is conservative (no energy sources or sinks).
Therefore, for a short time period, all combat dynamics can be derived from the so-called battle Hamiltonian (total combat energy function in an isolated region of battlespace)-at a certain entropy level.In the next short time period, we again have the conservative combat dynamics-at a higher entropy level, etc.
• Generalization/relaxation of Liouville's theorem: the so-called Hamilton-Langevin framework has been proposed in [3], to include: delta-strikes, dissipation and random external forces.If the magnitude of these non-

Conclusion
We have presented the Kähler dynamics approach to battlefields.It is the complex-dynamics synthesis of the combat dynamics series called the tensor-centric warfare, which includes tensor generalization of classical Lanchester-type combat equations, entropic Lie-dragging for modeling warfare uncertainty and symmetry, various (both deterministic and random) delta-strikes and missiles, and deep-learning at the battlefield.This synthesis is performed in the form of the

Appendix: Kähler Manifolds and Kähler-Ricci Flow
In this section, we give a brief review of Kähler manifolds (the main reference is [9]) and the Kähler-Ricci flow on them, which constitutes the geometric framework for the complex Red-Blue battle-vectorfield.

Geometry and Dynamics of Kähler Manifolds
Let n =   be a compact (i.e., closed and bounded) complex n-manifold 5 of complex dimension n (see [9] [10]).To be able to write various tensors on the manifold  , we chose a local point p ∈  with the neighborhood chart U that includes: 1) the holomorphic coordinates and their complex-conjugates: ) the natural basis of vectorfields in the tangent space p T  at p: { } and 3) the dual basis of co-vectorfields (i.e., holomorphic 1-forms) 6 in the cotangent space p T *  at p: { } To make the complex manifold ( ) ( )    into a Kähler n-manifold, we need to specify on it a Kähler metric g and its associated Kähler form ω , as follows (compare with [11] [12] [13] [14]).Consider a Hermitian metric 7 g defined at each point { } , for , 1, , i j n =  , as: 8   5   Recall that a complex-valued function satisfies the Cauchy-Riemann relations for each holomorphic coordinate, ( ) , , . .An almost complex structure J is defined on a complex manifold  as: The is a complex manifold  with a Hermitian metric tensor ( ) is a positive-definite Hermitian matrix function, , g  becomes a Kähler manifold iff the almost complex structure J on it satisfies the condition: , defined later in this section. 8 The metric g given by ( 7) has the following dynamical interpretation: it defines the complex kinetic-energy Lagrangian ( ) from which the conservative Lagrangian equations read in the contravariant form:  and in the standard covariant form (with indices denoting partial derivatives): such that ( ) ij g is a positive-definite Hermitian matrix.Its inverse ij g is given by the matrix ( ) ( ) . Associated to the Hermitian metric g, there is a real positive-definite exterior (1,1)-form ( ) on  , defined by: 9 i 0.
If the form ω is closed, 0 dω = , then g is called the Kähler metric and ω is called the Kähler form.The fundamental closure condition: 0 dω = is called the Kähler condition, the global condition for any Kähler manifold  , which is locally in ( ) z z U ∈ ⊂  equivalent to the following metric symme- tries: 10   and , (independent of the choice of local holomorphic coordinates ( ) From the Kähler form (8) with i z as canonical momenta, the Hamiltonian formalism can be derived from the complex kinetic-energy Hamiltonian ( ) from which the conservative Hamiltonian equations read (with indices denoting partial derivatives): , .
g dz dz g dz dz dz g dz dz dz g g dz dz dz g g dz dz dz which implies: and .: : ( They play the major role in the famous " ∂∂ -lemma" and the associated Dolbeault cohomology (from the Hodge theory).Any function i , i , such that : 0, 0, 0. i i i .
We remark that the two differential expressions with the Kähler potential ϕ , 13  and i g is another holomorphic coordinate system on  , then on the overlap { } { } The Kähler metric g induces the Levi-Civita connection on ( ) , which act on smooth functions f on  as: and consequently, the solutions of the Einstein equation: for some real constant λ ) can be defined by solving the scalar equation: ∇ ∇ act in the following way: 15   , , In general, the Christoffel symbols i jk Γ are chosen so that both covariant derivatives of the metric tensor vanish: Hermitian manifold ( ) , g  is a Kähler manifold iff the almost complex structure J satisfies: The Laplacian (or, rather Laplace-Beltrami) operator ∆ is defined in local coordinates ( ) ∆ -action on smooth functions f ∈  is given by: ( ) ⋅ is the trace operator (i.e., contraction with ji g ). 16More generally, ∆ -action on an arbitrary tensor T is defined in normal coordinates 17   for g on ( ) , g  as: ( ) A Kähler metric g defines a corresponding Riemannian metric R g on ( ) , g  , defined via its real and imaginary parts, as follows.In local coordinates 15 We can extend the covariant derivatives ( ) ( )   , , to act naturally on any type of tensor field on ( ) , g  ; e.g., in case of covariant 2-tensors ij S and ij S , we have: Similarly, if T is a mixed 3-tensor with components ij k T then we define its two covariant derivatives , as: For example, the trace ( ) Tr ⋅ of a real ( ) ∈ is defined by: ( ) Here, n ω is the volume form on  (see below), such that for any smooth function f ∈  , its L p -norm (with respect to a Kähler metric g) is given by: ( ) A normal coordinate system for g centered at a point p ∈  is a holomorphic coordinate system that satisfies: z x y = + , so that ( ) The Riemann curvature tensor Rm of the Kähler metric ( ) is very simply defined in two forms, mixed and covariant, respectively: and .
Using ( 9) and ( 10), we have locally (in an open chart { } The Riemann curvature tensor i jkl R on ( ) , g  has the following three symmetries: 18   1 For any two nonzero vectorfields ( ) , v u on T  , we say that ( ) , g  has positive holomorphic bisectional curvature and positive holomorphic sectional curvature, respectively, if 0 and 0.
The trace of the Riemann curvature tensor i jkl R is the Ricci curvature tensor Rc, defined as: ,  Similarly, the trace of the Ricci curvature is the scalar curvature: A Kähler metric g defines a pointwise norm g ⋅ on any tensor field on ( ) , g  ; e.g., the squared norm of functions f on  reads: and similarly for the vectorfields ( ) α β * ∈  we have: 19   18   While the symmetries 1) and 2) follow immediately from ( 9) and (11), to show 3) we need to compute at a point ( ) The norm g ⋅ extends to any tensor field on  ; e.g., if T is a tensor with components ij k T , its squared norm is given by: , .
Associated to the Ricci curvature tensor Rc is the Ricci form, ( ) ( ) , a real closed (1,1)-form on  , similar to the Kähler form ω , given by: ( ) ( ) ( ) which implies that ( ) Ric ω is closed: ( ) The Riemann curvature tensor i jkl R arises when commuting covariant derivatives ( ) ( )   , , . Using the standard commutator definition: , , we have the following commutation formulae for the vectorfields ( ) , v u on T  and co-vectorfields ( ) which can naturally be extended to tensors of any type on  .Also, when acting on any tensor, the covariant derivatives commute as: , , 0 Now we come to the essential notion of cohomology of a Kähler manifold ( ) ,ω  , which is defined using the formalism of ( ) defined as the quotient space: 23   20   The closure relation between these three derivative operators reads: ( ) ( ) ( ) For general p-forms, the de Rham cohomology group ( ) is defined as the kernel over the image of the exterior derivative d: Ker : , .Im : This is the ∂ -Poincaré lemma for complex manifolds, which says that a ∂ -closed (0,1)--form is locally ∂ -exact.

23
For general ( ) , p q -forms, the Dolbeault cohomology group ( ) is defined as the kernel over the image of the ∂ -operator: , .Im : In other words, a real ( )  The simplest example of the Kähler potential is the case of the simplest Kähler n-manifold n  -the complex Euclid n-space, in which n ϕ ∈  is given by: ( ) Another standard example is the following entropic Kähler potential (see [16]) with its Taylor expansion: The ∂∂ -lemma is an extension of the two Poincaré lemmas (for d and ∂ ).Basically, the ∂∂ -lemma says that a closed ( )
In particular, the extensions of the Kähler-Einstein (KE) metric: ( ) ( ) At the end of this section, we remark that it was shown by [17] [23] that the KRF (19) can be rewritten as the (parabolic, complex) Monge-Ampère equation: 33 is defined by: 33 On n  the complex Monge-Ampère operator for the smooth potential ϕ is defined as the de- terminant of the complex Hessian: ( ) ( ) 2 det det , with 0.
However, on a compact Kähler manifold ( ) , g  , the condition ( ) would imply that ϕ is constant, so the Hessian ( ) For more technical details on the Kähler-Ricci flow, see e.g., [12] [14] and the references therein.
)-(2), t t ∂ ≡ ∂ ∂ and the Red and Blue forces are defined as vectors following terms (placed on the right-hand side of Equations (1)):  Linear Lanchester-type terms, and operational capabilities of the Red and Blue forces (see[1]);  Entropic Lie-dragging of the opposite side terms, -equilibrium battlefield entropy grows, 0 t S ∂ > ; in case of non-resistance, the Lie derivatives vanish,

M
, intentionally without specifying any geometric structures on these manifolds.In the present paper, we use the most sophisticated geometric structure of Kähler manifolds, which allows development of both Lagrangian and Hamiltonian formalisms on it.Here we summarize and reformulate the two Red-and-Blue dynamical systems (1)-(2) in the form of a unique Kähler dynamical system, together with its specific geometrical underpinning called the Kähler-Ricci flow.This sophisticated geometric framework gives a new insight into deep mathematical and physical structures of battlefields and also provides a convenient computational wargaming framework.

)
In the form of the Monge-Ampère equation (with Dolbeault's ( ) dynamics for the Red force, derived from the Lagrangian energy function, dynamics for the Blue force, derived from the Hamiltonian energy function, T M * →  -or vice versa.
for Hamiltonian mechanics) represents Rayleigh's dissipation function (describing internal frictional forces proportional to velocity).So, let us try to formally match the Red and Blue vectorfields from Equations (1) with the general Lagrangian Equations (5) and the general Hamiltonian Equations (6): V. Ivancevic et al.DOI: 10.4236/ica.2018.94010129 Intelligent Control and Automation ( ) since our Red and Blue vectorfields are pure velocity-vectorfields without internal or external force co-vectorfields, both Lagrangian and Hamiltonian equations are reduced to the 1st-order systems of ODEs: in Lagrangian formulation the 2nd-order (inertial force) term vanishes, and in Hamiltonian formulation the whole force equation vanishes (momenta still exist but their time derivatives vanish).

Figure 1 .
Figure 1.A prototype of the computational TCW-wargame called the Entropy Battle.The wargame is currently under development in the Joint and Operations Analysis Division, Defence Science & Technology Group, Australia, using the C# language, with Irrlicht 3D graphics engine and Bullet 3D physics engine.It implements a simplified version of the battle-vectorfield

V•
. Ivancevic et al.DOI: 10.4236/ica.2018.94010131 Intelligent Control and Automation forces) comes from Hamiltonian mechanics (when W.R. Hamilton formally unified Lagrangian mechanics and optics).It is also used in statistical mechanics.Besides, the 2D phase plane was the main analytical tool of H. Poincaré in his qualitative analysis of differential equations, from which both topology and chaos theory emerged.Finally, L. Boltzmann defined the entropy by coarse-graining the phase space.Every kind of entropy (including Boltzmann, Gibbs, Shannon, Kolmogorov-Sinai, Rényi, Bekenstein-Hawking, Kosko fuzzy, entanglement, topological, partition-function based, path-integral based, etc.) is essentially a logarithm of some more fundamental underlying (probabilistic, phase space or topological) measure, therefore it is itself an additive measure, which in our combat case gives: Total combat entropy Red-entropy Blue-entropy.= + The cornerstone of Hamiltonian and statistical mechanics (as well as ergodic dynamics) is the key concept related to the Warfare Entropy and "Combat Signatures" in the Battlespace.It is the famous Liouville's theorem: The flow of a conservative Hamiltonian vectorfield preserves the phase-space Volume (technically, Hamiltonian flow is a symplectomorphism: the Lie derivative of the volume form: dRed^dBlue along the [Red, Blue] vectorfield vanishes).This volume preservation necessarily implies various shape distortions (called "combat signatures") and therefore uncertainty!• Liouville's theorem-based interpretation of the Warfare Entropy and "Combat Signatures": If dynamics in the Red-Blue phase-plane stretches in the Red direction, it necessarily shrinks in the Blue direction, and vice versa.The stretching and shrinking distortions of the Combat Area cause rapid entropy growth and combat signatures in the 2D phase-plane.More generally, in higher Red-Blue phase-space dimensions, Liouville's theorem causes Hamiltonian chaos, because there are so many possible ways for stretching and shrinking, each one reflected by entropy growth.
complex battle-vectorfield, defined using the global framework of Kähler battle-manifolds.The proposed Red-Blue combat dynamics model is defined on the Kähler battle-manifold by a unique battle-vectorfield which is underpinned by the geometric Kähler-Ricci flow.This complex synthesis gives a new insight into the "physics of warfare" in terms of "hidden" Lagrangian and Hamiltonian structures of the battlefields.It also provides a convenient and efficient computational framework for entropic wargaming, in which the Entropy Battle is currently under development.

Ω
 of exterior p-forms on the complex manifold  , which generalizes standard gradient, curl and divergence operators from vector calculus.Being the additive components of the exterior derivative: d = ∂ + ∂ , so that df f f = ∂ + ∂ for any smooth real function f on  , Dolbeault's differential operators ( ) , ∂ ∂ are the maps on the space Cauchy-Riemann equations).In a local k z -coordinate chart U ⊂  , for any holomorphic function f U ∈ , ( ) , ∂ ∂ -operators are given by: T and co-vectorfields 12 Potential definition of the Kähler metric, ij g ϕ = ∂∂ , implies that the Ricci curvature ( )

13 For
a real function : ϕ →   the real (1,1)-form i i j k ϕ ϕ ∂∂ ≡ ∂ ∂is the complex Hessian of ϕ .14 The Christoffel symbols i jk Γ on a Kähler manifold  do not have their conjugate part.Because of this unique feature of Kähler manifolds, the Riemann and Ricci curvature tensors (defined below) are much simpler on  than in standard Riemannian geometry.Intelligent Control and Automation ( ) , α β on T *  , the covariant derivatives ( ) , k k T T =which in case of the Ricci tensor Rc and the Riemann tensor Rm become, respectively:

.
form ω on  then we say that α is a Kähler class (and write 0 α > ).Therefore, the Kähler class of ω is its coho- Usually, all this is simply written: the Kähler class of ω is the cohomology class [ ] ( )

1 c
 , defined as the cohomology class of the Ricci form:

(
manifold, the basic geometric object in string theory.a Ricci-flat metric.In that 27 are the Kähler-Ricci (KR) solitons: a time-dependent Kähler metric is called a gradient KR soliton if there exists a real smooth Kähler potential ϕ on ( ) Similar to the real Ricci flow case, this soliton is called shrinking if 0 and the gradient vectorfield ϕ ∇ is holomorphic.If the Kähler manifold ( ) , g  admits a KE metric (or, a KR soliton) g then the first Chern class


with scalar curvature R and for any smooth function u on  , the Kähler-Perelman (KP) entropy et al.

•
Entropic Red-Blue commutators, DOI: 10.4236/ica.2018.94010125 Intelligent Control and Automation is not overwhelming, the entropic stretchingand-shrinking effect of Liouville's theorem is still visible.•Whileslowchanges of the battlefield are governed by Liouville's theorem, fast changes are governed by Onsager 2 -Prigogine's 3 entropic, irreversible, non-equilibrium thermodynamics with the arrow-of-time.4Suddenentropy growths in open combat Red-Blue systems reflect sudden energy dissipations due to impulsive Red-Blue crashes.
are Dolbeault's differential operators, which are the additive components of the exterior derivative (de Rham differential) d on  : d = ∂ + ∂ .11Inthat case, as shown by E. Kähler j ∂ ≡ ∂ Ivancevic et al.
which implies that the Christoffel symbols i 26An immediate consequence is that if ω and ϕ ω are two Kähler forms in the same for some smooth Kähler potential ϕ (which is uniquely determined up to a constant).In other words, two Kähler metrics ij g and ij g  on ( )