Geometric Dynamics on Riemannian Manifolds

: The purpose of this paper is threefold: (i) to highlight the second order ordinary differential equations (ODEs) as generated by ﬂows and Riemannian metrics (decomposable single-time dynamics); (ii) to analyze the second order partial differential equations (PDEs) as generated by multi-time ﬂows and pairs of Riemannian metrics (decomposable multi-time dynamics); (iii) to emphasise second order PDEs as generated by m -distributions and pairs of Riemannian metrics (decomposable multi-time dynamics). We detail ﬁve signiﬁcant decomposed dynamics: (i) the motion of the four outer planets relative to the sun ﬁxed by a Hamiltonian, (ii) the motion in a closed Newmann economical system ﬁxed by a Hamiltonian, (iii) electromagnetic geometric dynamics, (iv) Bessel motion generated by a ﬂow together with an Euclidean metric (created motion), (v) sinh-Gordon bi-time motion generated by a bi-ﬂow and two Euclidean metrics (created motion). Our analysis is based on some least squares Lagrangians and shows that there are dynamics that can be split into ﬂows and motions transversal to the ﬂows.


Statement of Geometric Dynamics Problems
The subject of dynamical systems concerns the evolution of systems in single-time or multi-time (multivariate). In continuous single-time, the systems may be modeled by ODEs, PDEs, or other type of equations. In the case of ODEs the phase space is finite-dimensional and for PDEs the phase space is infinite-dimensional. In continuous multi-time, the systems may be modeled by PDEs or other type of equations.
The evolution parameter need not be the physical time or the time vector; for example, a time-stationary solution of a PDE is parameterized by spatial variables.
The fundamental problem discussed in this paper is threefold: (i) let us highlight again the technique of transforming a flow into dynamics by the geometry of the space in which the flow takes place, (ii) emphasize the decomposability of some important motions (dynamics) into flow trajectories and transversal to flow trajectories (depending on initial conditions); (iii) extend the previous ideas to harmonic maps between a source manifold and a target manifold. Example 1. Equations of mechanics may appear different in form fromẋ(t) = X(x(t)), as they often involve higher time derivatives, but an equation that is second or higher order in time can always be rewritten as a set of first order equations.
The ODEs of the form F(x(t),ẋ(t),ẍ(t), ... x (t)) = 0 which contain third order derivatives in them, are sometimes called jerk equations. It has been shown that a jerk equation is in a mathematically well defined sense the minimal setting for solutions showing chaotic behaviour. A jerk equation is equivalent to a system of three first-order ordinary non-linear differential equationṡ x(t) = y(t),ẏ(t) = z(t),ż(t) = φ(x(t), y(t), z(t)).
The extremals are solutions of the Euler-Lagrange ODE system If the Lagrangian L i is associated to ODE L i (t, x(t),ẋ(t)) = 0, then the extremals contain the solutions of that equation and the dynamics is decomposable.

Example 2.
Let u(x, t) be the density of the diffusing material at location x ∈ R n and time t ∈ R. Let g ij (u(x, t), x), i, j = 1, n, be the collective diffusion coefficient (matrix) for density u at location x. The diffusion PDE is . If the diffusion coefficient depends on the density, then the diffusion equation is nonlinear, otherwise it is linear. More generally, when g ij (u(x, t), x) is a symmetric positive definite matrix (Riemannian metric), the equation describes anisotropic diffusion.
The diffusion PDE is equivalent to the first-order non-linear PDEs where the parameter of evolution (x, t) is (n + 1)-dimensional. A Riemannian metric h ij (u(x, t), x) produces a least squares Lagrangian on the jet space of coordinates (x, t, u, v, u x , u t , v x , v t ). It appears the associated geometric dynamics (Euler-Lagrange equations) Example 3. Let T be an orientable manifold with the coordinates t = (t 1 , ..., t m ) and M be a manifold with the coordinates x = (x 1 , ..., x n ). Using m vector fields X α (t, x) of class C ∞ on T × M, we introduce the distribution described by Pfaff equations Using some metric tensors h αβ (t), g ij (t), and the components ∂x i ∂t α (t) − X i α (t, x) of the pullbacks, we build the least squares Lagrangian Suppose the integral manifolds of the distribution have the dimension 1 ≤ p < m. Then we can introduce another least squares Lagrangian constructed from ODEs/PDEs that describes the integral manifolds and the action is an integral with the volume element on the p parameters which define the integral manifold (decomposable dynamics).
More generally, being given nm Lagrangians L i α (t, x(t), x γ (t)), i = 1, n, α = 1, m, x(t) = (x 1 (t), ..., x n (t)), t = (t 1 , ..., t m ) ∈ I ⊂ T, then the associated least squares Lagrangian density with respect to the Riemannian metrics If T ⊂ R m , the extremals are solutions of the Euler-Lagrange PDE system If the Lagrangian L i α is associated to the PDE L i α (t, x(t), x γ (t)) = 0, then the extremals contain the solutions of that equation and the dynamics is decomposable.
The ingredients needed to solve these problems are the Riemannian metrics, techniques of least squares Lagrangians and the idea of dynamics transversal decomposition.
The topics of the papers that inspired us in developing the theory of decomposable dynamics can be classified as follows: Applied ODEs and PDEs [11][12][13][14] are evolution equations modeling systems evolving with respect to a "time" parameter. When solving such evolution equations, the appropriate formulation of the problem is usually as an initial value, or Cauchy, problem. More specifically, certain initial data are given, representing the state of the system at some initial time. The goal, then, is to "predict the future", that is, to find the solution of the ODE or PDE, which represents the behaviour of the system at all moments.
Hamiltonian approach [15][16][17] of dynamics is sometimes more subtle than the Lagrangian approach because this point of view changes the dynamics into a Hamiltonian flow. Hamiltonian dynamics are based on ODEs or PDEs of the first order constructed from ODEs or PDEs of second order Euler-Lagrange equations. The transition from the Euler-Lagrange-type equations to Hamilton-type first order equations is based on the Legendre transformation.
Dynamics, winds and flows [18,19] are often mysteriously coupled. The most interesting cases are those of decomposable dynamics or winds.
Variational principles [20] are alternative methods for determining the state or dynamics of a physical system, by identifying it as a critical point (minimum, maximum or saddle point) of a functional.
Combining the previous ideas with the thought that certain flows accompanied by space geometry (Riemannian metric and Riemannian connection -derivation), generate what is now called the geometric dynamics. This geometric dynamics have been discovered by us first for magnetic flow and then for any other flow. It is in fact a geodesic motion in a gyroscopic field of forces. In time the ideas were extended by our research team to m-flows and to m-distributions [1][2][3][4][5][6][7][8].
The main aim of this paper is to give necessary and sufficient conditions for the decomposition of a general (single-time or multi-time) dynamics into a flow and a transversal movement.

First and Second Order ODEs on Manifolds
Let M be a differentiable manifold and I ⊂ R be a nontrivial interval. A (time dependent) non-autonomous first order differential equation on a manifold M [13] is given by , identically on I. The Cauchy problem is the following: find a solution of the ODE (1) which satisfies the initial condition x(t 0 ) = x 0 . Answer: the solution of this Cauchy problem exists and it is unique.
Let F : R × TM → R n be a C ∞ map. An equality of the typë is called a (time dependent) second order differential equation on M [13], provided that the associated vector field is tangent to TM, i.e., (y, F(t, x, y)) ∈ T (x,y) TM for all (t, x, y) ∈ R × TM. A solution of the differential Equation (2) is a C 2 curve x : I → R n , in such a way that x(t) ∈ M andẍ(t) = F(t, x(t),ẋ(t)), identically on I. The Cauchy problem: find a solution of the ODE (2) which satisfies the initial conditions Answer: the solution of this Cauchy problem exists and it is unique. If we use the components, the relations (1) and (2) are called respectively first order and second order ODE systems.

Single-Time Geometric Dynamics
We start with the triple (M, g, X), where M is a manifold of dimension n, g(x) = (g ij (x)), i, j = 1, ..., n, is a Riemannian metric and X(t, x) = (X i (t, x)) a time dependent C ∞ vector field, on the manifold M. Suppose the Levy-Civita connection ∇ of (M, g) has the components G i jk , i, j, k = 1, ..., n.

Definition 1.
We use the notations A function F : R × TM → R n is said to be generated by the pair (X, g) if it is of the form If F is generated by X and g, then the ODE (2) represents a single-time geometric dynamics or a geodesic motion in a gyroscopic field of forces [1][2][3][4][5][6][7][8]. By analogy with the reduction of the force system in mechanics, resultant and momentum, the decomposition of the set of solutions returns to the flow and the movement in the gyroscopic field of forces. Theorem 1. If F : R × TM → R n is generated by the pair (X, g), then the set of maximal solutions of ODE (2) is decomposable into a subset corresponding to the initial values solutions which are reducible to solutions of the ODE (1), and a subset of solutions corresponding to the initial values transversal to the solutions of the ODE (1). The converse is also true.
Proof. Based on existence and uniquennes theorem, each solution x = x(t) of any second order prolongation of first order ODE system has the property: A flow X and a Riemannian metric g determines a least squares Lagrangian The Euler-Lagrange ODEs represent a geometric prolongation of the flow. The Euler-Lagrange ODEs constitute just a decomposable dynamics (geodesic motion in a gyroscopic fields of forces)≡ (set of flow trajectories)⊕ (set of transversal trajectories imposed by the geometry of the space).

Theorem 2.
Suppose that X is an autonomous vector field. If the function F : TM → R n is generated by X and g, then the set of maximal solutions of ODE (2) splits into three categories: Proof. These statements are based on the Hamiltonian and the associated Hamilton ODEs.

Remark 1.
(i) Any normal ODE generates in the phase space a flow, which together with the phase space geometry give a geometric dynamics. This statement is true for any ODE, but then appears a flow with constraint.
(ii) Let us consider the quadruple (M, X, g, Γ), where M is a manifold, X is a flow on M, g is a fundamental tensor field and Γ is a symmetric connection (derivation). The triple (X, g, Γ) generates an extended geometric dynamics on M determined by ODEs (iii) On the Riemannian manifold ((0, ∞), g(x) = 1), let us take the flowẋ = 1. We attach the least squares Lagrangian L 1 = (ẋ − 1) 2 , with Euler-Lagrange equationẍ = 0. On any other Riemannian manifold ((0, ∞), g(x)), we find the least squares Lagrangian L 2 = g(x)(ẋ − 1) 2 , with Euler-Lagrange equation 2g(x) is a linear connection. We can extend the previous ODE to the ODE systemẍ with possible chaos in velocities.

Fundamental Tensor Field
Let M be a differentiable manifold of dimension n and I ⊂ R be a nontrivial interval. If the ODE system (2) is an Euler-Lagrange system on M for a regular Lagrangian L(t, x,ẋ), then there exists a fundamental d-tensor field g = (g ij ) on TM such that Conversely, given g ij (t, x,ẋ), to determine L(t, x,ẋ), we need complete integrability conditions. In these conditions, using two successive curvilinear integrals of the second type, we can write The pair (M, g) is called a Lagrangian manifold.

First Examples of Generated Geometric Dynamics
Let us show that the movement of planets and motion in closed Newmann economical systems are generated by flows and Riemannian metrics.
Given a function u(x), x ∈ M and a Riemannian metric g on M, let us consider the Hamiltonian H = 1 2 g(ẋ,ẋ) − u(x), as in [5]. If u(x) > 0, then the vector field X(x) = 2u(x) E(x) (Galilei formula), where E is an arbitrary unit vector field with respect to the metric g, satisfies g(X, X) = 2u(x). Consequently such a Hamiltonian, equal to the difference between the kinetic energy and a positive function, is coming from a vector field (flow) and a Riemannian metric, corresponding to a perfect square Lagrangian. Theorem 3. If u is positive, then the motion described by the Hamiltonian H = 1 2 g(ẋ,ẋ) − u(x) is generated by a flow and a Riemannian metric.

Motion of the Four Outer Planets
Let m 1 , m 2 , m 3 , m 4 be the masses of the four outer planets (Jupiter, Saturn, Uranus, Neptune), relative to the sun mass m 0 = 1. They produce an Euclidean metric g = diag(m 0 , m 1 , m 2 , m 3 , m 4 ). Corollary 1. The motion of the four outer planets relative to the sun is generated by a flow E(q) and the Euclidean metric g.

Proof. The motion of the four outer planets relative to the sun is described by the Hamiltonian
where p = (p 0 , p 1 , p 2 , p 3 , p 4 ) and q = (q 0 , q 1 , q 2 , q 3 , q 4 ) are velocity and position (supervectors) with p i , q i ∈ R 3 , G is the gravitational constant, and m 1 , m 2 , m 3 , m 4 are masses relative to the sun mass m 0 = 1. We apply the previous statement with a generic versor where ϕ i = ϕ i (q). One associated flow requires the fixing of the versor field E(q).
Since these planets are rather large, their orbits can affect one another (and possibly even the Sun).

Motion in Closed Newmann Economical Systems
A closed economical system is one that has no trade activity with outside economies. The closed economical system is self-sufficient, that means no imports come into the system and no exports leave the system.
We apply the previous statement with a generic versor field.

Comparison between Lorentz's Law and Geometric Dynamics
Lorentz law Let x = (x 1 , x 2 , x 3 ) ∈ R 3 and t ∈ R. For the potential vector magnetic field A(x, t), a particle moving with velocity v =ẋ and particle charge e has the potential momentum eA(x, t), so its potential energy is eA(x, t) ·ẋ. For a φ(x, t) field, the particle's potential energy is eφ(x, t). Using the total potential energy V = eφ − eA ·ẋ and the kinetic energy T = m 2ẋ ·ẋ, we built the Lorentz Lagrangian It is well-known that the movement of a charged particle into an electromagnetic field is described by the Euler-Lagrange ODE system (universal Lorentz law) mẍ = e (E +ẋ × rot A) .

Geometric dynamics
For the preservation of traditional formulas, we will refer to the magnetic flow generated by the vector potential "−A", using homogeneous dimensional relationships. The magnetic trajectories are the solutions of the ODE system mẋ = −eA. This system together with the Euclidean metric produce the least squares Lagrangian The Euler-Lagrange ODEs of L 2 are is the energy density associated to the vector field A. In this way we obtain a single-time geometric dynamics, which is in fact a geodesic motion in a gyroscopic field of forces. The associated Hamiltonian is ·ẋ − e 2 f A .

Remark 2.
(i) Generally, the single-time geometric dynamics produced by the potential vector field "−A" is different from the classic universal Lorentz law because In other words the Lagrangians L 1 and L 2 are not in the same equivalence class of Lagrangians.
(ii) The magnetic force F = eẋ × rot A do no work on the moving charge, being a gyroscopic force (the mechanical work produced by F is zero). Any gyroscopic force has the same property.

Geometric Dynamics Induced by Bessel Kinematics
Let ODE be an arbitrary differential equation and g be a Riemannian metric. The pair (ODE, g) generates a geometric dynamics.
Of course, we are interested in meaningful pairs. For example, let us start by recalling the Bessel ODE,ẍ It is well known that Bessel functions are very important in many problems of wave propagation and static potentials. The associated Bessel flow (kinematics) iṡ on the Riemannian manifold (R 2 , δ ij ). Consequently the associated least squares Lagrangian is By a direct computation, we get the corresponding Euler-Lagrange equations These ODEs are the same as the Bessel geometric dynamics ODEs where x 1 := x, x 2 := y and f (t, x, y) := 1 2 X 2 is the energy of the vector field A direct calculation in (4) gives us the Euler-Lagrange ODEs. Therefore, we have Theorem 4. (see [5,7]) The second order ODEs system (4) is an Euler-Lagrange prolongation of the first order ODEs system (3) if and only if the Lagrangian L is of least squares type, modulo a total derivative term.

(5)
Theorem 5. (existence and uniqueness) [20] The Cauchy problem consisting in the PDE system (5) and the initial condition x(t 0 ) = x 0 has a unique solution if and only if the system is completely integrable.
An equality of the type is called a (time dependent) second order elliptical PDE (system) on M.
Let Γ : G(t) = 0 be a hypersurface in T, containing the point t 0 and Λ(t) be a unit vector field along Γ, transversal (non-tangent) to Γ. Denote ϕ 0 (t) and ϕ 1 (t) as vector functions with n components on Γ, the first being of class C 1 and the second of class C 0 . The Cauchy problem attached to PDE (6) is (see [14], p. 208): find, in an unilateral or bilateral neighborhood of Γ, the solution of the PDE (6) satisfying the Cauchy conditions Answer: the solution of this Cauchy problem exists and it is unique. Knowing the Cauchy conditions, one can find the values of all first order partial derivatives of the function x(t) on the Cauchy surface Γ: firstly, ∂x ∂t α Γ = ∂ϕ 0 ∂t α (t), α = 1, ..., m − 1 and then the equalities Lemma 1. The initial conditions (7) are equivalent either to the initial conditions together with the complete integrability conditions and the compatibility condition to ϕ 0 .

Multi-Time Geometric Dynamics
The multi-time geometric dynamics was introduced in our papers [1,5,8] like Multi-time World Force Law involving field potentials (components of the d-tensor), gravitational potentials (components of the two Riemannian metrics), and the Yang-Mills potentials (components of the Riemannian connections and the nonlinear connection). This evolution can be called also harmonic maps deformation in a gyroscopic field of forces.

Definition 2.
Using the vector fields X α , the metric tensors h αβ , g ij , and the Christoffel symbols H α βγ , G i jk , we define The function F : J 1 (T, M) → R n is said to be generated by the triplet (X α , h, g) if it is of the form If F is generated by X α , h and g, then the PDE (4) represents a multi-time geometric dynamics. Suppose that the PDE system (3) is completely integrable (m-flow).
Theorem 6. If F : J 1 (T, M) → R n is generated by the triplet (X α , h, g), then the set of maximal solutions of PDE (6) is decomposable into a subset corresponding to the initial values solutions which are reducible to solutions of PDE (5), and a subset of solutions corresponding to the initial values transversal to the solutions of PDE (5). The converse is also true.

Proof.
Each solution x = x(t) of any second order prolongation of the first order PDE system has the property: x α (t 0 ) = X α (t 0 , x(t 0 )) implies x α (t) = X α (t, x(t)), ∀t ∈ T.
Any m-flow X α and two Riemannian metrics h and g determine a least squares Lagrangian density The Euler-Lagrange PDEs represent a prolongation of the m-flow and just a decomposable dynamics.
Remark 3. (i) Any normal PDE generates in the phase space a multidimensional flow, which together with the phase space geometry gives a geometric dynamics. This statement is true for any PDE, but then appears a multidimensional flow with constraints.
(ii) Let us consider the triple (T, h, H), where T is a manifold, h is a fundamental tensor field and H is a symmetric connection (derivation). We add the quadruple (M, X α , g, G), where M is a manifold, X α is an m-flow on M, g is a fundamental tensor field and G is a symmetric connection (derivation). The quintuple (X α ; h, H; g, G) generates an extended geometric dynamics on T × M.

Fundamental Tensor Field
Let M be a differentiable manifold of dimension n and I ⊂ R m be a nontrivial interval. If the ODE system (4) is an Euler-Lagrange system on M for a regular Lagrangian L(t, x, x γ ), then there exists a fundamental tensor field x, x γ ), i, j = 1, ..., n; α, β = 1, ..., m.
Conversely, given G αβ ij (t, x, x γ ), to determine L(t, x, x γ ), we need complete integrability conditions. In these conditions, using two successive curvilinear integrals of the second type, we can write The fundamental tensor field is said to be Kronecker decomposable if G = h ⊗ g. The pair (M, G) is called a Lagrangian manifold.

Geometric Dynamics Induced by sinh-Gordon Kinematics
This Section was elaborated in our research group. Any triple (PDE, h, g) generates a geometric dynamics, but we are interested in meaningful triples. For example, we use the sinh-Gordon equation ∂ 2 u ∂x∂t (x, t) = sinh u(x, t). The sinh-Gordon equation is a nonlinear partial differential equation that has applications in physics and hydrodynamics. It is known for its soliton solutions and arises as a special case of the Toda lattice equation.
Case 1 The equivalent sinh-Gordon diagonal flow (kinematics) is given by We attach a least squares Lagrangian on the Riemannian manifolds (R 2 , δ αβ ) and (R 2 , δ ij ). Its Euler-Lagrange PDEs are Case 2 A complete sinh-Gordon flow (kinematics) is given by This PDE system is completely integrable if and only if f x = sinh u, g t = u x cosh u. We use the least squares Lagrangian on the Riemannian manifolds (R 2 , δ αβ ) and (R 2 , δ ij ). The Euler-Lagrange equations associated to this Lagrangian are If the PDE system (10) is completely integrable, then the set of solutions of the Euler-Lagrange PDEs includes the set of solutions of PDE (10); otherwise, it does not. In the case when the PDE system (10) is not completely integrable, the solutions of the Euler-Lagrange PDE are solutions of the least square approximation for PDE (10).
On the other hand, the sinh-Gordon geometric dynamics is described by (see definition) Therefore, we have Theorem 7 (see [5,7]). This last second order PDEs system is an Euler-Lagrange prolongation of the first order PDEs system (10) if and only if the Lagrangian L is of least squares type, modulo a divergence type term.

How Are Disasters Favored?
Besides developing theories on ODE and PDE systems, previous research (transforming the flow into dynamics and conversely) offers a partial explanation of some disasters such as catastrophic winds (tornadoes, typhoons), catastrophic floods, crashes without an apparent cause of aircraft etc. Indeed, the geometry of space created by nature or by human activities, conscious or not, generates gyroscopic forces that lead to spiraling (uncontrolled) evolutions. Some of the catastrophes can be avoided by using sensors that highlight spiral or exotic movements.
If in a flow we have chaos (sensitive dependence on initial conditions, three or more dimensions), there is chaos in the associated geometric dynamics. The Riemannian structure can generate also chaos in geometric dynamics.
Author Contributions: Conceptualization, C.U. and I.T.; methodology, C.U. and I.T.; writing-original draft preparation, C.U. and I.T.; writing-review and editing, C.U. and I.T. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.