First, let us look for generating decomposable motions using flows (first order ODE system) and Riemannian metrics. In the multi-time case, things are a little more complicated since a set of (eventually linearly independent) vector fields of class generates an m-dimensional distribution (first order ODE system or first order PDE system), and for the PDE system, we need pairs of Riemannian metrics.
1. 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.
An 1-flow is generated by one vector field. A distribution of dimension m is generated by m vector fields of class . If the distribution is completely integrable, then it is called m-flow. The geometry on a manifold is summarized by a Riemannian metric whose components play the role of gravitational potentials. Here, the focus is not on finding precise solutions to a given 1-flow or m-flow or m-distribution (which is often hopeless), but rather to answer questions like “can a flow or an m-flow or an m-distribution be changed into a dynamics and what is the main ingredient that accomplishes this?”, or “can a dynamics be decomposable into flow trajectories and transversal to flow trajectories and what would be the structure of decomposability?”.
By single-time dynamics we understand an ODE similar to the second law of Newton. A multi-time (multivariate) dynamics is described by a second order elliptic PDE.
Now we return to our previous theories, finding that it is important to emphasize the existence of decomposable movements and the necessary and sufficient conditions in which the decomposition takes place. The theory is general because in any sufficiently large dimension any ODE is transformed into an 1-flow or any PDE is transformed into an m-flow. The geometry of space transforms the 1-flow into a geodesic motion in a gyroscopic field of forces. The geometry of two spaces (source, target) transforms the m-flow (or integral manifolds of an m-distribution) into harmonic maps deformed by gyroscopic field of forces.
The first contribution of this paper is to propose a geometrical condition for expressing the impact between: (i) an 1-flow and the geometry of underlying manifold; (ii) an m-flow and the geometries of two (source and target) underlying manifolds; (iii) an m-distribution and the geometries of two (source and target) underlying manifolds. The second contribution of this paper is to use these well-formedness and strictness conditions to prove the decomposability of a motion. Our third contribution is to show the generality of our theory for writing ODEs and PDEs models whose set of solutions are decomposable. Topics include meaning of ODEs and PDEs on manifolds, both from geometric and systems theory view point. Also, least squares Lagrangians, and decomposable dynamics are included.
Although our Geometric Dynamics theory has deep connections with classical areas of mathematics such as the calculus of variations and the theory of ODEs or PDEs, it did not become a field in its own right until the late 2000s and early 2019s. At that time, some problems arising in engineering and economics were recognized as variants of problems in ODEs or PDEs and in the calculus of variations, although they were not covered by existing theories. Summing, our Geometric Dynamics is a branch of mathematics applying geometric methods to Dynamical Systems and the name was created for a talk at Second Conference of Balkan Society of Geometers, Aristotle University of Thessaloniki, 23–26 June 1998 and after for the book [
2] and for the paper [
3] (it was confirmed by J. Marsden who named it “geodesic motion in a gyroscopic field of forces”, in the review “
001k: 70016/MR70G45 (34A26 37C10 37J05 37K05 74E20)” made to our book [
2]).
The name of Geometric Dynamics has been used by other authors for topics different than ours (see for example Proceedings of the International Symposium held at Instituto de Matematica Pura e Aplicada, Rio de Janeiro, Brasil, 1981 [
9]; Workshop “Geometric Dynamics Days”, Ruhr-Universitat Bochum, Westfalische Wilhelms-Universitat Munster and the Technische Universitat Dortmund, 2013; and the paper [
10], which investigates a family of dynamical systems arising from an evolutionary re-interpretation of certain optimal control and optimization problems).
Example 1. Equations of mechanics may appear different in form from , 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 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 equationsThis motivates a least squares Lagrangian of interest in jerk systems, namelyon the jet space of coordinates , and its associated geometric dynamics (Euler-Lagrange equations) More generally, being given n Lagrangians , , , , the associated least squares Lagrangian with respect to the Riemannian metric isThe extremals are solutions of the Euler-Lagrange ODE systemIf the Lagrangian is associated to ODE , then the extremals contain the solutions of that equation and the dynamics is decomposable. Example 2. Let be the density of the diffusing material at location and time . Let be the collective diffusion coefficient (matrix) for density u at location x. The diffusion PDE isIf the diffusion coefficient depends on the density, then the diffusion equation is nonlinear, otherwise it is linear. More generally, when is a symmetric positive definite matrix (Riemannian metric), the equation describes anisotropic diffusion. The diffusion PDE is equivalent to the first-order non-linear PDEswhere the parameter of evolution is -dimensional. A Riemannian metric produces a least squares Lagrangianon the jet space of coordinates . It appears the associated geometric dynamics (Euler-Lagrange equations) Example 3. Let T be an orientable manifold with the coordinates and M be a manifold with the coordinates . Using m vector fields of class on , we introduce the distribution described by Pfaff equationsUsing some metric tensors , and the components of the pullbacks, we build the least squares Lagrangian(non-decomposable dynamics). Suppose the integral manifolds of the distribution have the dimension . 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 Lagrangians , , , , , then the associated least squares Lagrangian density with respect to the Riemannian metrics , is If , the extremals are solutions of the Euler-Lagrange PDE system If the Lagrangian is associated to the PDE , 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.
3. 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
and a Riemannian metric
g on
M, let us consider the Hamiltonian
, as in [
5]. If
, then the vector field
(Galilei formula), where
E is an arbitrary unit vector field with respect to the metric
g, satisfies
. 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 is generated by a flow and a Riemannian metric.
3.1. Motion of the Four Outer Planets
Let be the masses of the four outer planets (Jupiter, Saturn, Uranus, Neptune), relative to the sun mass . They produce an Euclidean metric .
Corollary 1. The motion of the four outer planets relative to the sun is generated by a flow and the Euclidean metric g.
Proof. The motion of the four outer planets relative to the sun is described by the Hamiltonian
where
and
are velocity and position (supervectors) with
,
G is the gravitational constant, and
are masses relative to the sun mass
. We apply the previous statement with a generic versor
where
. One associated flow requires the fixing of the versor field
.
Since these planets are rather large, their orbits can affect one another (and possibly even the Sun). □
3.2. 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.
Corollary 2. The motion in closed Newmann economical systems is generated by a flow and a Riemannian metric .
Proof. We use a Hamiltonian [
21,
22] which relates
n capital goods
,…,
and the net capital formations
, …,
, namely
We apply the previous statement with a generic versor field. □
4. Comparison between Lorentz’s Law and Geometric Dynamics
Lorentz law
Let
and
. For the potential vector magnetic field
, a particle moving with velocity
and particle charge
e has the potential momentum
, so its potential energy is
. For a
field, the particle’s potential energy is
. Using the total potential energy
and the kinetic energy
, 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)
The Lorentz Hamiltonian is
Geometric dynamics
For the preservation of traditional formulas, we will refer to the magnetic flow generated by the vector potential “
”, using homogeneous dimensional relationships. The magnetic trajectories are the solutions of the ODE system
. This system together with the Euclidean metric produce the least squares Lagrangian
The Euler-Lagrange ODEs of
are
where
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
Remark 2. (i) Generally, the single-time geometric dynamics produced by the potential vector field “” is different from the classic universal Lorentz law becauseand the force field is not the electric field . In other words the Lagrangians and are not in the same equivalence class of Lagrangians. (ii) The magnetic force 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.
7. Geometric Dynamics Induced by sinh-Gordon Kinematics
This Section was elaborated in our research group.
Any triple generates a geometric dynamics, but we are interested in meaningful triples. For example, we use the sinh-Gordon equation 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
and
. 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
We use the least squares Lagrangian
on the Riemannian manifolds
and
. 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.