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Calculus of Variations and Partial Differential Equations of First Order
... This is related to the presence or absence of a set of points where ρ(q, t) = 0. But, a priori, we have no knowledge of this set (which we will call N). Furthermore the fields λ(q, t) have to satisfy, in any case, some consistency conditions, since they are related, through (19), to the variables ρ and j. However in the equation (20) we don't have any term depending on ρ (and then on the condition ρ(q, t) > 0). ...
... Therefore, as a result of this decoupling of λ(q, t) from ρ, the lack of knowledge of N has no influence. The consistency conditions can be solved automatically by promoting the λ(q, t) to a global function S(q, t), thus satisfying the global equation ∂ t S(q, t)+H q, ∂S(q, t) = 0 (q, t ∈ R n ×[0, τ ]) (21) From (19) we deduce that j/ρ can be promoted to a global variable determined by S(q, t). Then we can write for ρ the global equation ...
... If we make use of trajectories in the probabilistic approach, as happens if we assume initially that j(q, t) = ρ(q, t)v(q, t)) (v ∈ R n ),with v(q, t) independent of ρ and regular in R n × [0, τ ], (a relation which is deduced in the previous scheme), then the equation (21) and (22) can be also obtained through alternative well known classical procedures [16,17,18]. In this respect we note also that (14) and (15), with j/ρ substituted by the so called "geodesic fields" (or control fields) have been already obtained by Carathéodory in his approach to existence problems for the extremals of the classical action (7) [1,2,19]. They have been called by him the fundamental equations of the Calculus of Variations [19]. ...
A theoretical scheme, based on a probabilistic generalization of the Hamilton's principle, is elaborated to obtain an unified description of more general dynamical behaviors determined both from a lagrangian function and by mechanisms not contemplated by this function. Within this scheme, quantum mechanics, classical field theory and a quantum theory for scalar fields are discussed. As a by-product of the probabilistic scheme for classical field theory, the equations of the De Donder-Weyl theory for multi-dimensional variational problems are recovered.
... One can also show that all the data of the formulated problem are homogeneous of degree one in state variables, which can be only positive numbers. This is a particular case of Noether's theorem in the calculus of variations about the problems whose data is invariant under a group of transformations [15]. Hence the dimension of phase space of the optimal control problem (2-3) can be lowered by one unit by the introduction of a new variable x = p/n. ...
... By the substitution of the values from (15) to the equation (16) we get ...
... which leads to the fact that k = 0 and, actually, there is no jump in conjugate variables. They keep the same values as (15) and A + m = A − m . But let one suppose that the mutant reacts on the decision of the resident and also changes its control on S r from u − m = 0 to u + m = 1. ...
In this work we study the process of mutant invasion on an example of a consumer-resource system with annual character of the behavior. Namely, individuals are active during seasons of fixed length separated by winter periods. All individuals die at the end of the season and the size of the next generation is determined by the number of offspring produced during the past season. The rate at which the consumers produce immature offspring depends on their internal energy which can be increased by feeding. The reproduction of the resource simply occurs at a constant rate. At the beginning, we consider a population of consumers maximizing their common fitness, all consumers being individuals having the same goal function and acting for the common good. We suppose that a small fraction of the consumer population may appear at the beginning of one season and start to behave as mutants in the main population. We study how such invasion occurs.
... We shall see that this approach is equivalent to that of [15]; as in [15], the implicit function theorem is at the core of our proof, but we are going to use it in a way that is closer to the original approach of [10]. ...
... The paper is organized as follows: section 1 contains the notation and a theorem of [11] about the relationship between differentiability on parametrizations and on measures; section 2 recalls the hypotheses used in [15] from section 6 onwards; in section 3 we recall the method of [10] for the minimum of (4), in section 4 we deal with the master equation (6). §1 ...
... In lemmas 3.2-3.5 below, we recall the method of [10] for the minimals of the value function; in lemma 3.1, we prove that the value functions on measures and on parametrizations coincide. ...
We give a different proof of a theorem of W. Gangbo and A. Swiech on the short time existence of solutions of the master equation.
... In the first part, section 2, it is presented the general Hamilton-Jacobi theory for singular and higher-order lagrangian systems, as it is the case of Podolsky's theory. The HJ formalism is well known from classical mechanics to be a road for the study of integrability of classical systems, but its fundamental role in dynamical systems was discovered only after Carathéodory's work [1] on variational principles and first-order partial differential equations (PDEs). Carathéodory built the HJ theory directly from Hamilton's principle, showing that it is actually the theory that relates first-order PDEs, first-order ordinary differential equations (ODEs), and lagrangian variational problems. ...
... with F µν = ∂ µ A ν − ∂ ν A µ as the components of the electromagnetic tensor field, and a being a parameter with dimension of the inverse of mass. The second-order derivative term in (1) results in a well defined electrostatic potential for r = 0, so the self-energy contribution may be computed, it is just proportional to q 2 /a for each charge. The theory may be interpreted as an effective theory for short distances [15], as a way to get rid of the problems related to the r = 0 singularity in QED. ...
... taken from the generalised wave equation, result of the field equations of the lagrangian (1). This relation indicates two kinds of photons, with modes p 2 = 0 e p 2 = m 2 γ respectively. ...
We develop the Hamilton-Jacobi formalism for Podolsky's electromagnetic theory on the null-plane. The main goal is to build the complete set of Hamiltonian generators of the system, as well as to study the canonical and gauge transformations of the theory.
... R 2 , . θ is the angle of the trajectory of the ship with the .x 1 -axis, and W is the variable wind which depends on time and position (see [26,Eq. 459.8]). ...
... 459.8]). In the 1930s, this problem received the attention of some very well-known mathematicians such as Levi-Civita, Von Mises, and Manià [59,60,77] and became one of the classical problems in the calculus of variations (see [26]). Zermelo problem can also be solved using optimal control theory (see the classical book [16] or [12,14,15,73] for recent developments), but our interest will focus on more geometrical methods, namely, the use of Finsler geometry to solve the problem. ...
... This approximation has been widely used since the experimental results by Anderson [3] and the subsequent PDE system developed by Richards [71] for the wavefront of a wildfire with an elliptical growth. 26 Richards' equations are still used nowadays by fire growth simulators such as FARSITE [38] and Prometheus [76], which even extend the elliptical approximation to the isotropy caused by the slope, i.e., the wildfire becomes a displaced ellipse in the upward direction (since the fire moves faster upward than downward). ...
Some links between Lorentz and Finsler geometries have been developed in the last years, with applications even to the Riemannian case. Our purpose is to give a brief description of them, which may serve as an introduction to recent references. As a motivating example, we start with Zermelo navigation problem, where its known Finslerian description permits a Lorentzian picture which allows for a full geometric understanding of the original problem. Then, we develop some issues including (a) the accurate description of the Lorentzian causality using Finsler elements, (b) the non-singular description of some Finsler elements (such as Kropina metrics or complete extensions of Randers ones with constant flag curvature), (c) the natural relation between the Lorentzian causal boundary and the Gromov and Busemann ones in the Finsler setting, and (d) practical applications to the propagation of waves and firefronts.
... Hamilton-Jacobi (HJ) partial differential equations (PDEs) find applications in various fields such as physics [5,15,16,25,69], optimal control [7,37,44,45,74], game theory [9,14,42,58], imaging sciences [26,29,32,31], and machine learning [21]. In existing literature, a variety of approaches have been explored to numerically address HJ PDEs. ...
... To solve the semi-discrete equation (14), we propose solving the saddle point problem (15), whose first order optimality condition is ...
... i,k is a number defined by Algorithm 4: The proposed algorithm for solving (15) Inputs : Stepsize τ, σ > 0, error tolerance δ > 0, inner maximal iteration number N inner and outer maximal iteration number N outer . Outputs: Solution to the saddle point problem (15). 1 For each i = 1, . . . ...
Hamilton-Jacobi (HJ) partial differential equations (PDEs) have diverse applications spanning physics, optimal control, game theory, and imaging sciences. This research introduces a first-order optimization-based technique for HJ PDEs, which formulates the time-implicit update of HJ PDEs as saddle point problems. We remark that the saddle point formulation for HJ equations is aligned with the primal-dual formulation of optimal transport and potential mean-field games (MFGs). This connection enables us to extend MFG techniques and design numerical schemes for solving HJ PDEs. We employ the primal-dual hybrid gradient (PDHG) method to solve the saddle point problems, benefiting from the simple structures that enable fast computations in updates. Remarkably, the method caters to a broader range of Hamiltonians, encompassing non-smooth and spatiotemporally dependent cases. The approach's effectiveness is verified through various numerical examples in both one-dimensional and two-dimensional examples, such as quadratic and L 1 Hamiltonians with spatial and time dependence.
... In this paper, we examine the geometrical considerations in obtaining a Hamilton-Jacobi framework for the RT model, as a second-order derivative singular system. This is done based on the equivalent Lagrangians method introduced by Carathéodory [22] for regular systems and later strengthened, from the physical point of view, for firstand second-order derivative singular systems by Güler [23][24][25] and other authors [26][27][28][29]. To date, there is no general HJ scheme for this type of gravity except when a Friedmann-Robertson-Walker geometry is assumed on the world volume [30]. ...
... For a singular system, with R constraints, such a family arises from solving a set of R + 1 HJ partial differential equations; in other words, a complete solution of the HJ equations defines a family of hyper-surfaces in the configuration space of the fields. A particular family is related to the mentioned congruence, the characteristics, which are solutions of the characteristic equations [22]. Additionally, this framework provides an interesting alternative to analyzing the constraints content and reduced phase space information of the physical systems which do not differentiate between first-and second-class constraints [15] such that we do not need any gauge fixing terms [28]. ...
... Guided by [40] where it has been developed the HJ analysis for affine in acceleration theories, the present HJ study for the RT gravity has been performed. The complete characteristic equations, provided by the fundamental differential (22), contain a complete set of arbitrary parameters where the equations of motion are included when the parameter dt is solely considered. The remaining transformations associated with the other parameters represent gauge infinitesimal changes, see (77-80). ...
In higher co-dimension, we discuss the Hamilton–Jacobi formalism for brane gravity described by the Regge–Teitelboim model, in higher co-dimension. Considering that it is originally a second-order in derivatives singular theory, we analyze its constraint structure by identifying the complete set of Hamilton–Jacobi equations, under Carathéodory’s equivalent Lagrangians method, which goes hand in hand with the study of the integrability for this type of gravity. Besides, we calculate the characteristic equations, including the one that satisfies the Hamilton principal function S. We find the presence of involutive and non-involutive constraints so that by properly defining a generalized bracket, the non-involutive constraints that originally arise in our framework are removed while the set of parameters related to time evolution and gauge transformations is identified. A detailed comparison is also made with a recent Ostrogradsky–Hamilton approach for constrained systems, developed for this brane gravity. Some facts about the gauge symmetries behind this theory are discussed.
... That is, one can derive eld equations from an action principle that entails a Lagrangian, not a Lagrangian density. This is clearly shown in Carathéodory's approach to the calculus of variations [17]. We present next a very short account of this approach. ...
... The existence of a Mayer eld requires that some integrability conditions are satised. Carathéodory's formulation [17] makes clear how these conditions relate to the Euler-Lagrange equations. Let us summarize Carathéodory's approach [17]. ...
... Carathéodory's formulation [17] makes clear how these conditions relate to the Euler-Lagrange equations. Let us summarize Carathéodory's approach [17]. Extremals satisfying δ´L(x,ẋ)ds = 0 are the same as those satisfying the so-called equivalent variational problem δ´(L(x,ẋ) −ẋ µ ∂ µ S(x))ds = 0, where S(x) is an auxiliary function. ...
Classical electrodynamics (CED) has achieved great success in its domain of application, but despite this success, it has remained a theory that lacks complete self-consistency. It is worthwhile trying to make CED a self-consistent theory, because many important phenomena lie within its scope, and because modern field theories have been modelled on it. Alternative approaches to CED might help finding a definite formulation, and they might also lead to the prediction of new phenomena. Here we report two main results. The first one derives from standard CED. It is shown that the motion of a charged particle is ruled not only by the Lorentz equation, but also by equations that are formally identical to Maxwell equations. The latter hold for a velocity field and follow as a strict logical consequence of Hamilton's action principle for a \emph{single} particle. We construct a tensor with the velocity field in the same way as the electromagnetic tensor is constructed with the four potential. The two tensors are shown to be proportional to one another. As a consequence, and without leaving the realm of standard CED, one can envision new phenomena for a charged particle, which parallel those involving electromagnetic fields. The second result refers to a field-free approach to CED. This approach confirms the simultaneous validity of Maxwell-like and Lorentz equations as rulers of charged particle motion.
... The surface M is split into rectangles r 0 < r < r 1 with weak current if kF 0 k g < 1 or strong current if kF 0 k g > 1. Such a problem is a generalization of the historical problem of the quickest nautical path analyzed by Carathéodory and Zermelo [18,34], which have provided a complete study in the case of a linear current. ...
... A founding problem in classical calculus of variations is the problem called quickest nautical path introduced by Carathéodory and Zermelo [18,34] for a ship navigating on a river and aiming to reach the opposite shore in minimum time. Hence, M is the 2d-Euclidean space with metric g = dx 2 + dy 2 in the coordinates q = (x, y), y being the distance to the shore. ...
... Remark 2. The result is clear in the abnormal case due to (17), since q 3 (·) is strictly positive, unless the geodesic curve is the reference geodesic. It was already observed by Carathéodory and Zermelo, see [18] where the abnormal geodesic are called "limit curves". ...
In this article, the historical study from Carathéodory-Zermelo about computing the quickest nautical path is generalized to Zermelo navigation problems on surfaces of revolution, in the frame of geometric optimal control. Using the Maximum Principle, we present two methods dedicated to analyzing the geodesic flow and to compute the conjugate and cut loci. We apply these calculations to investigate case studies related to applications in hydrodynamics, space mechanics and geometry.
... In this paper we examine the geometrical considerations in obtaining a Hamilton-Jacobi framework for the RT model, as a second-order derivative singular system. This is done on the basis of the equivalent Lagrangians method introduced by Carathéodory [22] for regular systems and later strengthened, from the physical point of view, for first-and second-order derivative singular systems by Güler [23][24][25] and other authors [26][27][28][29]. To date, there is not a general HJ scheme for this type of gravity except when a Friedmann-Robertson-Walker geometry is assumed on the worldvolume [30]. ...
... On the other hand, this approach has a deep geometrical meaning that establishes that the necessary and sufficient condition for an action to be minimized, in a region of the configuration space, is the existence of a family of surfaces everywhere orthogonal to a congruence of curves. For a singular system, with R constraints, such a family arises from solving a set of R + 1 HJ partial differential equations where the complete solution is nothing but a congruence of curves, called characteristics, that form a dynamical system with several independent variables [22]. Additionally, this framework provides an interesting alternative to analyse the constraints content and reduced phase space information of the physical systems which does not differentiate between first-and second-class constraints [15] such that we do not need any gauge fixing terms [28]. ...
... In view of (20) and (22) we find that ...
We discuss the Hamilton-Jacobi formalism for brane gravity described by the Regge-Teitelboim model, in higher co-dimension. Being originally a second-order in derivatives singular theory, we analyzed its constraint structure by identifying the complete set of Hamilton-Jacobi equations, under the Carath\'eodory's equivalent Lagrangians method, which goes hand by hand with the study of the integrability for this type of gravity. Besides, we calculate the characteristic equations including the one that satisfy the Hamilton principal function S. We find the presence of involutive and non-involutive constraints so that by properly defining a generalized bracket, the non-involutive constraints that originally arise in our framework, are removed while the set of parameters related to the time evolution and the gauge transformations, are identified. A detailed comparison with a recent Ostrogradsky-Hamilton approach for constrained systems, developed for this brane gravity, is also made. Some remarks on the gauge symmetries behind this theory are commented upon.
... are related by global homeomorphisms: ξ = −η. However x = ξ 2 4 sgn(ξ) is not a diffeomorphism. Conceptually there is no problem in performing all operations shown. ...
... If the input to the time varying system is impulsive of the form u(t) = ∞ i=0 g i δ (i−1) (t−τ ), then the state jumps instantaneously, at time τ by the amount C ∞ (τ )g, where g = [g 1 , g 2 , · · · ] . Likewise, if the system is at time τ in the state x(τ ), then the successive derivatives (2) . . . ...
... This is a set of n partial differential equations of first order in n variables. It is a special case of a Mayer-Lie system [1,2,5]. It is known that such a system of equations is not generically solvable. ...
This paper introduces a novel approach towards balancing of nonlinear systems. It differs from other approaches in two key ideas: The first is the observation that linear time-invariant systems have a single equilibrium point (the origin), and that the balanced realization for the class of a linear time-invariant systems essentially relates to this equilibrium. This sparked the idea to consider balancing in general as a property associated with invariant sets of a given nominal flow. The second idea is to define a notion of global balancing as the one that commutes with linearization. The existence and construction of balanced and uncorrelated realizations are discussed. Uncorrelatedness is a relaxed notion of bal-ancedness, introduced here. Several examples, including the Van der Pol oscillator are shown.
... It is also worth mentioning that as we show in the following our conjecture helps to prove the fact mentioned by C. Caratheodory that a variational problem can be assigned to any first order partial differential equation [7]. ...
... Caratheodory that a variational problem can be assigned to any first order partial differential equation [7] in the context of differential geometry where the action function which is given usually by an integral belongs to zero forms. In other words we will prove the relation δω 0 = 0 ↔ dω 0 = 0 or equivalently ω 0 ∈ Harm 0 with the help of our conjecture. ...
... For the relevance of our stand point to consider the spectrum of Laplacian as the calculus of variation note that a variational problem can be assigned to any first order partial differential equation [7]. Nevertheless on a suitable compact manifold one can assign according to the Hodge decomposition theorem a second order differential equation to any first order differential equation, i. e. ...
The Hodge-de Rham theory of differential topology is considered as a generalization of Maxwell's differential equations of electrodynamics for general differential forms but the relation between integral Maxwell equations and Hodge-de Rahm theory is not yet considered. To bridge this gap we conjecture an integral representation for the adjoint exterior derivative that is based on Maxwell's integral equations and on the achievements of the theory of pseudo-differential operators. It establishes a direct relation between certain differential-and integral operators and explains the relation between the spectrum of Laplacian and the length of geodesics in Selberg trace formula. Our conjecture help to prove Caratheodory's statement that a variational problem can be assigned to any first order partial differential equation.
... The sought-after extremal curve must be in fact a member of a whole field of such curves. This is the imbedding theorem in the calculus of variations [48][49][50], basically a consequence of continuity assumptions. One usually focuses on a single extremal curve, even though this curve must be embedded in a whole family of extremals. ...
... The existence of the Mayer field imposes some integrability conditions. Carathéodory's "royal road" to the Calculus of Variations [49] leads at once to Euler-Lagrange, Hamilton and Hamilton-Jacobi equations. ...
... We seek for a field v(x) whose integral curves rendeŕ Lds extremal. To this end, we introduce an auxiliary function S(x) and require that the following, fundamental equations are satisfied [49][50][51]: ...
We present a local-realistic description of both wave-particle duality and Bohmian trajectories. Our approach is relativistic and based on Hamilton's principle of classical mechanics, but departs from its standard setting in two respects. First, we address an ensemble of extremal curves, the so-called Mayer field, instead of focusing on a single extremal curve. Second, we assume that there is a scale, below which we can only probabilistically assess which extremal curve in the ensemble is actually realized. The continuity equation ruling the conservation of probability represents a subsidiary condition for Hamilton's principle. As a consequence, the ensemble of extremals acquires a dynamics that is ruled by Maxwell equations. These equations are thus shown to also rule some non-electromagnetic phenomena. While particles follow well-defined trajectories, the field of extremals can display wave behavior.
... where x 1 , x 2 are the coordinates of R 2 , θ is the angle of the trajectory of the ship with the x 1 -axis and W is the variable wind which depends on time and position (see [26,Eq. 459.8]). ...
... 459.8]). In the thirties of the past century this problem received the attention of some very well-known mathematicians such as Levi-Civita, Von Mises and Manià [59,66,60] and became one of the classical problems in the Calculus of Variations (see [26]). Zermelo problem can also be solved using Optimal Control Theory (see the classical book [16] or [12,14,15,74] for recent developments), but our interest will focus on more geometrical methods, namely, the use of Finsler Geometry to solve the problem. ...
... This approximation has been widely used since the experimental results by Anderson [3] and the subsequent PDE system developed by Richards [72] for the wavefront of a wildfire with an elliptical growth. 26 Richards' equations are still used nowadays by fire growth simulators such as FARSITE [38] and Prometheus [77], which even extend the elliptical approximation to the isotropy caused by the slope, i.e., the wildfire becomes a displaced ellipse in the upward direction (since the fire moves faster upwards than downwards). ...
Some links between Lorentz and Finsler geometries have been developed in the last years, with applications even to the Riemannian case. Our purpose is to give a brief description of them, which may serve as an introduction to recent references. As a motivating example, we start with Zermelo navigation problem, where its known Finslerian description permits a Lorentzian picture which allows for a full geometric understanding of the original problem. Then, we develop some issues including: (a) the accurate description of the Lorentzian causality using Finsler elements, (b) the non-singular description of some Finsler elements (such as Kropina metrics or complete extensions of Randers ones with constant flag curvature), (c) the natural relation between the Lorentzian causal boundary and the Gromov and Busemann ones in the Finsler setting, and (d) practical applications to the propagation of waves and firefronts.
... Arbitrary "Lagrangian" functions, describing the distribution of certain particle properties may be transformed to "Eulerian" functions describing corresponding fields. This is a general method [9] frequently applied in 3−dimensional fluid mechanics [3] and applied here to 2n−dimensional phase space; in section 6 we will try to apply it to n−dimensional configuration space Σ = R n q . The mathematical realization of the Lagrangian to Eulerian transition is very simple. ...
... If we represent the initial values with the help of Eq. (7) as a function of the coordinates q(t), p(t) and the time t, we obtain a function F (q, p, t) = F 0 (Q 0 (t, q, p), P 0 (t, q, p), t). (9) Taking the fact into accout that q, p do also depend on time, the (total) derivative of F (q, p.t) with respect to t is given by ...
... The number L is referred to as "class" of the vector field M = {M 1 , ..., M n }. A detailed treatment of Pfaff's problem may be found, e.g., in Caratheodory's book [9]. ...
This work is based on the idea that the classical counterpart of quantum theory (QT) is not mechanics but probabilistic mechanics. We therefore choose the theory of statistical ensembles in phase space as starting point for a reconstruction of QT. These ensembles are described by a probability density and an action variable . Since the state variables of QT only depend on q and t, our first step is to carry out a projection from phase space to configuration space. We next show that instead of the momentum components one must introduce suitable potentials as dynamic variables. The quasi-quantal theory resulting from the projection is only locally valid. To correct this failure, we have to perform as a second step a linearization or randomization, which ultimately leads to QT. In this work we represent M as an irrotational field, where all components may be derived from a single function . We obtain the usual Schr\"odinger equation for a nonspinning particle. However, space is three-dimensional and M must be described by 3 independent functions. In the fourth work of this series, a complete representation of M will be used, which explains the origin of spin. We discuss several fundamental questions that do not depend on the form of M and compare our theory with other recent reconstructions of QT.
... The Hamilton-Jacobi formalism presented here follows the approach of Güler [15], which is an extension of Caratheodory's equivalent Lagrangian method in the calculus of variations [16]. This formalism is characterized by a set of Hamilton-Jacobi differential equations called Hamiltonians. ...
... where α = 0, ..., k.. The Cauchy's method [16] is employed to find the characteristic equations related to the above first order equations ...
In this work we perform the Hamilton-Jacobi constraint analysis of the four dimensional Background Field (BF) model with cosmological term. We obtain the complete set of involutive Hamiltonians that guarantee the integrability of the system and identify the reduced phase space. From the fundamental differential we recover the equations of motion and obtain the generators of the gauge and shift transformations.
... This method was already suggested by Cauchy and was mentioned in Carathéodory's book [4]. In order to have the general theory and some perspective in the research that has been done, it is possible to consider [3,7,16]. ...
... In particular, for x 0 = 1 2 and t = 1 the curve γ ζ (1, 4 is plotted in the following figure: while in the next two figures there are some half-spaces (corresponding to ζ = 1 0 , ...
The method of characteristics is extended to set-valued Hamilton-Jacobi equations. This problem arises from a calculus of variations’ problem with a multicriteria Lagrangian function: through an embedding into a set-valued framework, a set-valued Hamilton-Jacobi equation is derived, where the Hamiltonian function is the Fenchel conjugate of the Lagrangian function. In this paper a method of characteristics is described and some results are given for the Fenchel conjugate.
... The first problem is one founding example of calculus of variations and was set originally in 1931 by Zermelo [18] and presented in details by Carathéodory [7], in particular in the case of linear wind for which we shall refer as the historical case along this paper. The problem is a ship navigating on a river with a current and aiming to reach the opposite shore. ...
... In the historical example a cusp singularity was observed in [7] and it will serve as a model to analyze the general case in the frame of singularity theory, since integrability is not a technical requirement. One needs to recall some elementary facts. ...
... In a time-optimal version of quantum control theory, maximum fidelity transformations are sought in the least possible time to reduce the impact of decoherence, which rapidly degrades the quality of quantum states in quantum information processing [19,20]. Following a number of precursor works [21][22][23][24][25], a formal time-optimal version of quantum control theory was formulated by Carlini and coworkers [26][27][28] in analogy to Bernoulli's classical brachistochrone problem [29], and has since then been referred to as the quantum brachistochrone problem. If the time evolution has no constraints, the solution to the quantum brachistochrone problem reduces to finding the time-independent Hamiltonian that generates maximum speed of evolution along a geodesic [30][31][32][33][34]. ...
... Even if the form of the control Hamiltonian can be relaxed, limitations may still arise in situations where the system is immersed in an external field that cannot be altered by the controller. In analogy with the classical problem posed by Zermelo [29], this last situation has been often called the quantum Zermelo navigation problem [39][40][41][42]. ...
Time-efficient control schemes for manipulating quantum systems are of great importance in quantum technologies, where environmental forces rapidly degrade the quality of pure states over time. In this Letter, we formulate an approach to time-optimal control that circumvents the boundary-value problem that plagues the quantum brachistochrone equation at the expense of relaxing the form of the control Hamiltonian. In this setting, a coupled system of equations, one for the control Hamiltonian and another one for the duration of the protocol, realizes an ansatz-free approach to quantum control theory. We show how driven systems, in the form of a Landau-Zener type Hamiltonian, can be efficiently maneuvered to speed up a given state transformation in a highly adiabatic manner and with a low energy cost.
... Early work (Barber et al., 1973;Carter et al., 1971;Nobes et al., 1969) focused on amorphous substrates and isotropic erosion without mentioning the HJE. Subsequent advances introduced the HJE (Carter et al., 1984;Katardjiev, 1989;Katardjiev et al., 1989;Nobes et al., 1987;Smith et al., 1986;Witcomb, 1975) and the eikonal equation (Carter, 2001) and used them to address the issue of anisotropic erosion. Perhaps most relevant to our theoretical development is the review article by Smith et al. (1986), which is also notable for its invocation of an erosional Hamiltonian, and the papers by Carter et al. (1984), Katardjiev (1989) and Katardjiev et al. (1989), which connect the HJE and its Hamiltonian to Huygens' principle and the concept of erosional wavelets (see also Sethian, 1995b, c, 1997;Sethian and Adalsteinsson, 1997). ...
... There is a corresponding structure for phase velocity, known as the figuratrix, which is typically used in its reciprocal speed or slowness form. The velocity indicatrix and slowness figuratrix are linked through mutual conjugacy: as such, they contain the same information about front propagation but in different forms (Carathéodory, 1999;Perlick, 2000;Rider, 1926;Rund, 1959). ...
The rate of erosion of a landscape depends largely on local gradient and material fluxes. Since both quantities are functions of the shape of the catchment surface, this dependence constitutes a mathematical straitjacket, in the sense that – subject to simplifying assumptions about the erosion process, and absent variations in external forcing and erodibility – the rate of change of surface geometry is solely a function of surface geometry. Here we demonstrate how to use this geometric self-constraint to convert a gradient-dependent erosion model into its equivalent Hamiltonian, and explore the implications of having a Hamiltonian description of the erosion process. To achieve this conversion, we recognize that the rate of erosion defines the velocity of surface motion in its orthogonal direction, and we express this rate in its reciprocal form as the surface-normal slowness. By rewriting surface tilt in terms of normal slowness components and deploying a substitution developed in geometric mechanics, we extract what is known as the fundamental metric function of the model phase space; its square is the Hamiltonian. Such a Hamiltonian provides several new ways to solve for the evolution of an erosion surface: here we use it to derive Hamilton's ray-tracing equations, which describe both the velocity of a surface point and the rate of change of the surface-normal slowness at that point. In this context, gradient-dependent erosion involves two distinct directions: (i) the surface-normal direction, which points subvertically downwards, and (ii) the erosion ray direction, which points upstream at a generally small angle to horizontal with a sign controlled by the scaling of erosion with slope. If the model erosion rate scales faster than linearly with gradient, the rays point obliquely upwards, but if erosion scales sublinearly with gradient, the rays point obliquely downwards. This dependence of erosional anisotropy on gradient scaling explains why, as previous studies have shown, model knickpoints behave in two distinct ways depending on the gradient exponent. Analysis of the Hamiltonian shows that the erosion rays carry boundary-condition information upstream, and that they are geodesics, meaning that surface evolution takes the path of least erosion time. Correspondingly, the time it takes for external changes to propagate into and change a landscape is set by the velocity of these rays. The Hamiltonian also reveals that gradient-dependent erosion surfaces have a critical tilt, given by a simple function of the gradient scaling exponent, at which ray-propagation behaviour changes. Channel profiles generated from the non-dimensionalized Hamiltonian have a shape entirely determined by the scaling exponents and by a dimensionless erosion rate expressed as the surface tilt at the downstream boundary.
... The first problem is one founding example of calculus of variations and was set originally in 1931 by Zermelo and presented in details by Carathéodory [18,7], in particular in the case of linear wind for which we shall refer as the historical case along this paper. The problem is a ship navigating on a river with a current and aiming to reach the opposite shore. ...
... In the historical example a cusp singularity was observed in [7] and it will serve as a model to analyze the general case in the frame of singularity theory, since integrability is not a technical requirement. One needs to recall some elementary facts. ...
In this article, we use two case studies from geometry and optimal control of chemical network to analyze the relation between abnormal geodesics in time optimal control, accessibility properties and regularity of the time minimal value function.
... As an alternative, the covariant formulation allows to consider a finite dimensional configuration space (the dimension of the symmetry group itself). Although its roots go back to De Donder [25], Weyl [26], Caratheodory [27], after J. M. Souriau in the seventies [28], the classical field theory has been only well understood in the late 20th century (see for example [29] for an extension from symplectic to multisymplectic form). It is therefore not surprising that, in this covariant or jet formulation setting, the geometric constructions needed for reduction have been presented even more recently. ...
Although acoustics is one of the disciplines of mechanics, its "geometrization" is still limited to a few areas. As shown in the work on nonlinear propagation in Reissner beams, it seems that an interpretation of the theories of acoustics through the concepts of differential geometry can help to address the non-linear phenomena in their intrinsic qualities. This results in a field of research aimed at establishing and solving dynamic models purged of any artificial nonlinearity by taking advantage of symmetry properties underlying the use of Lie groups. The geometric constructions needed for reduction are presented in the context of the "covariant" approach.
... The Finsler metric in Eq. (2) defines a dynamical system through the minimisation of the 'action' functional I(γ) = τ2 τ1 Λ(x, y) dτ for a path γ and given initial and final conditions (γ(τ 1 ) and γ(τ 2 )), when the following three conditions are fulfilled [6,50] (i ) positive homogeneity of degree one in the second argument, Λ(x, ky) = kΛ(x, y), k > 0, (ii ) Λ(x, y) > 0 with i (y i ) 2 = 0, and ...
Self-interference embodies the essence of the particle-wave interpretation of quantum mechanics (QM). According to the Copenhagen particle-wave interpretation of QM, self-interference by a double slit requires a large transverse coherence of the incident wavepacket such that it covers the separation between the slits. Bohmian dynamics provides a first step in the separation of the particle-wave character of particles by introducing deterministic trajectories guided by a pilot wave that follows the time-dependent Schr\"odinger equation. In this work, I present a theory for quantum dynamics that incorporates all quantum (wave) effects into the geometry of the underlying phase space. This geometrical formulation of QM is consistent with quantum measurements and provides an alternative interpretation of quantum mechanics in terms of deterministic trajectories. In particular, it removes the need for the concept of wavefunction collapse (of the Copenhagen interpretation) to explain the emergence of the classical world. All three QM formulations (Schr\"odinger, Bohmian, and geometrical) are applied to the description of the scattering of a free electron by a hydrogen atom and a double slit.
... The function G is, in principle, determined by the form of G s and by the solutions of the equations of motion. Alternatively, a partial differential equation for G may be derived [8]. It may be found using the fact that the initial values are "constants of motion". ...
Koopman and von Neumann (KvN) extended the Liouville equation by introducing a phase space function whose physical meaning is unknown. We show that a different S(q,p,t), with well-defined physical meaning, may be introduced without destroying the attractive "quantum-like" mathematical features of the KvN theory. This new S(q,p,t) is the classical action expressed in phase space coordinates. It defines a mapping between observables and operators which preserves the Lie bracket structure. The new evolution equation reduces to Schr\"odinger's equation if functions on phase space are reduced to functions on configuration space. This new kind of "quantization" does not only establish a correspondence between observables and operators, but provides in addition a derivation of quantum operators and evolution equations from corresponding classical entities. It is performed by replacing by 0 and p by , thus providing an explanation for the common quantization rules.
... Historically Jacobi equations were related to the so-called accessory problem (see, e.g. [3,22]), where they are directly obtained as the variation of the Euler-Lagrange equations of a given Lagrangian. Thus they can be characterized via the Second Noether Theorem. ...
When a gauge-natural invariant variational principle is assigned, to determine {\em canonical} covariant conservation laws, the vertical part of gauge-natural lifts of infinitesimal principal automorphisms -- defining infinitesimal variations of sections of gauge-natural bundles -- must satisfy generalized Jacobi equations for the gauge-natural invariant Lagrangian. {\em Vice versa} all vertical parts of gauge-natural lifts of infinitesimal principal automorphisms which are in the kernel of generalized Jacobi morphisms are generators of canonical covariant currents and superpotentials. In particular, only a few gauge-natural lifts can be considered as {\em canonical} generators of covariant gauge-natural physical charges.
... Herglotz's work was motivated by ideas from S. Lie [17,18] and others. For historical remarks through 1935 see Caratheodory [2]. The contact transformations, which can be derived from the generalized variational principle, have found applications in thermodynamics. ...
This paper reviews the generalized Hamiltonian system and its connection to contact transformations. The generalized Hamiltonian system is related to Herglotz variational principle in the same way in which the Hamiltonian system is related to the classical variational principle. We prove a criterion for the integrability of the generalized Hamiltonian system in terms of a complete set of first integrals, and a method of generating such first integrals. These results are due to Gustav Herglotz.
... We also note when there is a continua of equilibria and the energy functional has some weak convexity conditions (satisfied by (22)), an argument in [SdlL17] -adapting to our context ideas of [Car67]-shows that the solutions are minimizers in the sense that compactly supported modifications increase the energy -considering only the terms which change -(class A minimizers). ...
We consider one dimensional chains of interacting particles subjected to one dimensional almost-periodic media. We formulate and prove two KAM type theorems corresponding to both short-range and long-range interactions respectively. Both theorems presented have an a posteriori format and establish the existence of almost-periodic equilibria. The new part here is that the potential function is given by some almost-periodic function with infinitely many incommensurate frequencies. In both cases, we do not need to assume that the system is close to integrable. We will show that if there exists an approximate solution for the functional equations, which satisfies some appropriate non-degeneracy conditions, then a true solution nearby is obtained. This procedure may be used to validate efficient numerical computations. Moreover, to well understand the role of almost-periodic media which can be approximated by quasi-periodic ones, we present a different approach -- the step by step increase of complexity method -- to the study of the above results of the almost-periodic models.
... A przecieø problem ten by≥ i pozostaje jednym z motorów rozwoju teorii rownaÒ róøniczkowych zarówno zwyczajnych jak i czπstkowych. åwiadczy o tym na przyk≥ad pierwszy tom traktatu [Fo90], [ESM13] i [Ca82]. Ponadto, lukπ wydaje siÍ byÊ pominiÍcie metod Hamiltona, Jacobiego i Lie ca≥kowania równaÒ róøniczkowych oraz ca≥kowite pominiÍcie prac M. Janeta 11 . ...
... In order to generalise the first two parts of proposition 1.2 we need the following results (see in part Caratheodory [2]): (d) λ k is the maximum value of Q on {p ∈ S n : x k+1 (p) = 0, . . . , x n (p) = 0}. ...
New expansionary and rotational quadratic forms are constructed for
E^n-endomorphisms. Relations amongst the various eigenvalues, eigendirections and matrix invariants are established , including propositions on complexity and geometric multiplicity. The underlying construction involves a novel, almost-orthogonal expansion based on two-plane rotations. The development is strongly geometric in flavour and has application to the theory of connections , of which the Frenet case on E^3 is given as a model.
2020 Mathematics Subject Classification:
Primary 15A04,15A63,53A45
Secondary 15A15,15A18,53Z50
Key words and phrases:
linear operators, matrix invariants, quadratic forms, rotation and expansion of real
endomorphisms, eigenspaces, eigenvalues, Frenet frame
... This elimination is actually an innocent looking standard technique of fluid mechanics known as Lagrangian to Eulerian transition [2,3]. It is based on the existence of a continuum of solutions of first order ordinary differential equations which fill and thus generate the space under consideration [4]; let us note for clarity that the Eulerian formulation is nothing but the standard formulation of physical fields. ...
As was recently shown, non-relativistic quantum theory can be derived by means of a projection method from a continuum of classical solutions for (massive) particles. In this paper we show that Maxwell's equations in empty space can be derived using the same method. In this case the starting point is a continuum of solutions of equations of motion for massless particles describing the structure of Galilean space-time. As a result of the projection, the space-time structure itself is changed by the appearance of a new fundamental constant c with the dimension of a velocity. This maximum velocity c, derived here for massless particles, is analogous to the accuracy limit derived earlier for massive particles. The projection method can thus be interpreted as a generalized quantization. We suspect that all fundamental fields can be traced back to continuous sets of particle trajectories, and that in this sense the particle concept is more fundamental than the field concept.
... Whereas the topological consideration of integrability includes a well structured general approach that merges all those isolated results. Nevertheless in view of the fact that Lagrange extracted those brackets from the "variation of constants" of celestial mechanics and that Lagrange-Poisson brackets are invariants of motion [1] hence they are obviously non-trivial differential topological invariants and should be generalized in the topological manner. ...
The general Lagrange-Poisson brackets with not-trivial brackets are introduced according to the original works of Lagrange and Poisson and topological methods satisfying the Jacobi identity. It is proved that their quantization results in QM commutators including a constant magnetic field and describes the effective form of QED and quantization as an integrability condition. Relations to integrability of differential equations and foliated manifolds are discussed in the light of topological methods.
... Whereas the topological consideration of integrability includes a well structured general approach that merges all those isolated results. Nevertheless in view of the fact that Lagrange extracted those brackets from the "variation of constants" of celestial mechanics and that Lagrange-Poisson brackets are invariants of motion [1] hence they are obviously non-trivial differential topological invariants and should be generalized in the topological manner. ...
The general Lagrange-Poisson brackets with not-trivial brackets are introduced according to the original works of Lagrange and Poisson and topological methods satisfying the Jacobi identity. It is proved that their quantization results in QM commutators including a constant magnetic field and describes the effective form of QED and quantization as an integrability condition. Relations to integrability of differential equations and foliated manifolds are discussed in the light of topological methods.
... The second one is related to Zermelo's navigation problem which consists in finding the paths between two points x 0 and x 1 that minimize the travel time of a ship or an airship moving under a wind in a Riemannian manifold (S, g 0 ) (see [26], [12] for the original formulation of the problem in the Euclidean space R 2 ). If the wind is time-independent then it can be represented by a vector field W on S. In the particular case when g 0 (W, W ) ≤ 1, the solutions of the problem are the pregeodesics of the Randers-Kropina metric, associated with the data g 0 and W , which are minimizers of its length functional, see [11,Corollary 6.18] ( 1 ). ...
We consider a geodesic problem in a manifold endowed with a Randers-Kropina metric. This is a type of a singular Finsler metric arising both in the description of the lightlike vectors of a spacetime endowed with a causal Killing vector field and in the Zermelo's navigation problem with a wind represented by a vector field having norm not greater than one. By using Lusternik-Schnirelman theory, we prove existence of infinitely many geodesics between two given points when the manifold is not contractible. Due to the type of non-holonomic constraints that the velocity vectors must satisfy, this is achieved thanks to some recent results about the homotopy type of the set of solutions of an affine control system associated with a totally non-integrable distribution.
... Almost a hundred years later, C. Carathéodory in the introduction to his famous book on the calculus of variations [5] remarks that "neither Jacobi, nor his students, nor the many other prominent men who so brilliantly represented and advanced this discipline during the nineteenth century, thought in any way of the relationship between the calculus of variations and partial differential equation". H. Poincaré also sidestepped this issue by treating canonical systems as the solutions of a dynamical system d dt ...
In his 1842 lectures on dynamics C.G. Jacobi summarized difficulties with differential equations by saying that the main problem in the integration of differential equations appears in the choice of right variables. Since there is no general rule for finding the right choice, it is better to introduce special variables first, and then investigate the problems that naturally lend themselves to these variables. This paper follows Jacobi’s prophetic observations by introducing certain “meta” variational problems on semi-simple reductive groups G having a compact subgroup K. We then use the Maximum Principle of optimal control to generate the Hamiltonians whose solutions project onto the extremal curves of these problems. We show that there is a particular sub-class of these Hamiltonians that admit a spectral representation on the Lie algebra of G. As a consequence, the spectral invariants associated with the spectral curve produce a large number of integrals of motion, all in involution with each other, that often meet the Liouville complete integrability criteria. We then show that the classical integrals of motion associated, with the Kowalewski top, the two-body problem of Kepler, and Jacobi’s geodesic problem on the ellipsoid can be all derived from the aforementioned Hamiltonian systems. We also introduce a rolling geodesic problem that admits a spectral representation on symmetric Riemannian spaces and we then show the relevance of the corresponding integrals on the nature of the curves whose elastic energy is minimal.
... Moreover, by (17) we obtain that τ ρ = o(ρ) as ρ → 0 + . On the other hand, by the first part of (20) and (13) we see that ...
We study the dynamics around closed orbits of autonomous Lagrangian systems. When the configuration space is two-dimensional and orientable we show that every closed orbit minimizing the free-period action functional is orbitally unstable. This result applies even when the minimizers are degenerate or nonisolated, but a particularly strong form of instability holds in the isolated case. Under some symmetry assumptions, free-period action minimizers are unstable also in the higher-dimensional case. Applications to geodesics and celestial mechanics are given.
... Jacobi sharpened Hamilton's formulation, by clarifying some mathematical issues. The resultant Hamilton-Jacobi theory and later developments are presented in several famous texts: [3,24,32,47,82,106,107]. For studies using modern PDE theory see [18,77,116]. ...
In this survey, we review the classical Hamilton–Jacobi theory from a geometric point of view in different geometric backgrounds. We propose a Hamilton-Jacobi equation for different geometric structures attending to one particular characterization: whether they fulfill the Jacobi and Leibniz identities simultaneously, or if at least they satisfy one of them. In this regard, we review the case of time-dependent (t-dependent in the sequel) and dissipative physical systems as systems that fulfill the Jacobi identity but not the Leibnitz identity. Furthermore, we review the contact-evolution Hamilton-Jacobi theory as a split off the regular contact geometry, and that actually satisfies the Leibniz rule instead of Jacobi. Furthermore, we include a novel result, which is the Hamilton-Jacobi equation for conformal Hamiltonian vector fields as a generalization of the well-known Hamilton-Jacobi on a symplectic manifold, that is retrieved in the case of a zero conformal factor. The interest of a geometric Hamilton–Jacobi equation is the primordial observation that if a Hamiltonian vector field can be projected into a configuration manifold by means of a 1-form dW, then the integral curves of the projected vector field can be transformed into integral curves of provided that W is a solution of the Hamilton-Jacobi equation. Geometrically, the solution of the Hamilton-Jacobi equation plays the role of a Lagrangian submanifold of a certain bundle. Exploiting these features in different geometric scenarios we propose a geometric theory for multiple physical systems depending on the fundamental identities that their dynamic satisfies. Different examples are pictured to reflect the results provided, being all of them new, except for one that is reassessment of a previously considered example.
... Note in view of the facts that firstly according to Caratheodory any first order differential equation describing a dynamical problem is related with a variational principle [5] that means the solving function can be given by the action function describing the dynamics. Secondly that as I proved any first order differential equation can be converted into a second order differential equation by iteration of relevant [6]. ...
We prove in this work the basic two dimensionality of physics, i. e. theoretical description of physical interactions as dynamical systems. It is proved that Hadamard's method of descent, Huygens principle and Cauchy data method to solve differential equations restricts the originally assumed number of static variables or dimensions as dependent variables in usual higher dimensional physical models to the number of degrees of freedom or independent variables. Further that the integarbility of dynamical systems as structurally stable systems restricts equivalently the underlying manifolds to compact ones with two degrees of freedom.
... Note in view of the facts that firstly according to Caratheodory any first order differential equation describing a dynamical problem is related with a variational principle [5] that means the solving function can be given by the action function describing the dynamics. Secondly that as I proved any first order differential equation can be converted into a second order differential equation by iteration of relevant [6]. ...
... In order to solve such a class of problems, the method of Lagrange multipliers is commonly used without the need to explicitly solve the fixed outside conditions and use them to eliminate the extra variables. Lagrange's method has been used exclusively to find and analyze the constrained local extrema [1][2][3][4][5][6]. But Lagrange's method has some disadvantages. ...
In this article, a new efficient numerical method is proposed to solve control theory problems in a finite difference setting. The method was originally devised by Gigena for determining and analyzing constrained local extrema without using Lagrange multipliers with the use of differential forms. He efficiently solved a number of algebraic problems and proved his method to be a better alternative as compared with already existing methods. Inspired by his work, we extended the scope of his work and generalized his technique to solve optimal control problems. The results are highly encouraging, and it can be seen that the proposed method has great advantages in terms of efficiency and computational cost in solving optimal control problems. This new approach provides a new way of looking at solving numerous other optimal control problems, such as discrete time models, with ease.
... Note in view of the facts that firstly according to Caratheodory any first order differential equation describing a dynamical problem is related with a variational principle [5] that means the solving function can be given by the action function describing the dynamics. Secondly that as I proved any first order differential equation can be converted into a second order differential equation by iteration of relevant [6]. ...
It is proved that different methods of integration of differential equations are related and can be unified applying differential topological methods which proves the basic tow degrees of freedom aspect of physics as the science of structurally stable dynamical systems.
... Jacobi sharpened Hamilton's formulation, by clarifying some mathematical issues. The resultant Hamilton-Jacobi theory and later developments are presented in several famous texts: [3,18,24,32,60,81,82]. For studies using modern PDE theory see [13,56,89]. ...
In this survey, we review the classical Hamilton Jacobi theory from a geometric point of view in different geometric backgrounds. We propose a Hamilton Jacobi equation for different geometric structures attending to one particular characterization: whether they fulfill the Jacobi and Leibniz identities simultaneously, or if at least they satisfy one of them. In this regard, we review the case of time dependent and dissipative physical systems as systems that fulfill the Jacobi identity but not the Leibnitz identity. Furthermore, we review the contact evolution Hamilton Jacobi theory as a split off the regular contact geometry, and that actually satisfies the Leibniz rule instead of Jacobi. Furthermore, we include a novel result, which is the Hamilton-Jacobi equation for conformal Hamiltonian vector fields as a generalization of the well known Hamilton Jacobi on a symplectic manifold, that is retrieved in the case of a zero conformal factor. The interest of a geometric Hamilton Jacobi equation is the primordial observation that if a Hamiltonian vector field can be projected into a configuration manifold by means of a 1-form dW, then the integral curves of the projected vector field can be transformed into integral curves of the Hamiltonian vector field provided that W is a solution of the Hamilton-Jacobi equation. Geometrically, the solution of the Hamilton Jacobi equation plays the role of a Lagrangian submanifold of a certain bundle. Exploiting these features in different geometric scenarios we propose a geometric theory for multiple physical systems depending on the fundamental identities that their dynamic satisfies. Different examples are pictured to reflect the results provided, being all of them new, except for one that is reassessment of a previously considered example.
In this paper we consider a three dimensional Kropina space and obtain the partial differential equation that characterizes a minimal surfaces with the induced metric. Using this characterization equation we study various immersions of minimal surfaces. In particular, we obtain the partial differential equation that characterizes the minimal translation surfaces and show that the plane is the only such surface.
We study, in a unified way, the following questions related to the properties of Pontryagin extremals for optimal control problems with unrestricted controls: i) How the transformations, which define the equivalence of two problems, transform the extremals? ii) How to obtain quantities which are conserved along any extremal? iii) How to assure that the set of extremals include the minimizers predicted by the existence theory? These questions are connected to: i) the Caratheodory method which establishes a correspondence between the minimizing curves of equivalent problems; ii) the interplay between the concept of invariance and the theory of optimality conditions in optimal control, which are the concern of the theorems of Noether; iii) regularity conditions for the minimizers and the work pioneered by Tonelli.
The objective of this article is to apply recent developments in geometric optimal control to analyze the time minimum control problem of dissipative two-level quantum systems whose dynamics is governed by the Lindblad equation. We focus our analysis on the case where the extremal Hamiltonian is integrable.
As was recently shown, non-relativistic quantum theory can be derived by means of a projection method from a continuum of classical solutions for (massive) particles. In this paper, we show that Maxwell’s equations in empty space can be derived using the same method. In this case, the starting point is a continuum of solutions of equations of motion for massless particles describing the structure of Galilean space-time. As a result of the projection, the space-time structure itself is changed by the appearance of a new fundamental constant c with the dimension of a velocity. This maximum velocity c , derived here for massless particles, is analogous to the accuracy limit ħ derived earlier for massive particles. The projection method can thus be interpreted as a generalized quantization. We suspect that all fundamental fields can be traced back to continuous sets of particle trajectories, and that in this sense, the particle concept is more fundamental than the field concept.
The notion of wind Finslerian structure Σ \Sigma is developed; this is a generalization of Finsler metrics (and Kropina ones) where the indicatrices at the tangent spaces may not contain the zero vector. In the particular case that these indicatrices are ellipsoids, called here wind Riemannian structures , they admit a double interpretation which provides: (a) a model for classical Zermelo’s navigation problem even when the trajectories of the moving objects (planes, ships) are influenced by strong winds or streams, and (b) a natural description of the causal structure of relativistic spacetimes endowed with a non-vanishing Killing vector field K K ( SSTK splittings ), in terms of Finslerian elements. These elements can be regarded as conformally invariant Killing initial data on a partial Cauchy hypersurface. The links between both interpretations as well as the possibility to improve the results on one of them using the other viewpoint are stressed.
The wind Finslerian structure Σ \Sigma is described in terms of two (conic, pseudo) Finsler metrics, F F and F l F_l , the former with a convex indicatrix and the latter with a concave one. Notions such as balls and geodesics are extended to Σ \Sigma . Among the applications, we obtain the solution of Zermelo’s navigation with arbitrary time-independent wind, metric-type properties for Σ \Sigma (distance-type arrival function, completeness, existence of minimizing, maximizing or closed geodesics), as well as description of spacetime elements (Cauchy developments, black hole horizons) in terms of Finslerian elements in Killing initial data. A general Fermat’s principle of independent interest for arbitrary spacetimes, as well as its applications to S STK\xspace spacetimes and Zermelo’s navigation, are also provided.
This paper mainly studies the contact extension of conservative or dissipative systems, including some old and new results for wholeness. Then extension of contact system is corresponding to the symplectification of contact Hamiltonian system. This is a reciprocal process and the relation between symplectic system and contact system has been discussed. We have an interesting discovery that by adding a pure variable p, the slope of the tangent of the orbit, every differential system can be regarded as an independent subsystem of contact Hamiltonian system defined on the projection space of contact phase space.
The use of oriented external electric fields (OEEFs) to promote and control chemical reactivity has motivated many theoretical and computational studies in the last decade to model the action of OEEFs on a molecular system and its effects on chemical processes. Given a reaction, a central goal in this research area is to predict the optimal OEEF (oOEEF) required to annihilate the reaction energy barrier with the smallest possible field strength. Here, we present a model rooted in catastrophe and optimum control theories that allows us to find the oOEEF for a given reaction valley in the potential energy surface (PES). In this model, the effective (or perturbed) PES of a polarizable molecular system is constructed by adding to the original, non-perturbed, PES a term accounting for the interaction of the OEEF with the intrinsic electric dipole and polarizability of the molecular system, so called the polarizable molecular electric dipole (PMED) model. We demonstrate that the oOEEF can be established by locating a point in the original PES with unique topological properties: the optimal barrier breakdown or bond-breaking point (oBBP). The essential feature of the oBBP structure is the fact that this point maintains its topological properties for all the applied OEEFs, also for the unperturbed PES, thus becoming much more relevant than the commonly used minima and transition state structures. The PMED model proposed here has been implemented in an open access package and is shown to successfully predict the oOEEF for two processes: an isomerization reaction of a cumulene derivative and the Huisgen cycloaddition reaction.
We introduce a symplectic bifoliation model of Information Geometry and Heat Theory based on Jean-Marie Souriau's Lie Groups Thermodynamics to describe transverse Poisson structure of metriplectic flow for dissipative phenomena. This model gives a cohomological characterization of Entropy, as an invariant Casimir function in coadjoint representation. The dual space of the Lie algebra foliates into coadjoint orbits identified with the Entropy level sets. In the framework of Thermodynamics, we associate a symplectic bifoliation structure to describe non-dissipative dynamics on symplectic leaves (on level sets of Entropy as constant Casimir function on each leaf), and transversal dissipative dynamics, given by Poisson transverse structure (Entropy production from leaf to leaf). The symplectic foliation orthogonal to the level sets of moment map is the foliation determined by hamiltonian vector fields generated by functions on dual Lie algebra. The orbits of a Hamiltonian action and the level sets of its moment map are polar to each other. The space of Casimir functions on a neighborhood of a point is isomorphic to the space of Casimirs for the transverse Poisson structure. Souriau’s model could be then interpreted by Mademoiselle Paulette Libermann's foliations, clarified as dual to Poisson Γ-structure of Haefliger, which is the maximum extension of the notion of moment in the sense of J.M. Souriau, as introduced by P. Molino, M. Condevaux and P. Dazord in papers of “Séminaire Sud-Rhodanien de Geometrie”. The symplectic duality to a symplectically complete foliation, in the sense of Libermann, associates an orthogonal foliation. Paulette Libermann proved that a Legendre foliation on a contact manifold is complete if and only if the pseudo-orthogonal distribution is completely integrable, and that the contact form is locally equivalent to the Poincaré-Cartan integral invariant. Paulette Libermann proved a classical theorem relating to co-isotropic foliations, which notably gives a proof of Darboux's theorem. Finally, we explore Edmond Fédida work on the theory of foliation structures in the language of fully integrable Pfaff systems associated with the Cartan’s moving frame.KeywordsBifoliationCoadjoint OrbitsMoment MapCasimir FunctionMetriplectic FlowTransverse Poisson StructureInformation GeometryMaurer-CartanMoving Frame
We introduce a topological description of variational principle and optimal control by differential topological methods which is related to the structural stability of dynamical systems under small perturbations and their integrability. The genuine relations between basic models, concepts and quantities of variational problem including optimal control such as compact manifolds, convex structures, Riemannian geometry, Hilbert space, probability aspects, geodesics and symplectic geometry are disclosed and described by topological methods.
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