In this article, we study isoperimetric problems of the calculus of variations with left and right Riemann-Liouville fractional derivatives. Both situations when the lower bound of the variational integrals coincide a...In this article, we study isoperimetric problems of the calculus of variations with left and right Riemann-Liouville fractional derivatives. Both situations when the lower bound of the variational integrals coincide and do not coincide with the lower bound of the fractional derivatives are considered.展开更多
This paper is devoted to outlining precisely the basic mathematics of a classical isoperimetric problem of the calculus of variations and showing how significant fluid mechanical problems in fluidization and spouting ...This paper is devoted to outlining precisely the basic mathematics of a classical isoperimetric problem of the calculus of variations and showing how significant fluid mechanical problems in fluidization and spouting can be addressed using this approach.展开更多
This paper deals with minimization problems in the calculus of variations set in a sequence of domains, the size of which tends to infinity in certain directions and such that the data only depend on the coordinates i...This paper deals with minimization problems in the calculus of variations set in a sequence of domains, the size of which tends to infinity in certain directions and such that the data only depend on the coordinates in the directions that remain constant. The authors study the asymptotic behavior of minimizers in various situations and show that they converge in an appropriate sense toward minimizers of a related energy functional in the constant directions.展开更多
It is known that Fourier’s heat equation, which is parabolic, implies an infinite velocity propagation, or, in other words, that the mechanism of heat conduction is established instantaneously under all conditions. T...It is known that Fourier’s heat equation, which is parabolic, implies an infinite velocity propagation, or, in other words, that the mechanism of heat conduction is established instantaneously under all conditions. This is unacceptable on physical grounds in spite of the fact that Fourier’s law agrees well with experiment. However, discrepancies are likely to occur when extremely short distances or extremely short time intervals are considered, as they must in some modern problems of aero-thermodynamics. Cattaneo and independently Vernotte proved that such process can be described by Heaviside’s telegraph equation. This paper shows that this fact can be derived using calculus of variations, by application of the Euler-Lagrange equation. So, we proved that the equation of heat conduction with finite velocity propagation of the thermal disturbance can be obtained as a solution to one variational problem.展开更多
The paper proposes an approximate solution to the classical (parabolic) multidimensional 2D and 3D heat conduction equation for a 5 × 5 cm aluminium plate and a 5 × 5 × 5 cm aluminum cube. An approximat...The paper proposes an approximate solution to the classical (parabolic) multidimensional 2D and 3D heat conduction equation for a 5 × 5 cm aluminium plate and a 5 × 5 × 5 cm aluminum cube. An approximate solution of the generalized (hyperbolic) 2D and 3D equation for the considered plate and cube is also proposed. Approximate solutions were obtained by applying calculus of variations and Euler-Lagrange equations. In order to verify the correctness of the proposed approximate solutions, they were compared with the exact solutions of parabolic and hyperbolic equations. The paper also presents the research on the influence of time parameters τ as well as the relaxation times τ ∗ to the variation of the profile of the temperature field for the considered aluminum plate and cube.展开更多
Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different ...Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different necessary conditions of invariance are obtained. As particular cases, we prove fractional versions of Noether's symmetry theorem. Invariant conditions for fractional optimal control problems, using the Hamiltonian formalism, are also investigated. As an example of potential application in Physics, we show that with conformable derivatives it is possible to formulate an Action Principle for particles under frictional forces that is far simpler than the one obtained with classical fractional derivatives.展开更多
First of all, Laurent series expansions of stress and displacement fields satisfying all the governing equations of plane problems in theory of elasticity are derived. Next, the variational method is applied to satisf...First of all, Laurent series expansions of stress and displacement fields satisfying all the governing equations of plane problems in theory of elasticity are derived. Next, the variational method is applied to satisfy the boundary conditions of a plate with a pin hole. Then, the coefficients in Laurent series and the stress concentration factor can be determined. Finally, the convergency tests and systematical results are given for engineering applications.展开更多
Isoperimetric problems consist in minimizing or maximizing a cost functional subject to an integral constraint.In this work, we present two fractional isoperimetric problems where the Lagrangian depends on a combined ...Isoperimetric problems consist in minimizing or maximizing a cost functional subject to an integral constraint.In this work, we present two fractional isoperimetric problems where the Lagrangian depends on a combined Caputo derivative of variable fractional order and we present a new variational problem subject to a holonomic constraint. We establish necessary optimality conditions in order to determine the minimizers of the fractional problems. The terminal point in the cost integral,as well as the terminal state, are considered to be free, and we obtain corresponding natural boundary conditions.展开更多
According to the necessary condition of the functional taking the extremum, that is its first variation is equal to zero, the variational problems of the functionals for the undetermined boundary in the calculus of va...According to the necessary condition of the functional taking the extremum, that is its first variation is equal to zero, the variational problems of the functionals for the undetermined boundary in the calculus of variations are researched, the functionals depend on single argument, arbitrary unknown functions and their derivatives of higher orders. A new view point is posed and demonstrated, i.e. when the first variation of the functional is equal to zero, all the variational terms are not independent to each other, and at least one of them is equal to zero. Some theorems and corollaries of the variational problems of the functionals are obtained.展开更多
The long-term stability of backfill material is the key to retaining roadways successfully. In order to study the rheological deformation of backfill material and its long-term stability, given the visco-elastoplastic...The long-term stability of backfill material is the key to retaining roadways successfully. In order to study the rheological deformation of backfill material and its long-term stability, given the visco-elastoplastic properties of this material, we introduced a softening and a hardening function for a new nonlinear the- ological model with time-varying parameters. Based on this, we presented the instability condition of this model by using the principle of minimum potential energy. Combined with engineering practice, we cal- culated the urlstable time period of backfill material. The results show that the time of instability of the backfill material relate to the initial parameters of the material, ＂the coefficients decided by temperature and the ratio of the plastic zone of the backfill material. Based on the results of our analysis from the point of view of energy, we can quickly obtain the time of instability of this model from our graphical analysis. The time of instability of the backfill material obtained from our investigation coincides with an actual project.展开更多
Energy methods and the principle of virtual work are commonly used for obtaining solutions of boundary value problems (BVPs) and initial value problems (IVPs) associated with homogeneous, isotropic and non-homogeneous...Energy methods and the principle of virtual work are commonly used for obtaining solutions of boundary value problems (BVPs) and initial value problems (IVPs) associated with homogeneous, isotropic and non-homogeneous, non-isotropic matter without using (or in the absence of) the mathematical models of the BVPs and the IVPs. These methods are also used for deriving mathematical models for BVPs and IVPs associated with isotropic, homogeneous as well as non-homogeneous, non-isotropic continuous matter. In energy methods when applied to IVPs, one constructs energy functional (<i>I</i>) consisting of kinetic energy, strain energy and the potential energy of loads. The first variation of this energy functional (<em>δI</em>) set to zero is a necessary condition for an extremum of <i>I</i>. In this approach one could use <i>δI</i> = 0 directly in constructing computational processes such as the finite element method or could derive Euler’s equations (differential or partial differential equations) from <i>δI</i> = 0, which is also satisfied by a solution obtained from <i>δI</i> = 0. The Euler’s equations obtained from <i>δI</i> = 0 indeed are the mathematical model associated with the energy functional <i>I</i>. In case of BVPs we follow the same approach except in this case, the energy functional <i>I</i> consists of strain energy and the potential energy of loads. In using the principle of virtual work for BVPs and the IVPs, we can also accomplish the same as described above using energy methods. In this paper we investigate consistency and validity of the mathematical models for isotropic, homogeneous and non-isotropic, non-homogeneous continuous matter for BVPs that are derived using energy functional consisting of strain energy and the potential energy of loads. Similar investigation is also presented for IVPs using energy functional consisting of kinetic energy, strain energy and the potential energy of loads. The computational approaches for BVPs and the IVPs designed using energy functional and principle of virtual work, their consistency and validity are also investigated. Classical continuum mechanics (CCM) principles <i>i.e.</i> conservation and balance laws of CCM with consistent constitutive theories and the elements of calculus of variations are employed in the investigations presented in this paper.展开更多
The fractional Pfaffian variational problem and Noether’s theorems were investigated in terms of Riemann-Liouville derivatives on the basis of El-Nabulsi fractional model.The problem of the calculus of variations wit...The fractional Pfaffian variational problem and Noether’s theorems were investigated in terms of Riemann-Liouville derivatives on the basis of El-Nabulsi fractional model.The problem of the calculus of variations with fractional derivatives is a hot topic recently.Firstly,within Riemann-Liouville derivatives,the ElNabulsi Pfaffian variational problem was presented,the fractional Pfaff-Birkhoff-d’Alembert principle was established,and the fractional Birkhoff equations and the corresponding transversality conditions were obtained.Then,the Noether’s theorems in terms of Riemann-Liouville derivatives for the Birkhoffian system on the basis of El-Nabulsi fractional model are investigated under the special and the general transformations respectively.Finally,an example is given to illustrate the methods and results appeared in this paper.展开更多
An exact approach for free transverse vibrations of a Timoshenko beam with ends elastically restrained against rotation and translation and arbitrarily located internal restraints is presented. The calculus of variati...An exact approach for free transverse vibrations of a Timoshenko beam with ends elastically restrained against rotation and translation and arbitrarily located internal restraints is presented. The calculus of variations is used to obtain the equations of motion, the boundary conditions and the transitions conditions which correspond to the described mechanical system. The derived differential equations are solved individually for each segment of the beam with the corresponding boundary and transitions conditions. The derived mathematical formulation generates as particular cases, and several mathematical models are used to simulate the presence of cracks. Some cases available in the literature and the presence of some errors are discussed. New results are presented for different end conditions and restraint conditions in the intermediate elastic constraints with their corresponding modal shapes.展开更多
The principle of least action, which has so successfully been applied to diverse fields of physics looks back at three centuries of philosophical and mathematical discussions and controversies. They could not explain ...The principle of least action, which has so successfully been applied to diverse fields of physics looks back at three centuries of philosophical and mathematical discussions and controversies. They could not explain why nature is applying the principle and why scalar energy quantities succeed in describing dynamic motion. When the least action integral is subdivided into infinitesimal small sections each one has to maintain the ability to minimize. This however has the mathematical consequence that the Lagrange function at a given point of the trajectory, the dynamic, available energy generating motion, must itself have a fundamental property to minimize. Since a scalar quantity, a pure number, cannot do that, energy must fundamentally be dynamic and time oriented for a consistent understanding. It must have vectorial properties in aiming at a decrease of free energy per state (which would also allow derivation of the second law of thermodynamics). Present physics is ignoring that and applying variation calculus as a formal mathematical tool to impose a minimization of scalar assumed energy quantities for obtaining dynamic motion. When, however, the dynamic property of energy is taken seriously it is fundamental and has also to be applied to quantum processes. A consequence is that particle and wave are not equivalent, but the wave (distributed energy) follows from the first (concentrated energy). Information, provided from the beginning, an information self-image of matter, is additionally needed to recreate the particle from the wave, shaping a “dynamic” particle-wave duality. It is shown that this new concept of a “dynamic” quantum state rationally explains quantization, the double slit experiment and quantum correlation, which has not been possible before. Some more general considerations on the link between quantum processes, gravitation and cosmological phenomena are also advanced.展开更多
Monopoles and vortices are well known magnetically charged soliton solutions of gauge field equations. Extending the idea of Dirac on monopoles, Schwinger pioneered the concept of solitons carrying both electric and m...Monopoles and vortices are well known magnetically charged soliton solutions of gauge field equations. Extending the idea of Dirac on monopoles, Schwinger pioneered the concept of solitons carrying both electric and magnetic charges, called dyons, which are useful in modeling elementary particles. Mathematically, the existence of dyons presents interesting variational partial differential equation problems, subject to topological constraints. This article is a survey on recent progress in the study of dyons.展开更多
Utilizing stress-energy tensors which allow for a divergence-free formulation, we establish Pohozaev＇s identity for certain classes of quasilinear systems with variational structure.
An approach was presented to intensify the mixing process. Firstly, a novel concept, the dissipation of mass transfer ability(DMA) associated with convective mass transfer, was defined via an analogy to the heat-work ...An approach was presented to intensify the mixing process. Firstly, a novel concept, the dissipation of mass transfer ability(DMA) associated with convective mass transfer, was defined via an analogy to the heat-work conversion. Accordingly, the focus on mass transfer enhancement can be shifted to seek the extremum of the DMA of the system. To this end, an optimization principle was proposed. A mathematical model was then developed to formulate the optimization into a variational problem. Subsequently, the intensification of the mixing process for a gas mixture in a micro-tube was provided to demonstrate the proposed principle. In the demonstration example, an optimized velocity field was obtained in which the mixing ability was improved, i.e., the mixing process should be intensified by adjusting the velocity field in related equipment. Therefore, a specific procedure was provided to produce a mixer with geometric irregularities associated with an ideal velocity.展开更多
Optical vortices arise as phase dislocations of light fields and they are of importance in modern optical physics.In this study,we employ the calculus of variations method to develop an existence theory for the steady...Optical vortices arise as phase dislocations of light fields and they are of importance in modern optical physics.In this study,we employ the calculus of variations method to develop an existence theory for the steady state vortex solutions of a nonlinear Schr?dinger type equation to model light waves that propagate in a medium with a new focusing-defocusing nonlinearity.First,we demonstrate the existence of positive radially symmetric solutions by constrained minimization,where we give some interesting explicit estimates related to vortex winding numbers and the wave propagation constant.Second,we establish the existence of saddle-point solutions through a mountain-pass argument.展开更多
In this study,we applied the variational model to fluidization of small spherical particles.Fluidization experiments were carried out for spherical particles with 13 diameters between dp=0.13 and 5.00 mm.We propose a ...In this study,we applied the variational model to fluidization of small spherical particles.Fluidization experiments were carried out for spherical particles with 13 diameters between dp=0.13 and 5.00 mm.We propose a generalized form of our variational model to predict the superficial velocity U and interphase drag coefficientβby introducing an exponent n to describe the different dependences of the drag force Fd on fluid velocity for different particle sizes(different flow regimes).By comparing the predictions with the experimental results,we conclude that n=1 should be used for small particles(dp<1 mm)and n=2 for larger particles(dp>1 mm).This conclusion is generalized by proposing n=1 for particles with Ret<160 and n=2 for particles with Ret>160.The average mean absolute error was 5.49%in calculating superficial velocity for different bed voidages using the modified variational model for all of the particles examined.The calculated values ofβwere compared with values of literature models for particles with dp<1.0 mm.The average mean absolute error of the modified variational model was 8.02%in calculatingβfor different bed voidages for all of the particles examined.展开更多
We derive the F-limit of scaled elastic energies h-4Eh（uh） associated with deformations Uh of a family of thin shells Sh = {z = x ＋ tn（x）; x ∈ S, -g1h（x） 〈 t 〈 g2h（x）}. The obtained yon Karman theory is va...We derive the F-limit of scaled elastic energies h-4Eh（uh） associated with deformations Uh of a family of thin shells Sh = {z = x ＋ tn（x）; x ∈ S, -g1h（x） 〈 t 〈 g2h（x）}. The obtained yon Karman theory is valid for a general sequence of boundaries h h gl, g2 converging to 0 in an appropriate manner as h vanishes. Our analysis relies on the techniques and extends the results in [10] and [11].展开更多
基金supported by FEDER funds through COMPETE - Operational Programme Factors of Competitiveness("Programa Operacional Factores de Competitividade")Portuguese funds through the Center for Research and Development in Mathematics and Applications(University of Aveiro) and the Portuguese Foundation for Science and Technology("FCT - Fundao para a Ciencia e a Tecnologia"),within project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690
文摘In this article, we study isoperimetric problems of the calculus of variations with left and right Riemann-Liouville fractional derivatives. Both situations when the lower bound of the variational integrals coincide and do not coincide with the lower bound of the fractional derivatives are considered.
文摘This paper is devoted to outlining precisely the basic mathematics of a classical isoperimetric problem of the calculus of variations and showing how significant fluid mechanical problems in fluidization and spouting can be addressed using this approach.
文摘This paper deals with minimization problems in the calculus of variations set in a sequence of domains, the size of which tends to infinity in certain directions and such that the data only depend on the coordinates in the directions that remain constant. The authors study the asymptotic behavior of minimizers in various situations and show that they converge in an appropriate sense toward minimizers of a related energy functional in the constant directions.
文摘It is known that Fourier’s heat equation, which is parabolic, implies an infinite velocity propagation, or, in other words, that the mechanism of heat conduction is established instantaneously under all conditions. This is unacceptable on physical grounds in spite of the fact that Fourier’s law agrees well with experiment. However, discrepancies are likely to occur when extremely short distances or extremely short time intervals are considered, as they must in some modern problems of aero-thermodynamics. Cattaneo and independently Vernotte proved that such process can be described by Heaviside’s telegraph equation. This paper shows that this fact can be derived using calculus of variations, by application of the Euler-Lagrange equation. So, we proved that the equation of heat conduction with finite velocity propagation of the thermal disturbance can be obtained as a solution to one variational problem.
文摘The paper proposes an approximate solution to the classical (parabolic) multidimensional 2D and 3D heat conduction equation for a 5 × 5 cm aluminium plate and a 5 × 5 × 5 cm aluminum cube. An approximate solution of the generalized (hyperbolic) 2D and 3D equation for the considered plate and cube is also proposed. Approximate solutions were obtained by applying calculus of variations and Euler-Lagrange equations. In order to verify the correctness of the proposed approximate solutions, they were compared with the exact solutions of parabolic and hyperbolic equations. The paper also presents the research on the influence of time parameters τ as well as the relaxation times τ ∗ to the variation of the profile of the temperature field for the considered aluminum plate and cube.
基金supported by CNPq and CAPES(Brazilian research funding agencies)Portuguese funds through the Center for Research and Development in Mathematics and Applications(CIDMA)the Portuguese Foundation for Science and Technology(FCT),within project UID/MAT/04106/2013
文摘Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different necessary conditions of invariance are obtained. As particular cases, we prove fractional versions of Noether's symmetry theorem. Invariant conditions for fractional optimal control problems, using the Hamiltonian formalism, are also investigated. As an example of potential application in Physics, we show that with conformable derivatives it is possible to formulate an Action Principle for particles under frictional forces that is far simpler than the one obtained with classical fractional derivatives.
文摘First of all, Laurent series expansions of stress and displacement fields satisfying all the governing equations of plane problems in theory of elasticity are derived. Next, the variational method is applied to satisfy the boundary conditions of a plate with a pin hole. Then, the coefficients in Laurent series and the stress concentration factor can be determined. Finally, the convergency tests and systematical results are given for engineering applications.
基金supported by Portuguese Funds through the Center for Research and Development in Mathematics and Applications(CIDMA)the Portuguese Foundation for Science and Technology(FCT)(UID/MAT/04106/2013)supported by FCT through the Ph.D. fellowship SFRH/BD/42557/2007
文摘Isoperimetric problems consist in minimizing or maximizing a cost functional subject to an integral constraint.In this work, we present two fractional isoperimetric problems where the Lagrangian depends on a combined Caputo derivative of variable fractional order and we present a new variational problem subject to a holonomic constraint. We establish necessary optimality conditions in order to determine the minimizers of the fractional problems. The terminal point in the cost integral,as well as the terminal state, are considered to be free, and we obtain corresponding natural boundary conditions.
文摘According to the necessary condition of the functional taking the extremum, that is its first variation is equal to zero, the variational problems of the functionals for the undetermined boundary in the calculus of variations are researched, the functionals depend on single argument, arbitrary unknown functions and their derivatives of higher orders. A new view point is posed and demonstrated, i.e. when the first variation of the functional is equal to zero, all the variational terms are not independent to each other, and at least one of them is equal to zero. Some theorems and corollaries of the variational problems of the functionals are obtained.
基金Project (No. 50874089) is supported by the National Natural Science Foundation of ChinaProject (No. 20096121110002) by the College of Doctoral Foundation of the Ministry of Education the Scientific Research Program Funded by Shaanxi Provincial Education Commission (No. 2010JK692)
文摘The long-term stability of backfill material is the key to retaining roadways successfully. In order to study the rheological deformation of backfill material and its long-term stability, given the visco-elastoplastic properties of this material, we introduced a softening and a hardening function for a new nonlinear the- ological model with time-varying parameters. Based on this, we presented the instability condition of this model by using the principle of minimum potential energy. Combined with engineering practice, we cal- culated the urlstable time period of backfill material. The results show that the time of instability of the backfill material relate to the initial parameters of the material, ＂the coefficients decided by temperature and the ratio of the plastic zone of the backfill material. Based on the results of our analysis from the point of view of energy, we can quickly obtain the time of instability of this model from our graphical analysis. The time of instability of the backfill material obtained from our investigation coincides with an actual project.
文摘Energy methods and the principle of virtual work are commonly used for obtaining solutions of boundary value problems (BVPs) and initial value problems (IVPs) associated with homogeneous, isotropic and non-homogeneous, non-isotropic matter without using (or in the absence of) the mathematical models of the BVPs and the IVPs. These methods are also used for deriving mathematical models for BVPs and IVPs associated with isotropic, homogeneous as well as non-homogeneous, non-isotropic continuous matter. In energy methods when applied to IVPs, one constructs energy functional (<i>I</i>) consisting of kinetic energy, strain energy and the potential energy of loads. The first variation of this energy functional (<em>δI</em>) set to zero is a necessary condition for an extremum of <i>I</i>. In this approach one could use <i>δI</i> = 0 directly in constructing computational processes such as the finite element method or could derive Euler’s equations (differential or partial differential equations) from <i>δI</i> = 0, which is also satisfied by a solution obtained from <i>δI</i> = 0. The Euler’s equations obtained from <i>δI</i> = 0 indeed are the mathematical model associated with the energy functional <i>I</i>. In case of BVPs we follow the same approach except in this case, the energy functional <i>I</i> consists of strain energy and the potential energy of loads. In using the principle of virtual work for BVPs and the IVPs, we can also accomplish the same as described above using energy methods. In this paper we investigate consistency and validity of the mathematical models for isotropic, homogeneous and non-isotropic, non-homogeneous continuous matter for BVPs that are derived using energy functional consisting of strain energy and the potential energy of loads. Similar investigation is also presented for IVPs using energy functional consisting of kinetic energy, strain energy and the potential energy of loads. The computational approaches for BVPs and the IVPs designed using energy functional and principle of virtual work, their consistency and validity are also investigated. Classical continuum mechanics (CCM) principles <i>i.e.</i> conservation and balance laws of CCM with consistent constitutive theories and the elements of calculus of variations are employed in the investigations presented in this paper.
基金National Natural Science Foundations of China(Nos.11572212,11272227,10972151)the Innovation Program for Scientific Research of Nanjing University of Science and Technology,Chinathe Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province,China(No.KYLX15_0405)
文摘The fractional Pfaffian variational problem and Noether’s theorems were investigated in terms of Riemann-Liouville derivatives on the basis of El-Nabulsi fractional model.The problem of the calculus of variations with fractional derivatives is a hot topic recently.Firstly,within Riemann-Liouville derivatives,the ElNabulsi Pfaffian variational problem was presented,the fractional Pfaff-Birkhoff-d’Alembert principle was established,and the fractional Birkhoff equations and the corresponding transversality conditions were obtained.Then,the Noether’s theorems in terms of Riemann-Liouville derivatives for the Birkhoffian system on the basis of El-Nabulsi fractional model are investigated under the special and the general transformations respectively.Finally,an example is given to illustrate the methods and results appeared in this paper.
文摘An exact approach for free transverse vibrations of a Timoshenko beam with ends elastically restrained against rotation and translation and arbitrarily located internal restraints is presented. The calculus of variations is used to obtain the equations of motion, the boundary conditions and the transitions conditions which correspond to the described mechanical system. The derived differential equations are solved individually for each segment of the beam with the corresponding boundary and transitions conditions. The derived mathematical formulation generates as particular cases, and several mathematical models are used to simulate the presence of cracks. Some cases available in the literature and the presence of some errors are discussed. New results are presented for different end conditions and restraint conditions in the intermediate elastic constraints with their corresponding modal shapes.
文摘The principle of least action, which has so successfully been applied to diverse fields of physics looks back at three centuries of philosophical and mathematical discussions and controversies. They could not explain why nature is applying the principle and why scalar energy quantities succeed in describing dynamic motion. When the least action integral is subdivided into infinitesimal small sections each one has to maintain the ability to minimize. This however has the mathematical consequence that the Lagrange function at a given point of the trajectory, the dynamic, available energy generating motion, must itself have a fundamental property to minimize. Since a scalar quantity, a pure number, cannot do that, energy must fundamentally be dynamic and time oriented for a consistent understanding. It must have vectorial properties in aiming at a decrease of free energy per state (which would also allow derivation of the second law of thermodynamics). Present physics is ignoring that and applying variation calculus as a formal mathematical tool to impose a minimization of scalar assumed energy quantities for obtaining dynamic motion. When, however, the dynamic property of energy is taken seriously it is fundamental and has also to be applied to quantum processes. A consequence is that particle and wave are not equivalent, but the wave (distributed energy) follows from the first (concentrated energy). Information, provided from the beginning, an information self-image of matter, is additionally needed to recreate the particle from the wave, shaping a “dynamic” particle-wave duality. It is shown that this new concept of a “dynamic” quantum state rationally explains quantization, the double slit experiment and quantum correlation, which has not been possible before. Some more general considerations on the link between quantum processes, gravitation and cosmological phenomena are also advanced.
文摘Monopoles and vortices are well known magnetically charged soliton solutions of gauge field equations. Extending the idea of Dirac on monopoles, Schwinger pioneered the concept of solitons carrying both electric and magnetic charges, called dyons, which are useful in modeling elementary particles. Mathematically, the existence of dyons presents interesting variational partial differential equation problems, subject to topological constraints. This article is a survey on recent progress in the study of dyons.
文摘Utilizing stress-energy tensors which allow for a divergence-free formulation, we establish Pohozaev＇s identity for certain classes of quasilinear systems with variational structure.
基金Supported by the National Basic Research Program of China("973" Program,No.2012CB720500)the National Natural Science Foundation of China(No.21176171)
文摘An approach was presented to intensify the mixing process. Firstly, a novel concept, the dissipation of mass transfer ability(DMA) associated with convective mass transfer, was defined via an analogy to the heat-work conversion. Accordingly, the focus on mass transfer enhancement can be shifted to seek the extremum of the DMA of the system. To this end, an optimization principle was proposed. A mathematical model was then developed to formulate the optimization into a variational problem. Subsequently, the intensification of the mixing process for a gas mixture in a micro-tube was provided to demonstrate the proposed principle. In the demonstration example, an optimized velocity field was obtained in which the mixing ability was improved, i.e., the mixing process should be intensified by adjusting the velocity field in related equipment. Therefore, a specific procedure was provided to produce a mixer with geometric irregularities associated with an ideal velocity.
基金the National Natural Science Foundation of China(No.11471099)the National Natural Science Foundation of He’nan Province of China(No.222300420416)。
文摘Optical vortices arise as phase dislocations of light fields and they are of importance in modern optical physics.In this study,we employ the calculus of variations method to develop an existence theory for the steady state vortex solutions of a nonlinear Schr?dinger type equation to model light waves that propagate in a medium with a new focusing-defocusing nonlinearity.First,we demonstrate the existence of positive radially symmetric solutions by constrained minimization,where we give some interesting explicit estimates related to vortex winding numbers and the wave propagation constant.Second,we establish the existence of saddle-point solutions through a mountain-pass argument.
基金This work was supported by the Serbian Ministry of Edu-cation,Science and Technological Development(grant number ON172022).
文摘In this study,we applied the variational model to fluidization of small spherical particles.Fluidization experiments were carried out for spherical particles with 13 diameters between dp=0.13 and 5.00 mm.We propose a generalized form of our variational model to predict the superficial velocity U and interphase drag coefficientβby introducing an exponent n to describe the different dependences of the drag force Fd on fluid velocity for different particle sizes(different flow regimes).By comparing the predictions with the experimental results,we conclude that n=1 should be used for small particles(dp<1 mm)and n=2 for larger particles(dp>1 mm).This conclusion is generalized by proposing n=1 for particles with Ret<160 and n=2 for particles with Ret>160.The average mean absolute error was 5.49%in calculating superficial velocity for different bed voidages using the modified variational model for all of the particles examined.The calculated values ofβwere compared with values of literature models for particles with dp<1.0 mm.The average mean absolute error of the modified variational model was 8.02%in calculatingβfor different bed voidages for all of the particles examined.
基金partially supported by Professor Marta Lewicka’s NSF grants DMS-0707275 and DMS-0846996her Polish MN grant N N201 547438
文摘We derive the F-limit of scaled elastic energies h-4Eh（uh） associated with deformations Uh of a family of thin shells Sh = {z = x ＋ tn（x）; x ∈ S, -g1h（x） 〈 t 〈 g2h（x）}. The obtained yon Karman theory is valid for a general sequence of boundaries h h gl, g2 converging to 0 in an appropriate manner as h vanishes. Our analysis relies on the techniques and extends the results in [10] and [11].
