The clamped plate problem describes the vibration of a clamped plate in the classical elastic mechanics, and the Xin-Laplacian is an important elliptic operator for understanding the geometric structure of translators of mean curvature flow(MCF for short). In this article, we investigate the clamped plate problem of the bi-Xin-Laplacian on Riemannian manifolds isometrically immersed in the Euclidean space. On one hand, we obtain some eigenvalue inequalities of the bi-Xin-Laplacian on some important Riemannian manifolds admitting some special functions. Let us emphasize that, this class of manifolds contains some interesting examples: Cartan-Hadamard manifolds, some types of warp product manifolds and homogenous spaces. On the other hand, we also consider the eigenvalue problem of the bi-Xin-Laplacian on the cylinders and obtain an eigenvalue inequality. In particular, we can give an estimate for the lower order eigenvalues on the cylinders.
{"title":"Eigenvalues of the bi-Xin-Laplacian on complete Riemannian manifolds","authors":"Xiaotian Hao, Lingzhong Zeng","doi":"10.3934/cam.2023009","DOIUrl":"https://doi.org/10.3934/cam.2023009","url":null,"abstract":"The clamped plate problem describes the vibration of a clamped plate in the classical elastic mechanics, and the Xin-Laplacian is an important elliptic operator for understanding the geometric structure of translators of mean curvature flow(MCF for short). In this article, we investigate the clamped plate problem of the bi-Xin-Laplacian on Riemannian manifolds isometrically immersed in the Euclidean space. On one hand, we obtain some eigenvalue inequalities of the bi-Xin-Laplacian on some important Riemannian manifolds admitting some special functions. Let us emphasize that, this class of manifolds contains some interesting examples: Cartan-Hadamard manifolds, some types of warp product manifolds and homogenous spaces. On the other hand, we also consider the eigenvalue problem of the bi-Xin-Laplacian on the cylinders and obtain an eigenvalue inequality. In particular, we can give an estimate for the lower order eigenvalues on the cylinders.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124517312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The dissipative (2 + 1)-dimensional AKNS equation is considered in this paper. First, the Lie symmetry analysis method is applied to the dissipative (2 + 1)-dimensional AKNS and six point symmetries are obtained. Symmetry reductions are performed by utilizing these obtained point symmetries and four differential equations are derived, including a fourth-order ordinary differential equation and three partial differential equations. Thereafter, the direct integration approach and the $ (G'/G^{2})- $expansion method are employed to solve the ordinary differential respectively. As a result, a periodic solution in terms of the Weierstrass elliptic function is obtained via the the direct integration approach, while six kinds of including the hyperbolic function types and the hyperbolic function types are derived via the $ (G'/G^{2})- $expansion method. The corresponding graphical representation of the obtained solutions are presented by choosing suitable parametric values. Finally, the multiplier technique and the classical Noether's theorem are employed to derive conserved vectors for the dissipative (2 + 1)-dimensional AKNS respectively. Consequently, eight local conservation laws for the dissipative (2 + 1)-dimensional AKNS equation are presented by utilizing the multiplier technique and five local conservation laws are derived by invoking Noether's theorem.
{"title":"Lie symmetry analysis, particular solutions and conservation laws for the dissipative (2 + 1)- dimensional AKNS equation","authors":"Sixing Tao","doi":"10.3934/cam.2023024","DOIUrl":"https://doi.org/10.3934/cam.2023024","url":null,"abstract":"The dissipative (2 + 1)-dimensional AKNS equation is considered in this paper. First, the Lie symmetry analysis method is applied to the dissipative (2 + 1)-dimensional AKNS and six point symmetries are obtained. Symmetry reductions are performed by utilizing these obtained point symmetries and four differential equations are derived, including a fourth-order ordinary differential equation and three partial differential equations. Thereafter, the direct integration approach and the $ (G'/G^{2})- $expansion method are employed to solve the ordinary differential respectively. As a result, a periodic solution in terms of the Weierstrass elliptic function is obtained via the the direct integration approach, while six kinds of including the hyperbolic function types and the hyperbolic function types are derived via the $ (G'/G^{2})- $expansion method. The corresponding graphical representation of the obtained solutions are presented by choosing suitable parametric values. Finally, the multiplier technique and the classical Noether's theorem are employed to derive conserved vectors for the dissipative (2 + 1)-dimensional AKNS respectively. Consequently, eight local conservation laws for the dissipative (2 + 1)-dimensional AKNS equation are presented by utilizing the multiplier technique and five local conservation laws are derived by invoking Noether's theorem.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115995442","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note we give a short proof of Arnold's conjecture for the zero section of a cotangent bundle of a closed manifold. The proof is based on some basic properties of Lagrangian spectral invariants from Floer theory.
{"title":"A short proof of cuplength estimates on Lagrangian intersections","authors":"Wenmin Gong","doi":"10.3934/cam.2023003","DOIUrl":"https://doi.org/10.3934/cam.2023003","url":null,"abstract":"<abstract><p>In this note we give a short proof of Arnold's conjecture for the zero section of a cotangent bundle of a closed manifold. The proof is based on some basic properties of Lagrangian spectral invariants from Floer theory.</p></abstract>","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133254838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper focuses on the spectral analysis of equations that describe the oscillations of a heavy pendulum partially filled with a homogeneous incompressible viscoelastic fluid. The constitutive equation of the fluid follows the simpler Oldroyd model. By examining the eigenvalues of the linear operator that describes the dynamics of the coupled system, it was demonstrated that under appropriate assumptions the equilibrium configuration remains stable in the linear approximation. Moreover, when the viscosity coefficient is sufficiently large the spectrum comprises three branches of eigenvalues with potential cluster points at $ 0 $, $ beta $ and $ infty $ where $ beta $ represents the viscoelastic parameter of the fluid. These three branches of eigenvalues correspond to frequencies associated with various types of waves.
{"title":"Analysis of small oscillations of a pendulum partially filled with a viscoelastic fluid","authors":"H. Essaouini, P. Capodanno","doi":"10.3934/cam.2023019","DOIUrl":"https://doi.org/10.3934/cam.2023019","url":null,"abstract":"This paper focuses on the spectral analysis of equations that describe the oscillations of a heavy pendulum partially filled with a homogeneous incompressible viscoelastic fluid. The constitutive equation of the fluid follows the simpler Oldroyd model. By examining the eigenvalues of the linear operator that describes the dynamics of the coupled system, it was demonstrated that under appropriate assumptions the equilibrium configuration remains stable in the linear approximation. Moreover, when the viscosity coefficient is sufficiently large the spectrum comprises three branches of eigenvalues with potential cluster points at $ 0 $, $ beta $ and $ infty $ where $ beta $ represents the viscoelastic parameter of the fluid. These three branches of eigenvalues correspond to frequencies associated with various types of waves.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128321702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we introduce the Laguerre bounded variation space and the Laguerre perimeter, thereby investigating their properties. Moreover, we prove the isoperimetric inequality and the Sobolev inequality in the Laguerre setting. As applications, we derive the mean curvature for the Laguerre perimeter.
{"title":"Laguerre BV spaces, Laguerre perimeter and their applications","authors":"Heming Wang, Yu Liu","doi":"10.3934/cam.2023011","DOIUrl":"https://doi.org/10.3934/cam.2023011","url":null,"abstract":"In this paper, we introduce the Laguerre bounded variation space and the Laguerre perimeter, thereby investigating their properties. Moreover, we prove the isoperimetric inequality and the Sobolev inequality in the Laguerre setting. As applications, we derive the mean curvature for the Laguerre perimeter.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126456016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, our main concern is to study the existence and multiplicity of solutions for the following sub-elliptic system with Hardy type potentials and multiple critical exponents on Carnot group begin{document}$ begin{equation*} left{begin{aligned} &-Delta_{mathbb{G}}u = frac{psi^{alpha}|u|^{2^*(alpha)-2}u}{d(z)^{alpha}}+ frac{p_{1}}{2^*(gamma)}frac{psi^{gamma}|u|^{p_{1}-2}u|v|^{p_{2}}}{d(z, z_{0})^{gamma}} +lambda h(z)frac{psi^{sigma}|u|^{q-2}u}{d(z)^{sigma}} , , & text{in } , , Omega, &-Delta_{mathbb{G}}v = frac{psi^{beta}|v|^{2^*(beta)-2}v}{d(z)^{beta}}+ frac{p_{2}}{2^*(gamma)}frac{psi^{gamma}|u|^{p_{1}}|v|^{p_{2}-2}v}{d(z, z_{0})^{gamma}} +lambda h(z)frac{psi^{sigma}|v|^{q-2}v}{d(z)^{sigma}}, , &text{in } , , Omega, &quad u = v = 0, , &text{on } , , partialOmega, end{aligned}right. end{equation*} $end{document} where $ -Delta_{mathbb{G}} $ is a sub-Laplacian on Carnot group $ mathbb{G} $, $ alpha, beta, gamma, sigmain [0, 2) $, $ d $ is the $ Delta_{mathbb{G}} $-natural gauge, $ psi = |nabla_{mathbb{G}}d| $ and $ nabla_{mathbb{G}} $ is the horizontal gradient associated to $ Delta_{mathbb{G}} $. The positive parameters $ lambda $, $ q $ satisfy $ 0 < lambda < infty $, $ 1 < q < 2 $, and $ p_{1} $, $ p_{2} > 1 $ with $ p_{1}+p_{2} = 2^*(gamma) $, here $ 2^*(alpha): = frac{2(Q-alpha)}{Q-2} $, $ 2^*(beta): = frac{2(Q-beta)}{Q-2} $ and $ 2^*(gamma) = frac{2(Q-gamma)}{Q-2} $ are the critical Hardy-Sobolev exponents, $ Q $ is the homogeneous dimension of the space $ mathbb{G} $. By means of variational methods and the mountain-pass theorem of Ambrosetti and Rabonowitz, we study the existence of multiple solutions to the sub-elliptic system.
In this work, our main concern is to study the existence and multiplicity of solutions for the following sub-elliptic system with Hardy type potentials and multiple critical exponents on Carnot group begin{document}$ begin{equation*} left{begin{aligned} &-Delta_{mathbb{G}}u = frac{psi^{alpha}|u|^{2^*(alpha)-2}u}{d(z)^{alpha}}+ frac{p_{1}}{2^*(gamma)}frac{psi^{gamma}|u|^{p_{1}-2}u|v|^{p_{2}}}{d(z, z_{0})^{gamma}} +lambda h(z)frac{psi^{sigma}|u|^{q-2}u}{d(z)^{sigma}} , , & text{in } , , Omega, &-Delta_{mathbb{G}}v = frac{psi^{beta}|v|^{2^*(beta)-2}v}{d(z)^{beta}}+ frac{p_{2}}{2^*(gamma)}frac{psi^{gamma}|u|^{p_{1}}|v|^{p_{2}-2}v}{d(z, z_{0})^{gamma}} +lambda h(z)frac{psi^{sigma}|v|^{q-2}v}{d(z)^{sigma}}, , &text{in } , , Omega, &quad u = v = 0, , &text{on } , , partialOmega, end{aligned}right. end{equation*} $end{document} where $ -Delta_{mathbb{G}} $ is a sub-Laplacian on Carnot group $ mathbb{G} $, $ alpha, beta, gamma, sigmain [0, 2) $, $ d $ is the $ Delta_{mathbb{G}} $-natural gauge, $ psi = |nabla_{mathbb{G}}d| $ and $ nabla_{mathbb{G}} $ is the horizontal gradient associated to $ Delta_{mathbb{G}} $. The positive parameters $ lambda $, $ q $ satisfy $ 0 < lambda < infty $, $ 1 < q < 2 $, and $ p_{1} $, $ p_{2} > 1 $ with $ p_{1}+p_{2} = 2^*(gamma) $, here $ 2^*(alpha): = frac{2(Q-alpha)}{Q-2} $, $ 2^*(beta): = frac{2(Q-beta)}{Q-2} $ and $ 2^*(gamma) = frac{2(Q-gamma)}{Q-2} $ are the critical Hardy-Sobolev exponents, $ Q $ is the homogeneous dimension of the space $ mathbb{G} $. By means of variational methods and the mountain-pass theorem of Ambrosetti and Rabonowitz, we study the existence of multiple solutions to the sub-elliptic system.
{"title":"On criticality coupled sub-Laplacian systems with Hardy type potentials on Stratified Lie groups","authors":"Jinguo Zhang, Shuhai Zhu","doi":"10.3934/cam.2023005","DOIUrl":"https://doi.org/10.3934/cam.2023005","url":null,"abstract":"In this work, our main concern is to study the existence and multiplicity of solutions for the following sub-elliptic system with Hardy type potentials and multiple critical exponents on Carnot group\u0000\u0000 begin{document}$ begin{equation*} left{begin{aligned} &-Delta_{mathbb{G}}u = frac{psi^{alpha}|u|^{2^*(alpha)-2}u}{d(z)^{alpha}}+ frac{p_{1}}{2^*(gamma)}frac{psi^{gamma}|u|^{p_{1}-2}u|v|^{p_{2}}}{d(z, z_{0})^{gamma}} +lambda h(z)frac{psi^{sigma}|u|^{q-2}u}{d(z)^{sigma}} , , & text{in } , , Omega, &-Delta_{mathbb{G}}v = frac{psi^{beta}|v|^{2^*(beta)-2}v}{d(z)^{beta}}+ frac{p_{2}}{2^*(gamma)}frac{psi^{gamma}|u|^{p_{1}}|v|^{p_{2}-2}v}{d(z, z_{0})^{gamma}} +lambda h(z)frac{psi^{sigma}|v|^{q-2}v}{d(z)^{sigma}}, , &text{in } , , Omega, &quad u = v = 0, , &text{on } , , partialOmega, end{aligned}right. end{equation*} $end{document} \u0000where $ -Delta_{mathbb{G}} $ is a sub-Laplacian on Carnot group $ mathbb{G} $, $ alpha, beta, gamma, sigmain [0, 2) $, $ d $ is the $ Delta_{mathbb{G}} $-natural gauge, $ psi = |nabla_{mathbb{G}}d| $ and $ nabla_{mathbb{G}} $ is the horizontal gradient associated to $ Delta_{mathbb{G}} $. The positive parameters $ lambda $, $ q $ satisfy $ 0 < lambda < infty $, $ 1 < q < 2 $, and $ p_{1} $, $ p_{2} > 1 $ with $ p_{1}+p_{2} = 2^*(gamma) $, here $ 2^*(alpha): = frac{2(Q-alpha)}{Q-2} $, $ 2^*(beta): = frac{2(Q-beta)}{Q-2} $ and $ 2^*(gamma) = frac{2(Q-gamma)}{Q-2} $ are the critical Hardy-Sobolev exponents, $ Q $ is the homogeneous dimension of the space $ mathbb{G} $. By means of variational methods and the mountain-pass theorem of Ambrosetti and Rabonowitz, we study the existence of multiple solutions to the sub-elliptic system.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"188 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121527746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article we revisit the global existence result of the wave-Klein-Gordon model of the system of the self-gravitating massive field. Our new observation is that, by applying the conformal energy estimates on hyperboloids, we obtain mildly increasing energy estimate up to the top order for the Klein-Gordon component, which clarify the question on the hierarchy of the energy bounds of the Klein-Gordon component in our previous work. Furthermore, a uniform-in-time energy estimate is established for the wave component up to the top order, as well as a scattering result. These improvements indicate that the partial conformal symmetry of the Einstein-massive scalar system will play an important role in the global analysis.
{"title":"Conformal-type energy estimates on hyperboloids and the wave-Klein-Gordon model of self-gravitating massive fields","authors":"Senhao Duan, Yue Ma, Weidong Zhang","doi":"10.3934/cam.2023007","DOIUrl":"https://doi.org/10.3934/cam.2023007","url":null,"abstract":"In this article we revisit the global existence result of the wave-Klein-Gordon model of the system of the self-gravitating massive field. Our new observation is that, by applying the conformal energy estimates on hyperboloids, we obtain mildly increasing energy estimate up to the top order for the Klein-Gordon component, which clarify the question on the hierarchy of the energy bounds of the Klein-Gordon component in our previous work. Furthermore, a uniform-in-time energy estimate is established for the wave component up to the top order, as well as a scattering result. These improvements indicate that the partial conformal symmetry of the Einstein-massive scalar system will play an important role in the global analysis.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"83 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126068696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is concerned with a nonlinear plate equation modeling the oscillations of suspension bridges. Under mixed boundary conditions consisting of simply supported and free boundary conditions, we obtain the global well-posedness of solutions in suitable function spaces. In addition, we use the perturbed energy method to prove the existence of a bounded absorbing set and establish a stabilizability estimate. Then, we derive the existence of a global attractor by verifying the asymptotic smoothness of the corresponding dissipative dynamical system.
{"title":"Global attractors for a nonlinear plate equation modeling the oscillations of suspension bridges","authors":"Yang Liu","doi":"10.3934/cam.2023021","DOIUrl":"https://doi.org/10.3934/cam.2023021","url":null,"abstract":"This paper is concerned with a nonlinear plate equation modeling the oscillations of suspension bridges. Under mixed boundary conditions consisting of simply supported and free boundary conditions, we obtain the global well-posedness of solutions in suitable function spaces. In addition, we use the perturbed energy method to prove the existence of a bounded absorbing set and establish a stabilizability estimate. Then, we derive the existence of a global attractor by verifying the asymptotic smoothness of the corresponding dissipative dynamical system.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132184792","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
R. Pinčák, A. Pigazzini, Saeid Jafari, Özge Korkmaz, C. Özel, E. Bartoš
The aim of this paper is to use a special type of Einstein warped product manifolds recently introduced, the so-called PNDP-manifolds, for the differential geometric study, by focusing on some aspects related to dark field in financial market such as the concept of dark volatility. This volatility is not fixed in any relevant economic parameter, a sort of negative dimension, a ghost field, that greatly influences the behavior of real market. Since the PNDP-manifold has a "virtual" dimension, we want to use it in order to show how the Global Market is influenced by dark volatility, and in this regard we also provide an example, by considering the classical exponential models as possible solutions to our approach. We show how dark volatility, combined with specific conditions, leads to the collapse of a forward price.
{"title":"A possible interpretation of financial markets affected by dark volatility","authors":"R. Pinčák, A. Pigazzini, Saeid Jafari, Özge Korkmaz, C. Özel, E. Bartoš","doi":"10.3934/cam.2023006","DOIUrl":"https://doi.org/10.3934/cam.2023006","url":null,"abstract":"The aim of this paper is to use a special type of Einstein warped product manifolds recently introduced, the so-called PNDP-manifolds, for the differential geometric study, by focusing on some aspects related to dark field in financial market such as the concept of dark volatility. This volatility is not fixed in any relevant economic parameter, a sort of negative dimension, a ghost field, that greatly influences the behavior of real market. Since the PNDP-manifold has a \"virtual\" dimension, we want to use it in order to show how the Global Market is influenced by dark volatility, and in this regard we also provide an example, by considering the classical exponential models as possible solutions to our approach. We show how dark volatility, combined with specific conditions, leads to the collapse of a forward price.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129374399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The quantization process was always tightly connected to the Hamiltonian formulation of classical mechanics. For non-Hamiltonian systems, traditional quantization algorithms turn out to be unsuitable. Numerous attempts to quantize non-Hamiltonian systems have shown that this problem is nontrivial and requires the development of new approaches. In this paper, we present the quantization methods that do not depend upon the Hamiltonian formulation of classical mechanics. Two approaches to the quantization of mechanical systems are considered: axiomatic and hydrodynamic. It is shown that the formal application of these approaches to the classical Hamilton-Jacobi theory allows obtaining the wave equation for the corresponding quantum system in natural way. Examples are considered that show the effectiveness of the proposed approaches, both for Hamiltonian and non-Hamiltonian systems. The spinor form of the relativistic Hamilton-Jacobi theory for classical particles is considered. It is shown that it naturally leads to the Dirac equation for the corresponding quantum particle and to its non-Hamiltonian generalization, the bispinor relativistic Kostin equation.
{"title":"Quantization of Hamiltonian and non-Hamiltonian systems","authors":"S. Rashkovskiy","doi":"10.3934/cam.2023014","DOIUrl":"https://doi.org/10.3934/cam.2023014","url":null,"abstract":"\u0000The quantization process was always tightly connected to the Hamiltonian formulation of classical mechanics. For non-Hamiltonian systems, traditional quantization algorithms turn out to be unsuitable. Numerous attempts to quantize non-Hamiltonian systems have shown that this problem is nontrivial and requires the development of new approaches. In this paper, we present the quantization methods that do not depend upon the Hamiltonian formulation of classical mechanics. Two approaches to the quantization of mechanical systems are considered: axiomatic and hydrodynamic. It is shown that the formal application of these approaches to the classical Hamilton-Jacobi theory allows obtaining the wave equation for the corresponding quantum system in natural way. Examples are considered that show the effectiveness of the proposed approaches, both for Hamiltonian and non-Hamiltonian systems. The spinor form of the relativistic Hamilton-Jacobi theory for classical particles is considered. It is shown that it naturally leads to the Dirac equation for the corresponding quantum particle and to its non-Hamiltonian generalization, the bispinor relativistic Kostin equation.\u0000","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124196814","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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