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Anirvan Dasgupta - One of the best experts on this subject based on the ideXlab platform.
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Revisiting umbra-Lagrangian–Hamiltonian Mechanics: Its variational foundation and extension of Noether's theorem and Poincare–Cartan integral
International Journal of Non-linear Mechanics, 2011Co-Authors: Amalendu Mukherjee, Vikas Rastogi, Anirvan DasguptaAbstract:Abstract This paper revisits an extension of the Lagrangian–Hamiltonian Mechanics that incorporates dissipative and non-potential fields, and non-integrable constraints in a compact form, such that one may obtain invariants of motion or possible invariant trajectories through an extension of Noether's theorem. A new concept of umbra-time has been introduced for this extension. This leads to a new form of equation, which is termed as the umbra-Lagrange's equation. The underlying variational principle, which is based on a recursive minimization of functionals, is presented. The introduction of the concept of umbra-time extends the classical manifold over which the system evolves. An extension of the Lagrangian–Hamiltonian Mechanics over vector fields in this extended space has been presented. The idea of umbra time is then carried forward to propose the basic concept of umbra-Hamiltonian, which is used along with the extended Noether's theorem to provide an insight into the dynamics of systems with symmetries. Gauge functions for umbra-Lagrangian are also introduced. Extension of the Poincare–Cartan integral for the umbra-Lagrangian theory is also proposed, and its implications have been discussed. Several examples are presented to illustrate all these concepts.
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revisiting umbra lagrangian Hamiltonian Mechanics its variational foundation and extension of noether s theorem and poincare cartan integral
International Journal of Non-linear Mechanics, 2011Co-Authors: Amalendu Mukherjee, Vikas Rastogi, Anirvan DasguptaAbstract:Abstract This paper revisits an extension of the Lagrangian–Hamiltonian Mechanics that incorporates dissipative and non-potential fields, and non-integrable constraints in a compact form, such that one may obtain invariants of motion or possible invariant trajectories through an extension of Noether's theorem. A new concept of umbra-time has been introduced for this extension. This leads to a new form of equation, which is termed as the umbra-Lagrange's equation. The underlying variational principle, which is based on a recursive minimization of functionals, is presented. The introduction of the concept of umbra-time extends the classical manifold over which the system evolves. An extension of the Lagrangian–Hamiltonian Mechanics over vector fields in this extended space has been presented. The idea of umbra time is then carried forward to propose the basic concept of umbra-Hamiltonian, which is used along with the extended Noether's theorem to provide an insight into the dynamics of systems with symmetries. Gauge functions for umbra-Lagrangian are also introduced. Extension of the Poincare–Cartan integral for the umbra-Lagrangian theory is also proposed, and its implications have been discussed. Several examples are presented to illustrate all these concepts.
Amalendu Mukherjee - One of the best experts on this subject based on the ideXlab platform.
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revisiting umbra lagrangian Hamiltonian Mechanics its variational foundation and extension of noether s theorem and poincare cartan integral
International Journal of Non-linear Mechanics, 2011Co-Authors: Amalendu Mukherjee, Vikas Rastogi, Anirvan DasguptaAbstract:Abstract This paper revisits an extension of the Lagrangian–Hamiltonian Mechanics that incorporates dissipative and non-potential fields, and non-integrable constraints in a compact form, such that one may obtain invariants of motion or possible invariant trajectories through an extension of Noether's theorem. A new concept of umbra-time has been introduced for this extension. This leads to a new form of equation, which is termed as the umbra-Lagrange's equation. The underlying variational principle, which is based on a recursive minimization of functionals, is presented. The introduction of the concept of umbra-time extends the classical manifold over which the system evolves. An extension of the Lagrangian–Hamiltonian Mechanics over vector fields in this extended space has been presented. The idea of umbra time is then carried forward to propose the basic concept of umbra-Hamiltonian, which is used along with the extended Noether's theorem to provide an insight into the dynamics of systems with symmetries. Gauge functions for umbra-Lagrangian are also introduced. Extension of the Poincare–Cartan integral for the umbra-Lagrangian theory is also proposed, and its implications have been discussed. Several examples are presented to illustrate all these concepts.
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Revisiting umbra-Lagrangian–Hamiltonian Mechanics: Its variational foundation and extension of Noether's theorem and Poincare–Cartan integral
International Journal of Non-linear Mechanics, 2011Co-Authors: Amalendu Mukherjee, Vikas Rastogi, Anirvan DasguptaAbstract:Abstract This paper revisits an extension of the Lagrangian–Hamiltonian Mechanics that incorporates dissipative and non-potential fields, and non-integrable constraints in a compact form, such that one may obtain invariants of motion or possible invariant trajectories through an extension of Noether's theorem. A new concept of umbra-time has been introduced for this extension. This leads to a new form of equation, which is termed as the umbra-Lagrange's equation. The underlying variational principle, which is based on a recursive minimization of functionals, is presented. The introduction of the concept of umbra-time extends the classical manifold over which the system evolves. An extension of the Lagrangian–Hamiltonian Mechanics over vector fields in this extended space has been presented. The idea of umbra time is then carried forward to propose the basic concept of umbra-Hamiltonian, which is used along with the extended Noether's theorem to provide an insight into the dynamics of systems with symmetries. Gauge functions for umbra-Lagrangian are also introduced. Extension of the Poincare–Cartan integral for the umbra-Lagrangian theory is also proposed, and its implications have been discussed. Several examples are presented to illustrate all these concepts.
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A review on extension of Lagrangian-Hamiltonian Mechanics
Journal of The Brazilian Society of Mechanical Sciences and Engineering, 2011Co-Authors: Vikas Rastogi, Amalendu Mukherjee, Anirban DasguptaAbstract:This paper presents a brief review on Lagrangian-Hamiltonian Mechanics and deals with the several developments and extensions in this area, which have been based upon the principle of D'Alambert or the other. It is not the intention of the authors to attempt to provide a detailed coverage of all the extensions of Lagrangian-Hamiltonian Mechanics, whereas detailed consideration is given to the extension of Noether's theorem for nonconservative systems only. The paper incorporates a candid commentary on various extensions including extension of Noether's theorem through differential variational principle. The paper further deals with an extended Lagrangian formulation for general class of dynamical systems with dissipative, non-potential fields with an aim to obtain invariants of motion for such systems. This new extension is based on a new concept of umbra-time, which leads to a peculiar form of equations termed as 'umbra-Lagrange's equation'. This equation leads to a simple and new fundamental view on Lagrangian Mechanics and is applied to investigate the dynamics of asymmetric and continuous systems. This will provide help to understand physical interpretations of various extensions of Lagrangian-Hamiltonian Mechanics.
Vikas Rastogi - One of the best experts on this subject based on the ideXlab platform.
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ICBGM@SummerSim - Extension of Lagrangian-Hamiltonian Mechanics: Umbra Poisson bracket using bond graphs
2016Co-Authors: Vikas RastogiAbstract:In mathematical as well as in classical Mechanics, the Poisson brackets are one of the binary operation properties in Hamiltonian Mechanics, which generally govern the Hamiltonian dynamics system. The present paper deals with the development of umbra-Poisson bracket for extended Lagrangian-Hamiltonian Mechanics, where a new time of umbra is applied in extended form and umbra-Lagrangian is obtained through bondgraphs. Some significant insight of the system has been achieved through some useful theorems of Poisson Brackets. It also proves that if Hamiltonian does not depend explicitly on time, the time derivative of umbra-Hamiltonian is zero. In all such analysis, the umbra-Lagrangian is generated through bond-graphs as it provided the interior as well as exterior information of the system. Further, it is also be proved that if any dynamical system with internal, compliant, dissipative, gyroscope elements and external source with sufficiently smooth real time variations of parameters has a umbra-Hamiltonian, then each of its component umbra-Poisson bracket with itself will be zero for all real time.
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Revisiting umbra-Lagrangian–Hamiltonian Mechanics: Its variational foundation and extension of Noether's theorem and Poincare–Cartan integral
International Journal of Non-linear Mechanics, 2011Co-Authors: Amalendu Mukherjee, Vikas Rastogi, Anirvan DasguptaAbstract:Abstract This paper revisits an extension of the Lagrangian–Hamiltonian Mechanics that incorporates dissipative and non-potential fields, and non-integrable constraints in a compact form, such that one may obtain invariants of motion or possible invariant trajectories through an extension of Noether's theorem. A new concept of umbra-time has been introduced for this extension. This leads to a new form of equation, which is termed as the umbra-Lagrange's equation. The underlying variational principle, which is based on a recursive minimization of functionals, is presented. The introduction of the concept of umbra-time extends the classical manifold over which the system evolves. An extension of the Lagrangian–Hamiltonian Mechanics over vector fields in this extended space has been presented. The idea of umbra time is then carried forward to propose the basic concept of umbra-Hamiltonian, which is used along with the extended Noether's theorem to provide an insight into the dynamics of systems with symmetries. Gauge functions for umbra-Lagrangian are also introduced. Extension of the Poincare–Cartan integral for the umbra-Lagrangian theory is also proposed, and its implications have been discussed. Several examples are presented to illustrate all these concepts.
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revisiting umbra lagrangian Hamiltonian Mechanics its variational foundation and extension of noether s theorem and poincare cartan integral
International Journal of Non-linear Mechanics, 2011Co-Authors: Amalendu Mukherjee, Vikas Rastogi, Anirvan DasguptaAbstract:Abstract This paper revisits an extension of the Lagrangian–Hamiltonian Mechanics that incorporates dissipative and non-potential fields, and non-integrable constraints in a compact form, such that one may obtain invariants of motion or possible invariant trajectories through an extension of Noether's theorem. A new concept of umbra-time has been introduced for this extension. This leads to a new form of equation, which is termed as the umbra-Lagrange's equation. The underlying variational principle, which is based on a recursive minimization of functionals, is presented. The introduction of the concept of umbra-time extends the classical manifold over which the system evolves. An extension of the Lagrangian–Hamiltonian Mechanics over vector fields in this extended space has been presented. The idea of umbra time is then carried forward to propose the basic concept of umbra-Hamiltonian, which is used along with the extended Noether's theorem to provide an insight into the dynamics of systems with symmetries. Gauge functions for umbra-Lagrangian are also introduced. Extension of the Poincare–Cartan integral for the umbra-Lagrangian theory is also proposed, and its implications have been discussed. Several examples are presented to illustrate all these concepts.
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A review on extension of Lagrangian-Hamiltonian Mechanics
Journal of The Brazilian Society of Mechanical Sciences and Engineering, 2011Co-Authors: Vikas Rastogi, Amalendu Mukherjee, Anirban DasguptaAbstract:This paper presents a brief review on Lagrangian-Hamiltonian Mechanics and deals with the several developments and extensions in this area, which have been based upon the principle of D'Alambert or the other. It is not the intention of the authors to attempt to provide a detailed coverage of all the extensions of Lagrangian-Hamiltonian Mechanics, whereas detailed consideration is given to the extension of Noether's theorem for nonconservative systems only. The paper incorporates a candid commentary on various extensions including extension of Noether's theorem through differential variational principle. The paper further deals with an extended Lagrangian formulation for general class of dynamical systems with dissipative, non-potential fields with an aim to obtain invariants of motion for such systems. This new extension is based on a new concept of umbra-time, which leads to a peculiar form of equations termed as 'umbra-Lagrange's equation'. This equation leads to a simple and new fundamental view on Lagrangian Mechanics and is applied to investigate the dynamics of asymmetric and continuous systems. This will provide help to understand physical interpretations of various extensions of Lagrangian-Hamiltonian Mechanics.
Gennadi Sardanashvily - One of the best experts on this subject based on the ideXlab platform.
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Noether's first theorem in Hamiltonian Mechanics
arXiv: Mathematical Physics, 2015Co-Authors: Gennadi SardanashvilyAbstract:Non-autonomous non-relativistic Mechanics is formulated as Lagrangian and Hamiltonian theory on fibre bundles over the time axis R. Hamiltonian Mechanics herewith can be reformulated as particular Lagrangian theory on a momentum phase space. This facts enable one to apply Noether's first theorem both to Lagrangian and Hamiltonian Mechanics. By virtue of Noether's first theorem, any symmetry defines a symmetry current which is an integral of motion in Lagrangian and Hamiltonian Mechanics. The converse is not true in Lagrangian Mechanics where integrals of motion need not come from symmetries. We show that, in Hamiltonian Mechanics, any integral of motion is a symmetry current. In particular, an energy function relative to a reference frame is a symmetry current along a connection on a configuration bundle which is this reference frame. An example of the global Kepler problem is analyzed in detail.
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Geometric quantization of relativistic Hamiltonian Mechanics
arXiv: General Relativity and Quantum Cosmology, 2002Co-Authors: Gennadi SardanashvilyAbstract:A relativistic Hamiltonian mechanical system is seen as a conservative Dirac constraint system on the cotangent bundle of a pseudo-Riemannian manifold. We provide geometric quantization of this cotangent bundle where the quantum constraint serves as a relativistic quantum equation.
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Geometric quantization of non-relativistic and relativistic Hamiltonian Mechanics
arXiv: Mathematical Physics, 2000Co-Authors: Giovanni Giachetta, Luigi Mangiarotti, Gennadi SardanashvilyAbstract:We show that non-relativistic and relativistic mechanical systems on a configuration space Q can be seen as the conservative Dirac constraint systems with zero Hamiltonians on different subbundles of the same cotangent bundle T^*Q. The geometric quantization of this cotangent bundle under the vertical polarization leads to compatible covariant quantizations of non-relativistic and relativistic Hamiltonian Mechanics.
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Quantum Jacobi fields in Hamiltonian Mechanics
arXiv: Quantum Physics, 2000Co-Authors: Giovanni Giachetta, Gennadi Sardanashvily, Luigi MangiarottiAbstract:Jacobi fields of classical solutions of a Hamiltonian mechanical system are quantized in the framework of vertical-extended Hamiltonian formalism. Quantum Jacobi fields characterize quantum transitions between classical solutions.
Kvetoslav Belda - One of the best experts on this subject based on the ideXlab platform.
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application of Hamiltonian Mechanics to control design for industrial robotic manipulators
International Conference on Methods and Models in Automation and Robotics, 2017Co-Authors: Vaclav Zada, Kvetoslav BeldaAbstract:The paper deals with a tracking control for robotic manipulators, where the robot dynamics is described by means of Hamiltonian Mechanics. This way leads to different physical descriptive quantities used in control design. In the paper, the model-oriented Lyapunov-based control is considered. It is introduced in the novel formulation using Hamiltonian Mechanics and compared with the conventional formulation based on Lagrangian Mechanics. The theoretical results, generally applicable to usual articulated industrial robotic manipulators, are demonstrated on one specific robot arm with three degrees of freedom.
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MMAR - Application of Hamiltonian Mechanics to control design for industrial robotic manipulators
2017 22nd International Conference on Methods and Models in Automation and Robotics (MMAR), 2017Co-Authors: Vaclav Zada, Kvetoslav BeldaAbstract:The paper deals with a tracking control for robotic manipulators, where the robot dynamics is described by means of Hamiltonian Mechanics. This way leads to different physical descriptive quantities used in control design. In the paper, the model-oriented Lyapunov-based control is considered. It is introduced in the novel formulation using Hamiltonian Mechanics and compared with the conventional formulation based on Lagrangian Mechanics. The theoretical results, generally applicable to usual articulated industrial robotic manipulators, are demonstrated on one specific robot arm with three degrees of freedom.
