In this paper we use Poincaré’s equations in group variables to de- scribe the motion of a holonomic mechanical system and to determine Jacobi's mul- tiplier for the equations of motion.
It is well known that the Poincaré gauge theories of gravity do not have the structure of a standard gauge theory. Nevertheless, we show that a general form of action for the gravitational gauge fields in the gau...It is well known that the Poincaré gauge theories of gravity do not have the structure of a standard gauge theory. Nevertheless, we show that a general form of action for the gravitational gauge fields in the gauge theory does possess local Poincaré invariance.展开更多
Relations between the experience of space technology and theory of space and time are found in this paper. A nontraditional approach to the concepts of space and time is introduced. The approach is based upon the the ...Relations between the experience of space technology and theory of space and time are found in this paper. A nontraditional approach to the concepts of space and time is introduced. The approach is based upon the the analysis of the detailed mechanism of radar measurement and nonlinear Doppler effects as measured by an astronaut. The Lorentz factor and four-dimensional interval may be interpreted from the point of view of a space technologist. A 'geometrical mean' notion for computing time interval is introduced parallelly with the usual arithmetic mean formulas, giving results comparable with those of special relativity theory. Space time relationships are demonstrated on the Poincare diagrams.展开更多
We briefly describe the importance of division algebras and Poincaré conjecture in both mathematical and physical scenarios. Mathematically, we argue that using the torsion concept one can combine the formalisms ...We briefly describe the importance of division algebras and Poincaré conjecture in both mathematical and physical scenarios. Mathematically, we argue that using the torsion concept one can combine the formalisms of division algebras and Poincaré conjecture. Physically, we show that both formalisms may be the underlying mathematical tools in special relativity and cosmology. Moreover, we explore the possibility that by using the concept of n-qubit system, such conjecture may allow generalization the Hopf maps.展开更多
Here concerned is a certain kind of non-standard measure defined on the n-dimensional Euclidean space (Rn), which (with n = 1) can be used to show that any standard linear point-set or the usual ordered field R of rea...Here concerned is a certain kind of non-standard measure defined on the n-dimensional Euclidean space (Rn), which (with n = 1) can be used to show that any standard linear point-set or the usual ordered field R of real numbers is of measure zero. The proposition just mentioned is basically consistent with Poincare's famous remark which renders a deep insight into the paradoxical structural nature of Cantor's continuum consisting precisely of all distinct real numbers.展开更多
With both additive and multiplicative noise excitations, the effect on the chaotic behaviour of the dynamical system is investigated in this paper. The random Melnikov theorem with the mean-square criterion that appli...With both additive and multiplicative noise excitations, the effect on the chaotic behaviour of the dynamical system is investigated in this paper. The random Melnikov theorem with the mean-square criterion that applies to a type of dynamical systems is analysed in order to obtain the conditions for the possible occurrence of chaos. As an example, for the Duffing system, we deduce its concrete expression for the threshold of multiplicative noise amplitude for the rising of chaos, and by combining figures, we discuss the influences of the amplitude, intensity and frequency of both bounded noises on the dynamical behaviour of the Duffing system separately. Finally, numerical simulations are illustrated to verify the theoretical analysis according to the largest Lyapunov exponent and Poincaré map.展开更多
A generalized first Noether theorem (GFNT) originating from the invariance under the finite continuous group for singular high-order Lagrangian and a generalized second Noether theorem (or generalized Noether identiti...A generalized first Noether theorem (GFNT) originating from the invariance under the finite continuous group for singular high-order Lagrangian and a generalized second Noether theorem (or generalized Noether identities (GNI)) for variant system under the infinite continuous group of field theory in canonical formalism are derived. The strong and weak conservation laws in canonical formalism are also obtained. It is pointed out that some variant systems also have Dirac constraint. Based on the canonical action, the generalized Poincaré-Cartan integral invariant (GPCⅡ) for singular high-order Lagrangian in the field theory is deduced. Some confusions in literafure are clarified. The GPCⅡ connected with canonical equations and canonical transformation are discussed.展开更多
文摘In this paper we use Poincaré’s equations in group variables to de- scribe the motion of a holonomic mechanical system and to determine Jacobi's mul- tiplier for the equations of motion.
基金supported by National Natural Science Foundation of China under Grant No.10675019
文摘It is well known that the Poincaré gauge theories of gravity do not have the structure of a standard gauge theory. Nevertheless, we show that a general form of action for the gravitational gauge fields in the gauge theory does possess local Poincaré invariance.
文摘Relations between the experience of space technology and theory of space and time are found in this paper. A nontraditional approach to the concepts of space and time is introduced. The approach is based upon the the analysis of the detailed mechanism of radar measurement and nonlinear Doppler effects as measured by an astronaut. The Lorentz factor and four-dimensional interval may be interpreted from the point of view of a space technologist. A 'geometrical mean' notion for computing time interval is introduced parallelly with the usual arithmetic mean formulas, giving results comparable with those of special relativity theory. Space time relationships are demonstrated on the Poincare diagrams.
文摘We briefly describe the importance of division algebras and Poincaré conjecture in both mathematical and physical scenarios. Mathematically, we argue that using the torsion concept one can combine the formalisms of division algebras and Poincaré conjecture. Physically, we show that both formalisms may be the underlying mathematical tools in special relativity and cosmology. Moreover, we explore the possibility that by using the concept of n-qubit system, such conjecture may allow generalization the Hopf maps.
基金Supperted by Special Foundation of Dalian Univ. of Technology.
文摘Here concerned is a certain kind of non-standard measure defined on the n-dimensional Euclidean space (Rn), which (with n = 1) can be used to show that any standard linear point-set or the usual ordered field R of real numbers is of measure zero. The proposition just mentioned is basically consistent with Poincare's famous remark which renders a deep insight into the paradoxical structural nature of Cantor's continuum consisting precisely of all distinct real numbers.
基金Project supported by the National Natural Science Foundation of China (Grant Nos 10472091 and 10332030)
文摘With both additive and multiplicative noise excitations, the effect on the chaotic behaviour of the dynamical system is investigated in this paper. The random Melnikov theorem with the mean-square criterion that applies to a type of dynamical systems is analysed in order to obtain the conditions for the possible occurrence of chaos. As an example, for the Duffing system, we deduce its concrete expression for the threshold of multiplicative noise amplitude for the rising of chaos, and by combining figures, we discuss the influences of the amplitude, intensity and frequency of both bounded noises on the dynamical behaviour of the Duffing system separately. Finally, numerical simulations are illustrated to verify the theoretical analysis according to the largest Lyapunov exponent and Poincaré map.
基金Project supported by the National Natural Science Foundation of China and Beijing Natural Science Foundation.
文摘A generalized first Noether theorem (GFNT) originating from the invariance under the finite continuous group for singular high-order Lagrangian and a generalized second Noether theorem (or generalized Noether identities (GNI)) for variant system under the infinite continuous group of field theory in canonical formalism are derived. The strong and weak conservation laws in canonical formalism are also obtained. It is pointed out that some variant systems also have Dirac constraint. Based on the canonical action, the generalized Poincaré-Cartan integral invariant (GPCⅡ) for singular high-order Lagrangian in the field theory is deduced. Some confusions in literafure are clarified. The GPCⅡ connected with canonical equations and canonical transformation are discussed.
