Time, Length, and Mass Are Derived Quantities

Fundamental units of measurements are kilograms, meters, and seconds—in regards to mass length, and time. All other measurements in mechanical quantities including kinetic quantities and dynamic quantities are called derived units. These derived units can be expressed in terms of fundamental units, such as acceleration, area, energy, force, power, velocity and volume. Derived quantities will be referred to as time, length, and mass. In order to explain that fundamental units are not equivalent with fundamental quantities, we need to understand the contraction of time and length in Special Relativity. If we choose the velocity of light as fundamental quantity and length and time as derived quantities, then we are able to construct three-dimensional space-time frames. Three-dimensional space-time frames representing time with polar coordination, time contraction and length contraction can be shown graphically.

Conformal invariance and Hojman conserved quantities of first order Lagrange systems

In this paper the conformal invariance by infinitesimal transformations of first order Lagrange systems is discussed in detail. The necessary and sufficient conditions of conformal invariance and Lie symmetry simultaneously by the action of infinitesimal transformations are given. Then it gets the Hojman conserved quantities of conformal invariance by the infinitesimal transformations. Finally an example is given to illustrate the application of the results.

Conformal invariance and conserved quantities of dynamical system of relative motion

This paper discusses in detail the conformal invariance by infinitesimal transformations of a dynamical system of relative motion. The necessary and sufficient conditions of conformal invariance and Lie symmetry are given simultaneously by the action of infinitesimal transformations. Then it obtains the conserved quantities of conformal invariance by the infinitesimal transformations. Finally an example is given to illustrate the application of the results.

Conformal invariance and conserved quantities of non-conservative Lagrange systems by point transformations

This paper studies the conformal invariance by infinitesimal point transformations of non-conservative Lagrange systems. It gives the necessary and sufficient conditions of conformal invariance by the action of infinitesimal point transformations being Lie symmetric simultaneously. Then the Noether conserved quantities of conformal invariance are obtained. Finally an illustrative example is given to verify the results.

Mei symmetry and new conserved quantities of Tzénoff equations for higher-order nonholonomic system

In this paper,the Mei symmetry of Tzénoff equations for the higher-order nonholonomic system and the new conserved quantities derived from that are researched,and the function expression of new conserved quantities and criterion equation which deduces these conserved quantities are presented.This result establishes the theory basis for further researches on conservation laws of Tzénoff equations of the higher-order nonholonomic constraint system.

Mei Symmetry and New Conserved Quantities of Tzénoff Equations for the Variable Mass Higher-Order Nonholonomic System

Operational systems of spacecraft are general variable mass mechanics systems,and their symmetries and conserved quantities imply profound physical rules of the space system.We study the Mei symmetry of Tzénoff equations for a variable mass nonholonomic system and the new conserved quantities derived.The function expression of the new conserved quantities and the criterion equation which deduces these conserved quantities are presented.This result has some theoretical values in further research of conservation laws obeyed by the variable mass system.

Computation of focus quantities of three-dimensional polynomial systems

Fix a collection of polynomial vector fields on R3with a singularity at the origin,for every one of which the linear part at the origin has two pure imaginary and one non-zero eigenvalue. Some such systems admit a local analytic first integral,which then defines a local center manifold of the system. Conditions for existence of a first integral are given by the vanishing certain polynomial or rational functions in the coefficients of the system called focus quantities. In this paper we prove that the focus quantities have a structure analogous to that in the two-dimensional case and use it to formulate an efficient algorithm for computing them.

Geometric quantities of lower doubly excited bound states of helium

Expectation values of single electron and interelectronic geometric quantities such as <γ>,<γ_(12)> ,<γ_(<)> , <γ_(>)>,<cosθ_(12)> and <θ_(12)> are calculated for doubly excited 2pnp ^(1)P^(e)(3 ≤ n ≤ 5), 2pnp ^(3)P^(e)(2 ≤ n ≤ 5)and 2pnd ^(1,3)D°(3 ≤ n ≤ 5) states of helium using Hylleraas-B-spline basis set. The energy levels converge to at least 10 significant digits in our calculations.The extrapolated values of geometric quantities except for <θ_(12)> reach 10 significant digits as well;<θ_(12)> reaches at least 7 significant digits using a multipole expansion approach. Our results provide a precise reference for future research.

Conformal invariance and generalized Hojman conserved quantities of mechanico-electrical systems

This paper studies conformal invariance and generalized Hojman conserved quantities of mechanico-electrical systems. The definition and the determining equation of conformal invariance for mechanico-electrical systems are provided. The conformal factor expression is deduced from conformal invariance and Lie symmetry under the infinitesimal single- parameter transformation group. The generalized Hojman conserved quantities from the conformal invariance of the system are given. An example is given to illustrate the application of the result.

Conformal invariance and Hojman conserved quantities of canonical Hamilton systems

This paper discusses the conformal invariance by infinitesimal transformations of canonical Hamilton systems. The necessary and sufficient conditions of conformal invarianee being Lie symmetrical simultaneously by the action of infinitesimal transformations are given. The determining equations of the conformal invariance are gained. Then the Hojman conserved quantities of conformal invariance by special infinitesimal transformations are obtained. Finally an illustrative example is given to verify the results.

Thermodynamical quantities around a RNAdS black hole

The entropy density, energy density, pressure and equation of state around the RNAdS black hole are calculated in the WKB approximation on the Teukolsky-type master equation. The appearance of spin-dependent terms is demonstrated. The existence of these terms shows that the black hole radiation is not exactly thermal radiation and the black hole entropy is not strictly proportional to the area of the event horizon.

Comparison between Cooling Rate Dependence of Macroscopic and Microscopic Quantities in Simulated Aluminium Glass

Constant-pressure molecular dynamics simulations and an analysis of the local atomic structures have been performed to study the cooling rate dependence of some macroscopic and microscopic quantities in aluminium glass.Macroscopic quantities,enthalpy and density,see an observable but small dependence on the cooling rate.Icosahedral ordering units exhibit strong cooling rate dependence,which is responsible for the dependence of the enthalpy and the density on the cooling rate;while the almost independence of some microstructural units such as the 1541,1431 and 1421 pairs of the cooling rate may lead to a small dependence of the enthalpy and the density on the cooling rate.

Conserved quantities and symmetries related to stochastic Hamiltonian systems

In this paper symmetries and conservation laws for stochastic dynamical systems are discussed in detail. Determining equations for infinitesimal approximate symmetries of Ito and Stratonovich dynamical systems are derived. It shows how to derive conserved quantities for stochastic dynamical systems by using their symmetries without recourse to Noether＇s theorem.