A manifestly gauge-invariant formulation of non-relativistic quantum mechanics is applied to the case of time-dependent harmonic oscillator in the magnetic dipole approximation. A general equation for obtaining gauge-...A manifestly gauge-invariant formulation of non-relativistic quantum mechanics is applied to the case of time-dependent harmonic oscillator in the magnetic dipole approximation. A general equation for obtaining gauge-invariant transition probability amplitudes is derived.展开更多
There are a number of interesting applications where modeling elastic and/or viscoelastic materials is fundamental,including uses in civil engineering,the food industry,land mine detection and ultrasonic imaging.Here ...There are a number of interesting applications where modeling elastic and/or viscoelastic materials is fundamental,including uses in civil engineering,the food industry,land mine detection and ultrasonic imaging.Here we provide an overview of the subject for both elastic and viscoelastic materials in order to understand the behavior of these materials.We begin with a brief introduction of some basic terminology and relationships in continuum mechanics,and a review of equations of motion in a continuum in both Lagrangian and Eulerian forms.To complete the set of equations,we then proceed to present and discuss a number of specific forms for the constitutive relationships between stress and strain proposed in the literature for both elastic and viscoelastic materials.In addition,we discuss some applications for these constitutive equations.Finally,we give a computational example describing the motion of soil experiencing dynamic loading by incorporating a specific form of constitutive equation into the equation of motion.展开更多
文摘A manifestly gauge-invariant formulation of non-relativistic quantum mechanics is applied to the case of time-dependent harmonic oscillator in the magnetic dipole approximation. A general equation for obtaining gauge-invariant transition probability amplitudes is derived.
基金This research was supported in part by the Air Force Office of Scientific Research under grant number FA9550-09-1-0226The efforts of ZRK were supported in part by the Department of Education with a GAANN Fellowship under grant number P200A070386。
文摘There are a number of interesting applications where modeling elastic and/or viscoelastic materials is fundamental,including uses in civil engineering,the food industry,land mine detection and ultrasonic imaging.Here we provide an overview of the subject for both elastic and viscoelastic materials in order to understand the behavior of these materials.We begin with a brief introduction of some basic terminology and relationships in continuum mechanics,and a review of equations of motion in a continuum in both Lagrangian and Eulerian forms.To complete the set of equations,we then proceed to present and discuss a number of specific forms for the constitutive relationships between stress and strain proposed in the literature for both elastic and viscoelastic materials.In addition,we discuss some applications for these constitutive equations.Finally,we give a computational example describing the motion of soil experiencing dynamic loading by incorporating a specific form of constitutive equation into the equation of motion.
