摘要:

Gantmacher formulas are generalized by including the effect of the centripetal and angular accelerations in addition to the usual Coriolis term. The integration is accomplished by using a method based on functions of matrices, which permits the summation of an infinite series in closed form through the use of the Cayley–Hamilton theorem. The problem is further extended by allowing arbitrary matrix coefficients in the vector equation of motion, provided they obey a certain commutativity condition. In addition to standard examples, the problem of a particle moving on a plane rotating with variable angular velocity is considered. This example differs from the ones considered in the literature because neither the centripetal nor the angular acceleration terms can be neglected. Solutions of various orders are developed by truncating the exact solution when appropriate.

ISSN:0002-9505    

DOI:10.1119/1.16621

出版商:American Association of Physics Teachers    

年代:1991

数据来源: AIP

摘要:

Different classical limits of quantum mechanics are discussed in the setting of a simple example, the two‐dimensional motion of a spinless electron in a uniform magnetic field. Both the time‐independent energy eigenstates and the time‐dependent states developing from initial localized states are exactly solvable. In the first example, an ℏ→0 limit with fixed energy leads to a simple ensemble of classical orbits. In the second, ℏ→0 with fixed initial mean position and momentum leads to a unique classical orbit.

ISSN:0002-9505    

DOI:10.1119/1.16622

出版商:American Association of Physics Teachers    

年代:1991

数据来源: AIP

摘要:

The method of successive approximations is applied to Maxwell’s equations to calculate the magnetic drag on a conducting disk rotating under the influence of a localizednonuniformmagnetic field. An expression for the damping torque produced by the magnetic field is obtained in the low‐velocity (quasi‐static) limit of the disk’s motion: The damping force, in the case of rectilinear motion, is also calculated. When the theoretical expression for the damping torque is specialized to the case of a uniform magnetic field, the result is found to be identical with that of an existing textbook treatment. In the Appendix, a simplified treatment of the magnetic drag problem suitable for an introductory‐level laboratory class is given. This treatment yields a final expression for the damping torque which is identical in form to the rigorous result except for a scaling factor.

ISSN:0002-9505    

DOI:10.1119/1.16623

出版商:American Association of Physics Teachers    

年代:1991

数据来源: AIP

摘要:

Experimental details concerning measurements of the deceleration of Al and Cu disks rotating under the influence of a localized, nonuniform magnetic field are discussed. For a specified magnetic field distribution, the deceleration of the disk is measured as a function of radial location of the field from the rotation axis and the magnetic drag coefficient at each point is calculated. The measured values of the drag coefficients are compared to theoretical predictions and excellent agreement is obtained.

ISSN:0002-9505    

DOI:10.1119/1.16829

出版商:American Association of Physics Teachers    

年代:1991

数据来源: AIP

摘要:

An important class of variational problems concerns properties of geometric curves (such as length, curvature, etc.) that are independent of the parametrization. When formulated in terms of an arbitrary parameter, many of the familiar tools appear to fail for these problems. For example, the associated Euler–Lagrange equations are not independent, the process of solving for the Hamiltonian breaks down, and, as a consequence, the form of the Hamilton–Jacobi equation is obscured. Several alternative methods have previously been developed specifically for the solution of problems of this form, but these relatively sophisticated techniques have no direct links to the methods familiar from the context of classical mechanics. It is shown here that it is possible to solve such problems by using the conventional tools of mechanics. In particular, the integration of the Euler–Lagrange equations is realized in terms of a specific parameter that is completely determined save for a crucial, arbitrary scale factor. By considering an unusual Legendre transform, a direct analogue of Hamilton’s canonical equations is shown to emerge. More importantly, a unique form for the analogue of the Hamilton–Jacobi equation is derived from the conventional Hamilton–Jacobi equations for members of a family of auxiliary variational problems. Fermat’s principle is discussed in some detail to illustrate the methods introduced here.

ISSN:0002-9505    

DOI:10.1119/1.16624

出版商:American Association of Physics Teachers    

年代:1991

数据来源: AIP

摘要:

For normal incidence onto a stack of multiple slabs, a method is described that relates the reflection and transmission coefficients in an inner slab and the reflection coefficients in two consecutive slabs, and that gives an expression for the reflection and transmission coefficients in the last slab. This method is more general than that traditionally used, and becomes easier and quicker as the number of slabs increases.

ISSN:0002-9505    

DOI:10.1119/1.16625

出版商:American Association of Physics Teachers    

年代:1991

数据来源: AIP

摘要:

A realistic expression for the net nuclear potential is obtained as the sum of the contributions due to the inner core and the outer shell. (AIP)

ISSN:0002-9505    

DOI:10.1119/1.16626

出版商:American Association of Physics Teachers    

年代:1991

数据来源: AIP

ISSN:0002-9505    

DOI:10.1119/1.16627

出版商:American Association of Physics Teachers    

年代:1991

数据来源: AIP

ISSN:0002-9505    

DOI:10.1119/1.16835

出版商:American Association of Physics Teachers    

年代:1991

数据来源: AIP

ISSN:0002-9505    

DOI:10.1119/1.16628

出版商:American Association of Physics Teachers    

年代:1991

数据来源: AIP