We derive an expression for the determination of the apsidal angles that holds good for arbitrary central potentials. This formula can be useful in the calculation of precession rates. Then we discuss under what conditions the apsidal angles remain independent of the mechanical energy and angular momentum in the central force problem and, as a consequence, an alternative and nonperturbative proof of Bertrand's theorem is obtained.

We apply the generalized Zeta function method to compute the Casimir energy and pressure between an unusual pair of parallel plates at finite temperature, namely, a perfectely conducting plate (ǫ → ∞) and an infinitely permeable one (µ → ∞). The high and low temperature limits of these quantities are discussed. Relationships between high and low temperature limits for the free energy are established by means of a modified version of the temperature inversion symmetry.

We evaluate the fermionic Casimir effect associated with a massive fermion confined within a planar (d + 1)-dimensional slab-bag, on which MIT bag model boundary conditions of the standard type, along a single spatial direction, are imposed. A simple and effective method for adding up the zero-point energy eigenvalues, corresponding to a quantum field under the influence of arbitrary boundary conditions, imposed on the field on flat surfaces perpendicular to a chosen spatial direction, is proposed. Using this procedure, an analytic result is obtained, from which small and large fermion mass limits, valid for an arbitrary number of dimensions, are derived. They match some known results in particular cases. The method can be easily extended to other configurations.

We compute the influence of an external magnetic field on the Casimir energy of a massive charged scalar field confined between two parallel infinite plates. For this case the obtained result shows that the magnetic field inhibits the Casimir effect.The Casimir effect can be generally defined as the effect of a non-trivial space topology on the vacuum fluctuations of relativistic quantum fields [1,2]. The corresponding change in the vacuum fluctuations appears as a shift in the vacuum energy and an associated vacuum pressure. This shift is known as the Casimir energy of the field due to the given space constraints. The original Casimir effect [3] is the attraction of two neutral perfectly conducting parallel plates placed in vacuum. The boundary conditions imposed by the metallic plates confine the vacuum fluctuations of the quantum electromagnetic field in the space between the plates. The effect of the boundary conditions can be viewed as a departure from the trivial topology of lR 3 to the topology of lR 2 × [0, a], where a is the distance between the plates. The resulting shift in the vacuum energy of the quantum electromagnetic field was computed by Casimir and is given by [3]:

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We re-evaluate the zero point Casimir energy for the case of a massive scalar field in R 1 × S 3 space, allowing also for deviations from the standard conformal value ξ = 1/6, by means of zero temperature zeta function techniques. We show that for the problem at hand this approach is equivalent to the high temperature regularisation of the vacuum energy.

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