Adiabatic Amplification of Energy and Magnetic Moment of a Charged Particle after the Magnetic Field Inversion.

We study the evolution of the energy and magnetic moment of a quantum charged particle placed in a homogeneous magnetic field, when this field changes its sign adiabatically. We show that after a single magnetic field passage through zero value, the famous adiabatic invariant ratio of energy to frequency is reestablished again, but with a proportionality coefficient higher than in the initial state. The concrete value of this proportionality coefficient depends on the power index of the frequency dependence on time near zero point. In particular, the adiabatic ratio of the initial ground state (with zero radial and angular quantum numbers) triplicates if the frequency tends to zero linearly as a function of time. If the Larmor frequency attains zero more than once, the adiabatic proportionality coefficient strongly depends on the lengths of the time intervals between zero points, so that the mean energy behavior can be quasi-stochastic after many passages through zero value. The original Born-Fock adiabatic theorem does not work after the frequency passes through zero. However, its generalization is found: the initial Fock state becomes a wide superposition of many instantaneous Fock states, whose weights do not depend on time in the new adiabatic regime.

Adiabatic Amplification of the Harmonic Oscillator Energy When the Frequency Passes through Zero.

We study the evolution of the energy of a harmonic oscillator when its frequency slowly varies with time and passes through a zero value. We consider both the classical and quantum descriptions of the system. We show that after a single frequency passage through a zero value, the famous adiabatic invariant ratio of energy to frequency (which does not hold for a zero frequency) is reestablished again, but with the proportionality coefficient dependent on the initial state. The dependence on the initial state disappears after averaging over the phases of initial states with the same energy (in particular, for the initial vacuum, the Fock and thermal quantum states). In this case, the mean proportionality coefficient is always greater than unity. The concrete value of the mean proportionality coefficient depends on the power index of the frequency dependence on a time near the zero point. In particular, the mean energy triplicates if the frequency tends to zero linearly. If the frequency attains zero more than once, the adiabatic proportionality coefficient strongly depends on the lengths of time intervals between zero points, so that the mean energy behavior becomes quasi-stochastic after many passages through a zero value. The original Born-Fock theorem does not work after the frequency passes through zero. However, its generalization is found: the initial Fock state becomes a wide superposition of many Fock states, whose weights do not depend on time in the new adiabatic regime. When the mean energy triplicates, the initial Nth Fock state becomes a superposition of, roughly speaking, 6N states, distributed nonuniformly. The initial vacuum and low-order Fock states become squeezed, as well as the initial thermal states with low values of the mean energy.

Nordic walking training in elderly, a randomized clinical trial. Part II: Biomechanical and metabolic adaptations.

BACKGROUND: Nordic walking is an attractive method of endurance training. Nevertheless, the biomechanic response due to the additional contribution of using poles in relation to free walking training has been less explored in the elderly. PURPOSE: This randomized parallel controlled trial aimed to assess the effects of 8 weeks of Nordic walking and free walking training on the walking economy, mechanical work, metabolically optimal speed, and electromyographic activation in elderly. METHODS: Thirty-three sedentary elderly were randomized into Nordic walking (n = 16) and free walking group (n = 17) with equalized loads. Submaximal walking tests were performed from 1 to 5 km h-1 on the treadmill. RESULTS: Walking economy was improved in both free and Nordic walking groups (x2 4.91, p = 0.014) and the metabolically optimal speed was increased by approximately 0.5 km h-1 changing the speed-cost profile. The electromyographic activation in lower and upper limbs, pendular recovery, and total, external, and internal mechanical work remained unchanged (p > 0.05). Interestingly, the internal mechanical work associated with arm movement was higher in the Nordic walking group than in the free walking group after training, while the co-contraction from upper limb muscles was reduced similarly to both groups. CONCLUSIONS: Eight weeks of Nordic walking training effectively improved the walking economy and functionality as well as maintained the gait mechanics, similar to free walking training in elderly people. This enhancement in the metabolic economy may have been mediated by a reduction in the co-contraction from upper limb muscles. TRIAL REGISTRATION: ClinicalTrails.gov NCT03096964.