Noether Symmetry Method for Hamiltonian Mechanics Involving Generalized Operators

Based on the generalized operators, Hamilton equation, Noether symmetry, and perturbation to Noether symmetry are studied. The main contents are divided into four parts, and every part includes two generalized operators. Firstly, Hamilton equations within generalized operators are established. Secondly, the Noether symmetry method and conserved quantity are studied. Thirdly, perturbation to the Noether symmetry and adiabatic invariant are presented. And finally, two applications are presented to illustrate the methods and results.

General formulas for adiabatic invariants in nearly periodic Hamiltonian systems

While it is well known that every nearly periodic Hamiltonian system possesses an adiabatic invariant, extant methods for computing terms in the adiabatic invariant series are inefficient. The most popular method involves the heavy intermediate calculation of a non-unique near-identity coordinate transformation, even though the adiabatic invariant itself is a uniquely defined scalar. A less well-known method, developed by S. Omohundro, avoids calculating intermediate sequences of coordinate transformations but is also inefficient as it involves its own sequence of complex intermediate calculations. In order to improve the efficiency of future calculations of adiabatic invariants, we derive generally applicable, readily computable formulas for the first several terms in the adiabatic invariant series. To demonstrate the utility of these formulas, we apply them to charged-particle dynamics in a strong magnetic field and magnetic field-line dynamics when the field lines are nearly closed.

Conserved Quantity and Adiabatic Invariant for Hamiltonian System with Variable Order

Hamiltonian mechanics plays an important role in the development of nonlinear science. This paper aims for a fractional Hamiltonian system of variable order. Several issues are discussed, including differential equation of motion, Noether symmetry, and perturbation to Noether symmetry. As a result, fractional Hamiltonian mechanics of variable order are established, and conserved quantity and adiabatic invariant are presented. Two applications, fractional isotropic harmonic oscillator model of variable order and fractional Lotka biochemical oscillator model of variable order are given to illustrate the Methods and Results.

Stellarator equilibria with reactor relevant energetic particle losses

Stellarator configurations with reactor relevant energetic particle losses are constructed by simultaneously optimizing for quasisymmetry and an analytically derived metric ( $\unicode[STIX]{x1D6E4}_{c}$ ), which attempts to align contours of the second adiabatic invariant, $J_{\Vert }$ with magnetic surfaces. Results show that with this optimization scheme it is possible to generate quasihelically symmetric equilibria on the scale of ARIES-CS which completely eliminate all collisionless alpha particle losses within normalized radius $r/a=0.3$ . We show that the best performance is obtained by reducing losses at the trapped–passing boundary. Energetic particle transport can be improved even when neoclassical transport, as calculated using the metric $\unicode[STIX]{x1D716}_{\text{eff}}$ , is degraded. Several quasihelically symmetric equilibria with different aspect ratios are presented, all with excellent energetic particle confinement.

Resonant damping of kink oscillations of thin cooling and expanding coronal magnetic loops

We have considered resonant damping of kink oscillations of cooling and expanding coronal magnetic loops. We derived an evolutionary equation describing the dependence of the oscillation amplitude on time. When there is no resonant damping, this equation reduces to the condition of conservation of a previously derived adiabatic invariant. We used the evolutionary equation describing the amplitude to study the competition between damping due to resonant absorption and amplification due to cooling. Our main aim is to investigate the effect of loop expansion on this process. We show that the loop expansion acts in favour of amplification. We found that, when there is no resonant damping, the larger the loop expansion the faster the amplitude growths. When the oscillation amplitude decays due to resonant damping, the loop expansion reduces the damping rate. For some values of parameters the loop expansion can fully counterbalance the amplitude decay and turn the amplitude evolution into amplification.