Jürgen Struckmeier

COVARIANT HAMILTONIAN FIELD THEORY

A consistent, local coordinate formulation of covariant Hamiltonian field theory is presented. Whereas the covariant canonical field equations are equivalent to the Euler–Lagrange field equations, the covariant canonical transformation theory offers more general means for defining mappings that preserve the form of the field equations than the usual Lagrangian description. It is proven that Poisson brackets, Lagrange brackets, and canonical 2-forms exist that are invariant under canonical transformations of the fields. The technique to derive transformation rules for the fields from generating functions is demonstrated by means of various examples. In particular, it is shown that the infinitesimal canonical transformation furnishes the most general form of Noethers theorem. Furthermore, we specify the generating function of an infinitesimal space-time step that conforms to the field equations.

Canonical transformations and exact invariants for time-dependent Hamiltonian systems

An exact invariant is derived for n-degree-of-freedomnon-relativistic Ham iltonian sys- tems with general time-dependent potentials. To work out the invariant, an infinitesimal canonical transformation is performed in the framework of the extended phase-space. We apply this ap- proach to derive the invariant for a specific class of Hamiltonian systems. For the considered class of Hamiltonian systems, the invariant is obtained equivalently performing in the extended phase- space a finite canonical transformation of the initially time-dependent Hamiltonian to a time-inde- pendent one. It is furthermore shown that the invariant can be expressed as an integral of an energy balance equation. The invariant itself contains a time-dependent auxiliary function xðtÞ that represents a solution of a linear third-order differential equation, referred to as the auxiliary equation. The coefficients of the auxiliary equation depend in general on the explicitly known configuration space trajectory defined by the systems time evolution. This complexity of the auxiliary equation reflects the gen- erally involved phase-space symmetry associated with the conserved quantity of a time-dependent non-linear Hamiltonian system. Our results are applied to three examples of time-dependent damped and undamped oscillators. The known invariants for time-dependent and time-indepen- dent harmonic oscillators are shown to follow directly from our generalized formulation.

Parametric sum envelope instability of periodically focused intense beams

The envelope instability is a second order parametric resonance with the periodic focusing and known to appear in space charge dominated beams near 90° phase advance per focusing period. We show in 2d approximation that space charge may also induce parametric “sum envelope instabilities” leading to simultaneous growth of envelopes or skew angles as well as emittances. This can happen by two-plane envelope coupling or by exciting a skew (“odd”) mode in an otherwise fully uncoupled linear lattice. At resonance, the two individual phase advances are split more or less symmetrically away from 90°, and exponential growth occurs. Results from perturbation theory are compared with full envelope models, particle-in-cell simulations, and smooth approximation stopband calculations, all showing very good agreement for realistic space charge parameters.

EXTENDED HAMILTON–LAGRANGE FORMALISM AND ITS APPLICATION TO FEYNMAN'S PATH INTEGRAL FOR RELATIVISTIC QUANTUM PHYSICS

We present a consistent and comprehensive treatise on the foundations of the extended Hamilton–Lagrange formalism — where the dynamical system is parametrized along a general system evolution parameter s, and the time t is treated as a dependent variable t(s) on equal footing with all other configuration space variables qi(s). In the action principle, the conventional classical action L1dt is then replaced by the generalized action L1ds, with L and L1 denoting the conventional and the extended Lagrangian, respectively. It is shown that a unique correlation of L1 and L exists if we refrain from performing simultaneously a transformation of the dynamical variables. With the appropriate correlation of L1 and L in place, the extension of the formalism preserves its canonical form. In the extended formalism, the dynamical system is described as a constrained motion within an extended space. We show that the value of the constraint and the parameter s constitutes an additional pair of canonically conjugate variables. In the corresponding quantum system, we thus encounter an additional uncertainty relation. As a consequence of the formal similarity of conventional and extended Hamilton–Lagrange formalisms, Feynmans nonrelativistic path integral approach can be converted on a general level into a form appropriate for relativistic quantum physics. In the emerging parametrized quantum description, the additional uncertainty relation serves as the means to incorporate the constraint and hence to finally eliminate the parametrization. We derive the extended Lagrangian L1 of a classical relativistic point particle in an external electromagnetic field and show that the generalized path integral approach yields the Klein–Gordon equation as the corresponding quantum description. We furthermore derive the space–time propagator for a free relativistic particle from its extended Lagrangian L1. These results can be regarded as the proof of principle of the relativistic generalization of Feynmans path integral approach to quantum physics.

Covariant Hamiltonian Representation of Noether’s Theorem and Its Application to SU(N) Gauge Theories

We present the derivation of the Yang-Mills gauge theory based on the covariant Hamiltonian representation of Noether’s theorem. As the starting point, we re-formulate our previous presentation of the canonical Hamiltonian derivation of Noether’s theorem (Struckmeier and Reichau, Exciting Interdisciplinary Physics, Springer, New York, p. 367, 2013, [1]). The formalism is then applied to derive the Yang-Mills gauge theory. The Noether currents of U(1) and SU(N) gauge theories are derived from the respective infinitesimal generating functions of the pertinent symmetry transformations which maintain the form of the respective Hamiltonian.