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By means of some one-dimensional examples we suggest a possible new relationship between Lagrangian and Hamiltonian formalisms for non-degenerate systems. [ABSTRACT FROM AUTHOR]

In joint work with J. L. Lebowitz, C. Mastrodonato, and N. Zanghì [2, 3, 4], we considered an isolated, macroscopic quantum system. Let H be a micro-canonical 'energy shell,' i.e., a subspace of the system's Hilbert space spanned by the (finitely) many energy eigenstates with energies between E and E+δE. The thermal equilibrium macro-state at energy E corresponds to a subspace Heq of H such that dimHeq/dimH is close to 1. We say that a system with state vector ψ ε H is in thermal equilibrium if ψ is 'close' to Heq. We argue that for 'typical' Hamiltonians, all initial state vectors ψ0 evolve in such a way that ψt is in thermal equilibrium for most times t. This is closely related to von Neumann's quantum ergodic theorem of 1929. [ABSTRACT FROM AUTHOR]

For a time-periodically perturbed 1D Hamiltonian system, we match the separatrix map and the resonance Hamiltonian dynamics for the frequency ranges where the separatrix chaotic layer (SCL) possesses the largest width. This allows us to describe the boundaries of the SCL in the phase plane, in particular high peaks in the frequency dependence of the SCL width in energy. [ABSTRACT FROM AUTHOR]

Motivated by two time physics theory we revisited the Noether theorem for Hamiltonian constrained systems. Our review presents a novel method to show that the gauge transformations are generated by the conserved quantities associated with the first class constraints. © 2006 American Institute of Physics [ABSTRACT FROM AUTHOR]

The paper provides the discrete Lagrangian and Hamiltonian formulations of mechanical systems for both non-singular and singular cases. The Lagrangians with linear velocities and with higher velocities are investigated and the corresponding difference Euler-Lagrange equations and Hamiltonians are found. © 2006 American Institute of Physics [ABSTRACT FROM AUTHOR]