Newton's method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized Jacobian instead of the derivative. This extension includes the B-derivative version of Newton's method as a special case. Convergence theorems are proved under the condition of semismoothness. It is shown that the gradient function of the augmented Lagrangian for C2-no...

Contents Introduction 5 Chapter 1. The theory in the smooth framework 11 1.1. The ordinary differential equation 11 1.2. Existence and uniqueness in the classical setting 11 1.3. The classical flow of a vector field 16 1.4. The transport equation and the continuity equation 19 Part 1. The Eulerian viewpoint 23 Chapter 2. Renormalized solutions and well-posedness of the PDE 25 2.1. A strategy to...

Lagrangian perturbation theory for cosmological fluid describes structure formation in the quasinonlinear stage well. In a previous paper, we presented a third-order perturbative equation for Lagrangian perturbation with pressure. There we considered only the longitudinal modes for the first-order perturbation. In this paper, we generalize the perturbation, i.e., we consider both the longitudin...

The Schwinger-Dyson equation for a scalar propagator is solved in Minkowski space with the help of an integral spectral representation, both for spacelike and timelike momenta. The equation is re-written into a form suitable for numerical solution by iterations. This procedure is described for a simple unphysical Lagrangian with a cubic interaction, with future extensions to more realistic theo...

We apply the relativistic chiral Lagrangian to the nuclear equation of state. An effective chiral power expansion scheme, which is constructed to work around nuclear saturation density, is presented. The leading and subleading terms are evaluated and are shown to provide an equation of state with excellent saturation properties. Our saturation mechanism is found to probe details of the nuclear ...

We develop a spinor equation of the electromagnetic field, which is equivalent to the Maxwell equation and has a similar form as the Dirac equation. The spinor is the very conjugate momentum of the vector potential in the Lagrangian mechanics. In this framework the electromagnetic field described by the spinor exhibits the SU(2) internal symmetry. The quantization and the problem of longinal an...

We study higher{ordervariational derivativesof a generic second{order Lagrangian L 0 = L 0 (x; ; @; @ 2) and in this context we discuss the Jacobi equation ensuing from the second variation of the action. We exhibit the diierent inte-grations by parts which may be performed to obtain the Jacobi equation and we show that there is a particular integration by parts which is invariant. We introduce...

Further results are reported for the one-component quaternionic wave equation recently introduced. A Lagrangian is found for the momentum-space version of the free equation; and another, nonlocal in time, is found for the complete equation. Further study of multiparticle systems has us looking into the mathematics of tensor products of Hilbert spaces. The principles of linearity and superpositi...

Using the quantum Hamilton-Jacobi equation within the framework of the equivalence postulate, we construct a Lagrangian of a quantum system in one dimension and derive a third order equation of motion representing a first integral of the quantum Newton’s law. We then integrate this equation in the free particle case and compare our results to those of Floydian trajectories. Finally, we propose ...

Marsden and Scheurle [1993] studied Lagrangian reduction in the context of momentum map constraints—here meaning the reduction of the standard Euler-Lagrange system restricted to a level set of a momentum map. This provides a Lagrangian parallel to the reduction of symplectic manifolds. The present paper studies the Lagrangian parallel of Poisson reduction for Hamiltonian systems. For the reduc...