Related papers: Non-differentiable variational principles
Differential operators usually result in derivatives expressed as a ratio of differentials. For all but the simplest derivatives, these ratios are typically not algebraically manipulable, but must be held together as a unit in order to…
The introduction of nonlinearities in the Schr\"odinger equation has been considered in the literature as an effective manner to describe the action of external environments or mean fields. Here, in particular, we explore the nonlinear…
We obtain Euler-Lagrange equations, transversality conditions and a Noether-like theorem for Herglotz-type variational problems with Lagrangians depending on generalized fractional derivatives. As an application, we consider a damped…
We extend the notion of variational integrator for classical Euler-Lagrange equations to the fractional ones. As in the classical case, we prove that the variational integrator allows to preserve Noether-type results at the discrete level.
The fractional operators together with exponential quantum in coordinate and momentum space corresponding to the power of observables are introduced. Based on an exponential relation between energy and momentum, the fractional Schr\"odinger…
A previous derivation of the single-particle Schr\"odinger equation from statistical assumptions is generalized to an arbitrary number $N$ of particles moving in three-dimensional space. Spin and gauge fields are also taken into account. It…
A new calculus based on fractal subsets of the real line is formulated. In this calculus, an integral of order $\alpha, 0 < \alpha \leq 1$, called $F^\alpha$-integral, is defined, which is suitable to integrate functions with fractal…
We show that nonlinear problems including nonlinear partial differential equations can be efficiently solved by variational quantum computing. We achieve this by utilizing multiple copies of variational quantum states to treat…
Classical statistical particle mechanics in the configuration space can be represented by a nonlinear Schrodinger equation. Even without assuming the existence of deterministic particle trajectories, the resulting quantum-like statistical…
Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains…
We introduce new fractional operators of variable order on isolated time scales with Mittag-Leffler kernels. This allows a general formulation of a class of fractional variational problems involving variable-order difference operators. Main…
The master equation describing non-equilibrium one-dimensional problems like diffusion limited reactions or critical dynamics of classical spin systems can be written as a Schr\"odinger equation in which the wave function is the probability…
We prove the Euler-Lagrange delta-differential equations for problems of the calculus of variations on arbitrary time scales with delta-integral functionals depending on higher-order delta derivatives.
Variational principles play a fundamental role in deriving evolution equations of physics. They are working well in case of nondissipative evolution but for dissipative systems they are not unique, not predictive and not constructive. With…
We study fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives, generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the…
A theoretical scheme, based on a probabilistic generalization of the Hamilton's principle, is elaborated to obtain an unified description of more general dynamical behaviors determined both from a lagrangian function and by mechanisms not…
The center of interest in this work are variational problems with integral functionals depending on special nonlocal gradients. The latter correspond to truncated versions of the Riesz fractional gradient, as introduced in [Bellido, Cueto &…
We propose a quantum algorithm to solve systems of nonlinear differential equations. Using a quantum feature map encoding, we define functions as expectation values of parametrized quantum circuits. We use automatic differentiation to…
Since the seminal work of Emmy Noether it is well know that all conservations laws in physics, \textrm{e.g.}, conservation of energy or conservation of momentum, are directly related to the invariance of the action under a family of…
We begin by presenting the classical deterministic problems of the calculus of variations, with emphasis on the necessary optimality conditions of Euler-Lagrange and the Noether theorem. As examples of application, we obtain the…