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Register a new userCalculus of variations and optimal control for generalized functions
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We present an extension of some results of higher order calculus of variations and optimal control to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions, while sharing many nonlinear properties with ordinary smooth functions. We prove the higher order Euler-Lagrange equations, the DAlembert principle in differential form, the du Bois-Reymond optimality condition and the Noethers theorem. We start the theory of optimal control proving a weak form of the Pontryagin maximum principle and the Noethers theorem for optimal control. We close with a study of a singularly variable length pendulum, oscillations damped by two media and the Pais-Uhlenbeck oscillator with singular frequencies.
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We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions while sharing many nonlinear properties with ordinary smooth functions. We prove full connections between extremals and Euler-Lagrange equations, classical necessary and sufficient conditions to have a minimizer, the necessary Legendre condition, Jacobis theorem on conjugate points and Noethers theorem. We close with an application to low regularity Riemannian geometry.
Divergence-free symmetric tensors seem ubiquitous in Mathematical Physics. We show that this structure occurs in models that are described by the so-called second variational principle, where the argument of the Lagrangian is a closed differential form. Divergence-free tensors are nothing but the second form of the Euler--Lagrange equations. The symmetry is associated with the invariance of the Lagrangian density upon the action of some orthogonal group.
In this paper we present the groundwork for an It^o/Malliavin stochastic calculus and Hidas white noise analysis in the context of a supersymmentry with Z3-graded algebras. To this end we establish a ternary Fock space and the corresponding strong algebra of stochastic distributions and present its application in the study of stochastic processes in this context.
We study the quantum evolution under the combined action of the exponentials of two not necessarily commuting operators. We consider the limit in which the two evolutions alternate at infinite frequency. This case appears in a plethora of situations, both in physics (Feynman integral) and mathematics (product formulas). We focus on the case in which the two evolution times are scaled differently in the limit and generalize standard techniques and results.
The Humbert-Bessel are multi-index functions with various applications in electromagnetism. New families of functions sharing some similarities with Bessel functions are often introduced in the mathematical literature, but at a closer analysis they are not new, in the strict sense of the word, and are shown to be expressible in terms of already discussed forms. This is indeed the case of the re-modified Bessel functions, whose properties have been analyzed within the context of coincidence problems in probability theory. In this paper we show that these functions are particular cases of the Humbert-Bessel ones.
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