A method for constructing general null Lagrangians and their higher harmonics is presented for dynamical systems with one degree of freedom. It is shown that these Lagrangians can be used to obtain non-standard Lagrangians, which give equations of motion for the law of inertia and some dissipative dynamical systems. The necessary condition for deriving equations of motion by using null Lagrangians is presented, and it is demonstrated that this condition plays the same role for null Lagrangians as the Euler-Lagrange equation plays for standard and non-standard Lagrangians. The obtained results and their applications establish a novel role of null Lagrangians in classical dynamics.
Null Lagrangians and their gauge functions are derived for given standard and non-standard Lagrangians. The obtained standard null Lagrangians generalize those previously found but the non-standard null Lagrangians are new. The gauge functions are used to make the action invariant and introduce the exact null Lagrangians, which form a new family of null Lagrangians. The conditions required for the action to be invariant are derived for all null Lagrangians presented in this paper, thus, making them exact. As a specific application, the exact null Lagrangians are derived for a second-order ordi…
Among different Lagrangians, null Lagrangians are known for having identically zero the Euler-Lagrange equation and, therefore, they have no effects on the resulting equations of motion. However, there is a special family of null Lagrangians that can be used to convert linear and undriven equations of motion into nonlinear and driven ones. To identify this special family, general null Lagrangians and their gauge functions are constructed for second-order ordinary differential equations of motion describing one-dimensional dynamical systems. The gauge functions corresponding to forces and nonli…
A method to construct general null Lagrangians and their exact gauge functions is developed. The functions are used to define classical forces independently from Newtonian dynamics. It is shown that the forces generated by the exact gauge functions allow for the conversion of the first Newton equation into the second one. The presented approach gives new insights into the origin of forces in Classical Mechanics.
Standard and non-standard Lagrangians that give the same equation of motion are significantly different in their forms, as the latter do not have terms that clearly discernable energy-like expressions. A special family of these Lagrangians are null Lagrangians and their gauge functions. It is shown that non-standard null Lagrangians and their gauge functions can be used to introduce dissipative forces to dynamical systems. With standard null Lagrangians being known for introducing non-dissipative forces, the presented results allow for a complete picture of novel roles played by null Lagrangia…
The space of Null Lagrangians is the least investigated territory in dynamics since they are identically sent to zero by their Euler-Lagrange operator and thereby having no effects on equations of motion. A humble effort to discover the relevance of these Null Lagrangians in dynamics is made by introducing a generalized procedure (with respect to the recent procedure introduced by the authors of this paper) that takes advantage of the null-ness of these Lagrangians to construct non-standard Lagrangians that represent a range of interesting dynamical systems. By using the generalized procedure,…
Knotted solutions to electromagnetism are investigated as an independent subsector of the theory. We write down a Lagrangian and a Hamiltonian formulation of Bateman's construction for the knotted electromagnetic solutions. We introduce a general definition of the null condition and generalize the construction of Maxwell's theory to massless free complex scalar, its dual two form field, and to a massless DBI scalar. We set up the framework for quantizing the theory both in a path integral approach, as well as the canonical Dirac method for a constrained system. We make several observations abo…
This manuscript provides a characterisation of the equivalence class of classical smooth Lagrangian densities that involve terms depending on two distinct points of the underlying Euclidean base space of the theory. Theories of this type are referred to as non-local field theories, which are of particular interest in the group field theory approach to quantum gravity. The notion of equivalence of Lagrangian densities is set by physical indistinguishability by means of their equations of motion whose derivation is shown briefly. We expect our results to give a more comprehensive view on the non…
A novel method to make Lagrangians Galilean invariant is developed. The method, based on null Lagrangians and their gauge functions, is used to demonstrate the Galilean invariance of the Lagrangian for Newton's law of inertia. It is suggested that this new solution of an old physics problem may have implications and potential applications to all gauge-based theories of physics.
In linear anisotropic elasticity, the elastic properties of a medium are described by the fourth rank elasticity tensor C. The decomposition of C into a partially symmetric tensor M and a partially antisymmetric tensors N is often used in the literature. An alternative, less well-known decomposition, into the completely symmetric part S of C plus the reminder A, turns out to be irreducible under the 3-dimensional general linear group. We show that the SA-decomposition is unique, irreducible, and preserves the symmetries of the elasticity tensor. The MN-decomposition fails to have these desirab…
We consider the Frenet-Serret geometry of null curves in a three and a four-dimensional Minkowski background. We develop a theory of deformations adapted to the Frenet-Serret frame. We exploit it to provide a Lagrangian description of the dynamics of geometric models for null curves.
In the context of asymptotically flat spacetimes we exploit techniques proper either of white noise analysis either of dynamical systems in order to develop the Lagrangian and the Hamiltonian approach to a BMS invariant field theory at null infinity.
The local motion of a null curve in Minkowski 3-space induces an evolution equation for its Lorentz invariant curvature. Special motions are constructed whose induced evolution equations are the members of the KdV hierarchy. The null curves which move under the KdV flow without changing shape are proven to be the trajectories of a certain particle model on null curves described by a Lagrangian linear in the curvature. In addition, it is shown that the curvature of a null curve which evolves by similarities can be computed in terms of the solutions of the second Painlevé equation.
Lagrangian formalism is established for differential equations with special functions of mathematical physics as solutions. Formalism is based on either standard or non-standard Lagrangians. This work shows that the procedure of deriving the standard Lagrangians leads to Lagrangians for which the Euler--Lagrange equation vanishes identically, and that only some of these Lagrangians become the null Lagrangians with the well-defined gauge functions. It is also demonstrated that the non-standard Lagrangians require that the Euler--Lagrange equations are amended by the auxiliary conditions, which …
We use a trick similar to Weinberg's for adiabatic modes, in a Manton approximation for general relativity on manifolds with spatial boundary. This results in a description of the slow-time dependent solutions as null geodesics on the space of boundary diffeomorphisms, with respect to a metric we prove to be composed solely of the boundary data. We show how the solutions in the bulk space is determined with the constraints of general relativity. To give our description a larger perspective, we furthermore identify our resulting Lagrangian as a generalized version of the covariantized Lagrangia…
We present a study of the properties of Bargmann Invariants (BI) and Null Phase Curves (NPC) in the theory of the geometric phase for finite dimensional systems. A recent suggestion to exploit the Majorana theorem on symmetric SU(2) multispinors is combined with the Schwinger oscillator operator construction to develop efficient operator based methods to handle these problems. The BI is described using intrinsic unitary invariant angle parameters, whose algebraic properties as functions of Hilbert space dimension are analysed using elegant group theoretic methods. The BI-geometric phase connec…
We study the geometric nature of the Jacobi equation. In particular we prove that Jacobi vector fields (JVFs) along a solution of the Euler-Lagrange (EL) equations are themselves solutions of the EL equations but considered on a non-standard algebroid (different from the tangent bundle Lie algebroid). As a consequence we obtain a simple non-computational proof of the relation between the null subspace of the second variation of the action and the presence of JVFs (and conjugate points) along an extremal. We work in the framework of skew-symmetric algebroids.
The aim of this comment is to call to the attention of DSR readers a basic fact. The introduction of noncommutative structures in problems like the one addressed in [1] is not necessary for the understanding of DSR physics. It can be described just as the relativistic free particle problem in a different parametrization.
Building towards a more covariant approach to canonical classical and quantum gravity we outline an approach to constrained dynamics that de-emphasizes the role of the Hamiltonian phase space and highlights the role of the Lagrangian phase space. We identify a "Lagrangian one-form" to replace the standard symplectic one-form, which we use to construct the canonical constraints and an associated constraint algebra. The method is particularly useful for generally covariant systems and systems with a degenerate canonical symplectic form, such as Einstein Cartan gravity, to which we apply the meth…
