We show that classical particle mechanics (Hamiltonian and Lagrangian consistent with relativistic electromagnetism) can be derived from three fundamental assumptions: infinite reducibility, deterministic and reversible evolution, and kinematic equivalence. The core idea is that deterministic and reversible systems preserve the cardinality of a set of states, which puts considerable constraints on the equations of motion. This perspective links different concepts from different branches of math and physics (e.g. cardinality of a set, cotangent bundle for phase space, Hamiltonian flow, locally Minkowskian space-time manifold), providing new insights. The derivation strives to use definitions and mathematical concepts compatible with future extensions to field theories and quantum mechanics.
We deal with Lagrangians which are not the standard scalar ones. We present a short review of tensor Lagrangians, which generate massless free fields and the Dirac field, as well as vector and pseudovector Lagrangians for the electric and magnetic fields of Maxwell's equations with sources. We introduce and analyse Lagrangians which are equivalent to the Hamilton-Jacobi equation and recast them to relativistic equations.
A proper choice for the sign of the Lorentz-covariant relativistic Lagrangian for a particle moving in an electromagnetic field is presented. This choice is based on which space-time metric is used in the representation of the relativistic particle dynamics.
We study and formulate the Lagrangian for the LC, RC, RL, and RLC circuits by using the analogy concept with the mechanical problem in classical mechanics formulations. We found that the Lagrangian for the LC and RLC circuits are governed by two terms i. e. kinetic energy-like and potential energy-like terms. The Lagrangian for the RC circuit is only a contribution from the potential energy-like term and the Lagrangian for the RL circuit is only from the kinetic energy-like term.
This work builds on the Volterra series formalism presented in [D. W. Dreisigmeyer and P. M. Young, J. Phys. A \textbf{36}, 8297, (2003)] to model nonconservative systems. Here we treat Lagrangians and actions as `time dependent' Volterra series. We present a new family of kernels to be used in these Volterra series that allow us to derive a single retarded equation of motion using a variational principle.
We reexamine the problem of having nonconservative equations of motion arise from the use of a variational principle. In particular, a formalism is developed that allows the inclusion of fractional derivatives. This is done within the Lagrangian framework by treating the action as a Volterra series. It is then possible to derive two equations of motion, one of these is an advanced equation and the other is retarded.
In classical mechanics, there are two objectivities: The ``isometric objectivity'' which concerns the constitutive laws of materials once expressed in a reference frame, and the ``covariant objectivity'' which concerns the universal laws of physics required to be observer independent. Covariant objectivity is the main topic in this manuscript. We therefore first introduce motions and Eulerian variables and functions, then we deduce Lagrangian variables and functions, then define the function ``deformation gradient'', from which we deduce simple definitions of the Lie derivatives. Then we detai…
A survey of topics of recent interest in Hamiltonian and Lagrangian dynamical systems, including accessible discussions of regularization of the central force problem; inequivalent Lagrangians and Hamiltonians; constants of central force motion; a general discussion of higher-order Lagrangians and Hamiltonians with examples from Bohmian quantum mechanics, the Korteweg-de Vries equation and the logistic equation; gauge theories of Newtonian mechanics; classical spin, Grassmann numbers, and pseudomechanics.
We present two types of relativistic Lagrangians for the Lorentz-Dirac equation written in terms of an arbitrary world-line parameter. One of the Lagrangians contains an exponential damping function of the proper time and explicitly depends on the world-line parameter. Another Lagrangian includes additional cross-terms consisting of auxiliary dynamical variables and does not depend explicitly on the world-line parameter. We demonstrate that both the Lagrangians actually yield the Lorentz-Dirac equation with a source-like term.
I extract some philosophical morals from some aspects of Lagrangian mechanics. (A companion paper will present similar morals from Hamiltonian mechanics and Hamilton-Jacobi theory.) One main moral concerns methodology: Lagrangian mechanics provides a level of description of phenomena which has been largely ignored by philosophers, since it falls between their accustomed levels--``laws of nature'' and ``models''. Another main moral concerns ontology: the ontology of Lagrangian mechanics is both more subtle and more problematic than philosophers often realize. The treatment of Lagrangian mechani…
It is shown that a given non-autonomous system of two first-order ordinary differential equations can be expressed in Hamiltonian form. The derivation presented here allow us to obtain previously known results such as the infinite number of Hamiltonians in the autonomous case and the Helmholtz condition for the existence of a Lagrangian.
A parameter-invariant variational problem with a manifestly covariant Lagrangian function of second order is considered, which covers the case of the free relativistic top at constraint manifold of constant acceleration
We develop a relativistic variational model for a nematic liquid crystal interacting with an electro- magnetic field. The constitutive relation for a general anisotropic uniaxial diamagnetic and dielectric medium is analyzed. We discuss light wave propagation in this moving uniaxial medium, for which the corresponding optical metrics are identified explicitly. A Lagrangian for the coupled system of a nematic liquid crystal and the electromagnetic field is constructed, from which a complete set of equations of motion for the system is derived. The canonical energy-momentum and spin tensors are …
The aim of this work is to show that particle mechanics, both classical and quantum, Hamiltonian and Lagrangian, can be derived from few simple physical assumptions. Assuming deterministic and reversible time evolution will give us a dynamical system whose set of states forms a topological space and whose law of evolution is a self-homeomorphism. Assuming the system is infinitesimally reducible---specifying the state and the dynamics of the whole system is equivalent to giving the state and the dynamics of its infinitesimal parts---will give us a classical Hamiltonian system. Assuming the syst…
In this article, we argue that two distinct types of time should be taken into account in relativistic physics: a geometric time, which emanates from the structure of spacetime and its metrics, and a causal time, indicating the flow from the past to the future. A particularity of causal times is that its values have no intrinsic meaning, as their evolution alone is meaningful. In the context of relativistic Lagrangian mechanics, causal times corresponds to admissible parameterizations of paths, and we show that in order for a langragian to not depend on any particular causal time (as its value…
An application of variational principle to bifurcation of periodic solution in Lagrangian mechanics is shown. A few higher derivatives of the action integral at a periodic solution reveals the behaviour of the action in function space near the solution. Then the variational principle gives a method to find bifurcations from the solution. The second derivative (Hessian) of the action has an important role. At a bifurcation point, an eigenvalue of Hessian tends to zero. Inversely, if an eigenvalue tends to zero, the zero point is a bifurcation point. The third and higher derivatives of the actio…
We describe a new method to formulate classical Lagrangian mechanics on a finite-dimensional Lie group. This new approach is much more pedagogical than many previous treatments of the subject, and it directly introduces students to generator matrices and their usefulness in many manipulations. The example of rigid body rotation, i.e. motion on the Lie group SO(3), is used as an example, and it is shown how to derive Euler's equations directly from the principle of least action. The techniques covered in this paper generalize to other Lie groups in a straightforward manner, which is discussed. …
Textbook treatments of classical mechanics typically assume that the Lagrangian is nonsingular. That is, the matrix of second derivatives of the Lagrangian with respect to the velocities is invertible. This assumption insures that (i) Lagrange's equations can be solved for the accelerations as functions of coordinates and velocities, and (ii) the definition of the conjugate momenta can be inverted for the velocities as functions of coordinates and momenta. This assumption, however, is unnecessarily restrictive -- there are interesting classical dynamical systems with singular Lagrangians. The …
The geodesic has a fundamental role in physics and in mathematics: roughly speaking, it represents the curve that minimizes the arc length between two points on a manifold. We analyze a basic but misinterpreted difference between the Lagrangian that gives the arc length of a curve and the one that describes the motion of a free particle in curved space. Although they provide the same formal equations of motion, they are not equivalent. We explore this difference from a geometrical point of view, where we observe that the non-equivalence is nothing more than a matter of symmetry. As application…
Guided by the symmetries of the Euler-Lagrange equations of motion, a study of the constrained dynamics of singular Lagrangians is presented. We find that these equations of motion admit a generalized Lie symmetry, and on the Lagrangian phase space the generators of this symmetry lie in the kernel of the Lagrangian two-form. Solutions of the energy equation\textemdash called second-order, Euler-Lagrange vector fields (SOELVFs)\textemdash with integral flows that have this symmetry are determined. Importantly, while second-order, Lagrangian vector fields are not such a solution, it is always po…
We give a general definition of \emph{relativistic force} in the context of Lagrangian mechanics. Once this is done we prove that the only relativistic forces which are linear on the velocities are those coming from differential 2-forms defined on the configuration space. In this sense, electromagnetic fields provide a mechanical system with the simplest type of relativistic forces.
