Related papers: A Brief Review of Helmholtz Conditions
In two recent papers necessary and sufficient conditions for a given system of second-order ordinary differential equations to be of Lagrangian form with additional dissipative forces were derived. We point out that these conditions are not…
The Helmholtz conditions are necessary and sufficient conditions for a system of second order differential equations to be variational, that is, equivalent to a system of Euler-Lagrange equations for a regular Lagrangian. On the other hand,…
The Lagrangian formulation of classical mechanics is widely applicable in solving a vast array of physics problems encountered in the undergraduate and graduate physics curriculum. Unfortunately, many treatments of the topic lack…
In this paper we provide generalized Helmholtz conditions, in terms of a semi-basic 1-form, which characterize when a given system of second order ordinary differential equations is equivalent to the Lagrange equations, for some given…
New, gauge-independent, second-order Lagrangian for the motion of classical, charged test particles is proposed. It differs from the standard, gauge-dependent, first order Lagrangian by boundary terms only. A new method of deriving…
We derive the equations of motion of an action-dependent version of the Einstein-Hilbert Lagrangian, as a specific instance of the Herglotz variational problem. Action-dependent Lagrangians lead to dissipative dynamics, which cannot be…
Starting from the classical Newton's second law which, according to our assumption, is valid in any instantaneous inertial rest frame of body that moves in Minkowskian space-time we get the relativistic equation of motion…
Equations of motion for a general relativistic post-Newtonian Lagrangian approach mainly refer to acceleration equations, i.e. differential equations of velocities. They are directly from the Euler-Lagrangian equations, and usually have…
We apply a simple decomposition to the energy of a moving particle. Based on this decomposition, we identify the potential and kinetic energies, then use them to give general definitions of momentum and the various kinds of forces exerted…
Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian…
Orbital motion of a body can be found from Newtonian equation of motion. However, it is useful to express the motion through time derivatives of Keplerian orbital elements, mainly if the motion is perturbed by small perturbing force. The…
I discuss the physical basis of classical mechanics, such as expressed commonly using the framework of Newton's Principia. Newton's formulation of the laws of motion is seen to have quite a few ambiguities and shortcomings. Therefore I…
Using Euler's equations of motion and the Hamiltonian formulation, we obtain the equations of motion of systems with internal angular momentum that are moving with respect to a given reference frame, when subjected to a torque which is…
The problem of derivation of the equations of motion from the field equations is considered. Einstein's field equations have a specific analytical form: They are linear in the second order derivatives and quadratic in the first order…
Interactions are explored through the observation of the dynamics of particles. On the classical level the basic underlying assumption in that scheme is that Newton's second law holds. Relaxing the validity of this axiom by, e.g., allowing…
This paper presents a general formulation of equations of motion of a pendulum with n point mass by use of two different methods. The first one is obtained by using Lagrange Mechanics and mathematical induction(inspection), and the second…
We derive the equations of motion of an extended test body in the context of Einstein's theory of gravitation. The equations of motion are obtained via a multipolar approximation method and are given up to the quadrupolar order. Special…
A conventional derivation of motion equations in mechanics and field equations in field theory is based on the principle of least action with a proper Lagrangian. With a time-independent Lagrangian, a function of coordinates and velocities…
Minimizing the Action integral of a Lagrangian provides the Euler-Lagrange equation of motion in the elegant machinery of Lagrangian Mechanics. However two relations define the divergence of current and energy-momentum, and provide an…
We derive the discrete version of the classical Helmholtz condition. Precisely, we state a theorem characterizing second order finite differences equations admitting a Lagrangian formulation. Moreover, in the affirmative case, we provide…