A brief historical review of the Principle of Least Action is given, and it's methodological importance is emphasized: the basic equations of Classical Mechanics can be derived from the conjunction of the Principle Least Action with the Galilean Principle of Relativity and the hypothesis of homogeneity and isotropy of space and the homogeneity of time. In classical electrodinamics, the situation is similar: one can build the action keeping now in mind the Einstein's Principle of Relativity. By means of the solution of the Euler-Lagrange equations one obtain the equations of motion of charges in given fields (Lorentz force) and the first pair of Maxwell equations. Also, starting from the sources of the field (charges and currents), the second pair of Maxwell equations is obtained.\nIn the case of the General Relativity, the trajectories of a free particle are given by the geodesic lines in space-time. The Einstein equations are also obtained by finding the extreme of some action. A brief historical review of quantum theory is given, starting from the introduction of the quantum of action in physics, to the Schrödinger equation, which can be obtained from a variational principle, and the Heisenberg Uncertainty Principle. Interesting is the role of the classical action in the formulation of quantum mechanics by means of the Feynman path integrals. In it, the classical trajectory in the motion of a system between two configurations, it corresponds the limit in which the Planck constant tends to zero. In quantum electrodinamics, it is required also to start from a Lagrangian together with the properties of space-time simmetry, the gauge invariance and the CPT invariance. Finally, mention is made to the electroweak theory and to quantum cromodynamics
A brief historical review of the Principle of Least Action is given, and it's methodological importance is emphasized: the basic equations of Classical Mechanics can be derived from the conjunction of the Principle Least Action with the Galilean Principle of Relativity and the hypothesis of homogeneity and isotropy of space and the homogeneity of time. In classical electrodinamics, the situation is similar: one can build the action keeping now in mind the Einstein's Principle of Relativity. By means of the solution of the Euler-Lagrange equations one obtain the equations of motion of charges in given fields (Lorentz force) and the first pair of Maxwell equations. Also, starting from the sources of the field (charges and currents), the second pair of Maxwell equations is obtained.\nIn the case of the General Relativity, the trajectories of a free particle are given by the geodesic lines in space-time. The Einstein equations are also obtained by finding the extreme of some action. A brief historical review of quantum theory is given, starting from the introduction of the quantum of action in physics, to the Schrödinger equation, which can be obtained from a variational principle, and the Heisenberg Uncertainty Principle. Interesting is the role of the classical action in the formulation of quantum mechanics by means of the Feynman path integrals. In it, the classical trajectory in the motion of a system between two configurations, it corresponds the limit in which the Planck constant tends to zero. In quantum electrodinamics, it is required also to start from a Lagrangian together with the properties of space-time simmetry, the gauge invariance and the CPT invariance. Finally, mention is made to the electroweak theory and to quantum cromodynamics
Investigations into the nature of the principle of least action have shown that there is an intrinsic relationship between geometrical and topological methods and the variational principle in classical mechanics. In this work, we follow and extend this kind of mathematical analysis into the domain of quantum mechanics. First, we show that the identification of the momentum of a quantum particle with the de Broglie wavelength in 2-dimensional space would lead to an interesting feature; namely the action principle S=0 would be satisfied not only by the stationary path, corresponding to the classical motion, but also by any path. Thereupon the Bohr quantum condition possesses a topological character in the sense that the principal quantum number is identified with the winding number, which is used to represent the fundamental group of paths. We extend our discussions into 3-dimensional space and show that the charge of a particle also possesses a topological character and is quantised and classified by the homotopy group of closed surfaces. We then discuss the possibility to extend our discussions into spaces with higher dimensions and show that there exist physical quantities that can be quantised by the higher homotopy groups. Finally we note that if Einstein’s field equations of general relativity are derived from Hilbert’s action through the principle of least action then for the case of n=2 the field equations are satisfied by any metric if the energy-momentum tensor is identified with the metric tensor, similar to the case when the momentum of a particle is identified with the curvature of the particle’s path.
We propose six principles as the fundamental principles of quantum mechanics: principle of space and time, Galilean principle of relativity, Hamilton's principle, wave principle, probability principle, and principle of indestructibility and increatiblity of particles. We deductively develop the formalism of quantum mechanics on the basis of them: we determine the form of the Lagrangian that satisfies requirements of these principles, and obtain the Schroedinger equation from the Lagrangian. We also derive the canonical commutation relations. Then we adopt the following four guide lines. First, we do not premise the relations between dynamical variables in classical mechanics. Second, we define energy, momentum, and angular momentum as the constants of motion that are derived from homogeneity and isotropy in space and time on the basis of principle of space and time. Since energy and momentum are quantitatively defined in classical mechanics, we define them in quantum mechanics so that the corresponding conservation laws are satisfied in a coupling system of a quantum particle and a classical particle. Third, we define Planck's constant and the mass of a particle as proportionality constants between energy and frequency due to one of Einstein-de Broglie formulas and between momentum and velocity, respectively. We shall obtain the canonical commutation relations and the Schroedinger equation for a particle in an external field in the definitive form. We shall also prove that relations between dynamical variables in quantum mechanics have the same forms for those in classical mechanics.
# Least Action in Classical Mechanics: # The Concept of Functional # The Principle of Least Action # Conservation of Energy. Constants of the Motion # Particles Interacting Through an Intermediate Oscillator # Least Action in Quantum Mechanics: # The Lagrangian in Quantum Mechanics # The Calculation of Matrix Elements in the Language of a Lagrangian # The Equations of Motion in Lagrangian Form # Translation to the Ordinary Notation of Quantum Mechanics # The Generalization to Any Action Function # Conservation of Energy. Constants of the Motion # The Role of the Wave Function # Transition Probabilities # Expectation Values for Observables # Application to the Forced Harmonic Oscillator # Particles Interacting Through an Intermediate Oscillator # Space-Time Approach to Non-Relativistic Quantum Mechanics # The Lagrangian in Quantum Mechanics
In this paper, we show that the difficulties of interpretation of the principle of least action concerning "final causes" or "efficient causes" are due to the existence of two different actions, the "Euler-Lagrange action" (or classical action) and the "Hamilton-Jacobi action". These two actions, which are not clearly differentiated in the texbooks, are solutions to the same Hamilton-Jacobi equation, but with very different initial conditions: smooth conditions for the Hamilton-Jacobi action, singular conditions for the Euler-Lagrange action. There are related by the Minplus Path Integral which is the analog in classical mechanics of the Feynmann Path Integral in quantum mechanics. Finally, we propose a clear-cut interpretation of the principle of least action: the Hamilton-Jacobi action does not use "final causes" and seems to be the action used by Nature; the Euler-Lagrange action uses "final causes" and is the action used by an observer to retrospectively determine the trajectory of the particle.
The Lagrangian could be written in terms of kinetic and potential energy, functions of coordinates and velocity and the amplitude of the fields at that point. The essential condition is that the Lagrangian assumed for the field, when studied with Lagrange’s equation, gives Maxwell’s equations as solution and leads to the same wave equation. There are many other fields of physics and many mechanical systems to which the Lagrangian approach can be applied. The goal of the present text is to introduce the beginner to the least action principle and the Lagrangian approach in an elementary way. The chapter argues that the Lagrangian chosen for the fields involved must be an expression that leads to those fields and equations. In wave mechanics, particle properties are described by a so-called wave representing the classical matter field ψ used in Schrodinger’s approach of wave mechanics.
The principle of least action is known to apply to many domains of Physics. Yet its justifications and derivations usually come from archaic classical principles like the Fermat or Maupertuis principles, or mathematical analytical frameworks used for classical mechanics like the Lagrange, Hamilton and D’Alembert formalisms. As is well known, it is widely applicable to other domains, without necessarily a clear justification for it, especially when applied to Physics where Lagrangians or Hamiltonians may not be defined or do not exist. In addition, the Feynman Path Integral formulation of Quantum Mechanics gave it a second life, and another way to justify its validity not only in classical, but also in quantum Physics. However, the conventional logic to reach these conclusions is at time circular, and the principle is applied without proof that it is applicable: it just works and everybody does it. It is this applicability that led us in the past to treat the Action path integral as God’s equation.The paper justifies the principle of least action and Feynman’s path integrals through different physical reasoning instead of the traditional mathematical formalism, and on that basis we confirm its validity in most situations, even if Hamiltonians or Lagrangians may be shaky to define, or the system may be dissipative. We also explain when, and why, sometimes, it may not conventionally apply. The derivation uncovers different links between Entropy and Action, which also opens a different way to look at the origin of the probability distribution of even just a single lone particle; especially when complemented with the W-type multi-fold hypothesis, and makes some considerations on Feynman diagrams, instantons, renormalons and resurgence. Also discontinuities in the paths of the path integrals imply that spacetime is discrete, and non-commutative, and supersymmetry in flat spacetime is unphysical. Surfaceology, as optimized computation of the Feynman diagram is shown to trivially predict that all gauge theories double copy dual to gravity share the same forbidden scatterings. Relying on a recently encountered paper, we show how the results of the multi-fold theory, in particular the multi-fold space time matter induction and scattering, built explicitly on the path integral formalism, its underlying principles, and the principle of least action, allows recovering almost everything in Today’s physics. It leaves out essentially only some of the coupling constants and mixing angles/parameters. This goes a long way towards the ambition of a Theory Of Everything (TOE), even if not presented as usually expected. It maybe the best that can be achieved considering Gödel's incompleteness theorems. Our justification of the principle of least action, and path integrals, and our work on multi-fold theory encounters relationships between Entropy and Action. We review these encounters, including how our early prediction that entanglement might be irreversible match recent published results that argue against the existence of a (second) Law of entanglement analogous to the second Law of Thermodynamics. Also, from our approach, we obtain a formulation of the evolution of entropy, including for irreversible systems, which matches Onsager and Prigogine’s famous models and principles.