Transformations In Classical Mechanics Research Articles

Singular Solutions of Clairaut Equations

A relationship is established between singular solutions of the Clairaut equation in the theory of ordinary differential equations and singular solutions of the Clairaut equation in the theory of partial differential equations. It is shown that exact relationships exist between Legendre transformations in classical mechanics and Clairaut equations.

Some reference formulas for the generating functions of canonical transformations

We study some properties of the canonical transformations in classical mechanics and quantum field theory and give a number of practical formulas concerning their generating functions. First, we give a diagrammatic formula for the perturbative expansion of the composition law around the identity map. Then, we propose a standard way to express the generating function of a canonical transformation by means of a certain "componential" map, which obeys the Baker-Campbell-Hausdorff formula. We derive the diagrammatic interpretation of the componential map, work out its relation with the solution of the Hamilton-Jacobi equation and derive its time-ordered version. Finally, we generalize the results to the Batalin-Vilkovisky formalism, where the conjugate variables may have both bosonic and fermionic statistics, and describe applications to quantum field theory.

Critical Landscape Topology for Optimization on the Symplectic Group

Optimization problems over compact Lie groups have been extensively studied due to their broad applications in linear programming and optimal control. This paper analyzes least square problems over a noncompact Lie group, the symplectic group $\Sp(2N,\R)$, which can be used to assess the optimality of control over dynamical transformations in classical mechanics and quantum optics. The critical topology for minimizing the Frobenius distance from a target symplectic transformation is solved. It is shown that the critical points include a unique local minimum and a number of saddle points. The topology is more complicated than those of previously studied problems on compact Lie groups such as the orthogonal and unitary groups because the incompatibility of the Frobenius norm with the pseudo-Riemannian structure on the symplectic group brings significant nonlinearity to the problem. Nonetheless, the lack of traps guarantees the global convergence of local optimization algorithms.

Semiclassical approach to Landau levels and Landau level degeneracy using canonical transformations in classical mechanics

We use the method of canonical transformations in classical mechanics to investigate the relationship between the classical and quantum dynamics of a charged particle in a uniform magnetic field. A simple application of Bohr-Sommerfeld quantization rules leads to a full description of the energy spectrum.

Non-point transformations in classical mechanics

Non-point transformations in classical mechanics

On the gauge transformations in classical mechanics

On the gauge transformations in classical mechanics

Generalized legendre transformations

A general theory of Toda chains is presented. The chains are treated as Lagrange dynamic systems with one continuous and one discrete time. The theory is based on the fact that the chains are invariant with respect to transformations that are analogous to the Legendre transformations in classical mechanics.

Relations of canonical and unitary transformations for a general time-dependent quadratic Hamiltonian system

We consider general time-dependent quadratic Hamiltonian systems which are connected by canonical transformations and give the same classical equations of motion. In those systems, we demonstrate that canonical transformations in classical mechanics correspond to unitary transformations in quantum mechanics. The wave functions and the propagators are evaluated using the invariant operator method. However, the values of some functions of the canonical variables q and p are not equal to the values of the same functions of the other canonical variables Q and P, but the values of the functions of q and the kinetic momentum ${\mathrm{p}}_{\mathrm{k}}$ are equal to those of the other Q and ${\mathrm{P}}_{\mathrm{k}}$ in classical mechanics. We prove that these also hold in the quantum treatment. The uncertainty relations of momentum and position are evaluated for the two Hamiltonians.

Systematic Treatment of General Time-Dependent Harmonic Oscillator in Classical and Quantum Mechanics

A systematic way of treating a general time-dependent harmonic oscillator in classical and quantum mechanics is given. By a general canonical transformation in classical mechanics, the time-dependent Hamiltonian can be transformed to a time-independent one (the Lewis-Riesenfeld invariant), explicitly separating a total time-derivative term. In quantum mechanics, one can obtain the phase of the wave function straightforwardly from this total-derivative term, together with the corresponding time-independent Hamiltonian. We solve analytically and algebraically the general time-dependent harmonic oscillator driven by a time-dependent inverse cubic force as an example.

Systematic Treatment of General Time-Dependent Harmonic Oscillator in Classical and Quantum Mechanics

A systematic way of treating a general time-dependent harmonic oscillator in classical and quantum mechanics is given. By a general canonical transformation in classical mechanics, the time-dependent Hamiltonian can be transformed to a time-independent one (the Lewis-Riesenfeld invariant), explicitly separating a total time-derivative term. In quantum mechanics, one can obtain the phase of the wave function straightforwardly from this total-derivative term, together with the corresponding time-independent Hamiltonian. We solve analytically and algebraically the general time-dependent harmonic oscillator driven by a time-dependent inverse cubic force as an example.

Correspondence between the classical and quantum canonical transformation groups from an operator formulation of the wigner function

An explicit expression of the “Wigner operator” is derived, such that the Wigner function of a quantum state is equal to the expectation value of this operator with respect to the same state. This Wigner operator leads to a representation-independent procedure for establishing the correspondence between the inhomogeneous symplectic group applicable to linear canonical transformations in classical mechanics and the Weyl-metaplectic group governing the symmetry of unitary transformations in quantum mechanics.

Quantum energy mapping — an interpretation based on classical orbits

Nonintegrable space and time transformations in classical mechanics are extended to the quasi-classical regime by applying Ehrenfest's adiabatic hypothesis. The resulting mapping between quasi-classical orbits induces an approximative mapping between discrete energy spectra of different quantum systems which is correct up to the first order of Planck's constant. A further generalization of this quasi-classical energy mapping to the full quantum mechanical case is often possible by performing a successive renormalization of parameters. In this way we construct a mapping of classical orbits which constitutes the basis for the quantum energy mapping.

Space and time transformations in classical mechanics and in quantum mechanics

The application of space and time transformations leads to general mappings between different systems of both classical mechanics and quantum mechanics. It turns out that the space and the time transformations can be independently chosen in classical mechanics whereas they have to be connected in a special way in quantum mechanics.

Gauge transformations in classical mechanics as canonical transformations

Gauge transformations for a classical charged particle are a special case of classical canonical transformations. The principle of form invariance under a gauge transformation follows as a special case of the form invariance of Hamilton’s equations under a canonical transformation.

On the use of the interaction picture in classical mechanics

The interaction picture is formulated as a canonical transformation in classical mechanics. It is shown that often great computational saving may be achieved by integrating Hamilton's equation in this representation.

Canonical scattering transformation in classical mechanics

The quasiclassical phase shift is identified as the generator of the classical canonical $S$ transformation. This illuminates the connection between resonances and looping trajectories and the classical meaning of the time delay and Levinson's theorem. Finally the Coulomb phase shift is derived using for the time delay a free motion with a modified time dependence.

Feynman path integral and gauge invariance

Gauge invariance in both quantum and classical mechanics is investigated using the Feynman path integral formulation of quantum mechanics. The Feynman path integral expresses the wave function in terms of the Lagrangian. A gauge transformation in classical mechanics is the addition of a total derivative of a gauge function to the Lagrangian. A gauge transformation of the second and first kind in quantum mechanics is obtained by adding and subtracting, respectively, a total derivative to the Lagrangian in the Feynman path integral.

Canonical Realizations of the Galilei Group

The general theory of the realizations of finite Lie groups by means of canonical transformations in classical mechanics, which has been developed in a preceding paper and already applied to the rotation group, is now applied to the Galilei group. Some complements to the general theory are introduced; in particular, a new kind of possible canonical realizations connected with the singularity surfaces of the functions Q(y),P(y),J(y) are discussed (singular realizations). In agreement with the situation encountered in quantum mechanics, the constants dρσ appearing in the fundamental Poisson bracket relations among the infinitesimal generators ({yρ,yσ}=cρστyτ+dρσ) cannot all be reduced to zero. There remains a single independent constant m, which, in the physically significant cases (m > 0), represents the mass of the system. No physical interpretation seems to be attachable to the realizations corresponding to m = 0. For m ≠ 0, two different kinds of irreducible realizations exist: one of a singular type which describes the free mass-point, and another of a regular type which describes a classical particle spin. A number of physical significant examples corresponding to nonirreducible realizations are thereafter discussed and the related typical forms are constructed: specifically, the cases dealt with are the rigid rod (linear rotator), the rigid body, and a system of two interacting mass points. It is shown that the problem of the construction of the variables of the typical form is equivalent to the determination of an appropriate solution of the time-independent Hamilton-Jacobi equation.

Canonical Transformation Governing Temporal Development in Classical Mechanics

A simple recursion formula is introduced for the derivation of the generating function of the canonical transformation in classical mechanics in the form of a power series in t, where the transformation relates a variable at the time t with its initial value given at t = 0. A discussion is given of the relationship of this approach to the Hamilton-Jacobi differential equation, with emphasis on the comparison of the formal aspects of classical and quantum mechanics.

Canonical Realizations of the Rotation Group

A general theory of the realizations of Lie groups by means of canonical transformations in classical mechanics, proposed in a preceding paper, is applied to the rotation group. A number of significant physical examples corresponding to nonirreducible realizations is treated in detail: specifically, the mass point, the rotator, and the rigid body with a fixed point. The explicit form of the possible irreducible realizations is worked out. Such realizations do not directly correspond to any realistic physical model but play a relevant role for the introduction of the spin in classical mechanics.