Sergio A. Hojman

A new conservation law constructed without using either Lagrangians or Hamiltonians

A new conservation theorem is derived. The conserved quantity is constructed in terms of a symmetry transformation vector of the equations of motion only, without using either Lagrangian or Hamiltonian structures (which may even fail to exist for the equations at hand). One example and implications of the theorem on the structure of point symmetry transformations are presented.

On the inverse problem of the calculus of variations

We consider the inverse problem of the calculus of variations for any system by writing its differential equations of motion in first‐order form. We provide a way of constructing infinitely many Lagrangians for such a system in terms of its constants of motion using a covariant geometrical approach. We present examples of first‐order Lagrangians for systems for which no second‐order Lagrangians exist. The Hamiltonian theory for first‐order (degenerate) Lagrangians is constructed using Dirac’s method for singular Lagrangians.

Equivalent Lagrangians: Multidimensional case

We generalize a theorem known for one‐dimensional nonsingular equivalent Lagrangians (L and ?) to the multidimensional case. In particular, we prove that the matrix Λ, which relates the left‐hand sides of the Euler–Lagrange equations obtained from L and ?, is such that the trace of all its integer powers are constants of the motion. We construct several multidimensional examples in which the elements of Λ are functions of position, velocity, and time, and prove that in some cases equivalence prevails even if detΛ = 0.

No Lagrangian - No quantization

This work starts with classical equations of motion and sets very general quantization conditions (commutation relations). It is proved that these conditions imply that the equations of motion are equivalent to the Euler–Lagrange equations of a Lagrangian L. The result is a generalization of work by Feynman, recently reported by Dyson [Am. J. Phys. 58, 209–211 (1990)]. The Lagrangian L need not be unique. Examples are given, including classical equations that do not come from a Lagrangian and therefore cannot be quantized consistently.

Symmetries of Lagrangians and of their equations of motion

A new kind of Lagrangian symmetry is defined in such a way that the resulting set of Lagrangian symmetries coincides with the set of symmetries of its equations of motion. Several constants of motion may be associated to each of the new symmetry transformations. One example is presented.

First‐order equivalent Lagrangians and conservation laws

We present a theorem for (first‐order) Lagrangian theories which associates several conserved quantities to one (s‐equivalence) symmetry transformation.

Affine collineations in Riemannian spaces

Sufficient conditions on the Christoffel symbols and metric tensors to insure the existence of affine collineations in Riemann spaces are obtained. A proof of the group property of affine collineations is given. Several examples of physically relevant Riemannian spaces admitting affine collineations are presented and the constants of motion are constructed.

An algorithm to relate general solutions of different bidimensional problems

A sufficient condition to relate the general solutions of two different bidimensional problems, both in classical and quantum mechanics, is found. The point transformation, as well as the explicit construction of the related potentials, is presented. The procedure is then used in geometrical optics, the vibrating membrane, and other physical systems in two dimensions. Several examples and applications are worked out.

Do electromagnetic waves always propagate along null geodesics

We find exact solutions to Maxwell equations written in terms of four-vector potentials in non--rotating, as well as in Godel and Kerr spacetimes. We show that Maxwell equations can be reduced to two uncoupled second-order differential equations for combinations of the components of the four-vector potential. Exact electromagnetic waves solutions are written on given gravitational field backgrounds where they evolve. We find that in non--rotating spherical symmetric spacetimes, electromagnetic waves travel along null geodesics. However, electromagnetic waves on Godel and Kerr spacetimes do not exhibit that behavior.

Equivalent Lagrangians in classical field theory

Two Lagrangians L and L′ are equivalent if the equations of motion derived from them have the same set of solutions. In that case, a matrix Λ may be defined which has the property that the trace of any analytic function of Λ is a constant of the motion. We extend this trace theorem to the case of classical field theory and discuss some of the implications for quantum theory and for procedures for finding equivalent Lagrangians.