A Generalized Analytical Mechanics in which Quantum Phenomena Appear

Abstract

We propose a mechanics of a massive particle in a potential field effective for both classical and quantum system as a modified classical analytical mechanics (modified CM). We transform, under coordinate transformation, the covariant tensor of order two in the Hamilton-Jacobi (H-J) eq. of CM, not with the classical action, but with extended action of diffeomorphism group. Then, the H-J eq., a first-order partial differential eq., is modified to a third-order one. The Euler-Lagrange (E-L) eq. of CM, a second-order ordinary differential eq., related to the H-J eq. through the action integral is accordingly modified to a fourth-order one. Thus obtained mechanics accommodates quantum phenomena due to the higher-order eqs., and always gives trajectory unlike quantum mechanics (QM) due to the E-L eq. Discrete energy levels of a particle in a confining potential are the same as those of QM because quantization criterion is equivalent. Particle distribution in an ensemble disagrees with that of QM even if initial distribution is set identical because dynamics is different; it however agrees with observed data to date within experimental uncertainty. The mechanics thus is a testable alternative to QM.Comment: 51 pages, 8 figures. A reason for 1-D energy quantization which works on real line (R1) is given; the one in previous versions works only on real projective line (RP1). Some other improvements were incorporate