Extending Newton's law from nonlocal-in-time kinetic energy

• Published in: Physics letters. A Vol. 373; no. 14; pp. 1201 - 1211
• Main Author: Suykens, J.A.K.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 23-03-2009
• Subjects: Discontinuous velocity jumps; Higher order Euler–Lagrange equation; Newton's second law of motion; Nonlocal-in-time kinetic energy; Pais–Uhlenbeck oscillators; Quantized nonlocality time extent; 11.10.Ef; 11.10.Lm; 11.27.+d; 03.65.Sq; Newton's second law of motion; Higher order Euler–Lagrange equation; Pais–Uhlenbeck oscillators; Discontinuous velocity jumps; Nonlocal-in-time kinetic energy; Quantized nonlocality time extent
• ISSN: 0375-9601; 1873-2429
• DOI: 10.1016/j.physleta.2009.01.065

We study a new equation of motion derived from a context of classical Newtonian mechanics by replacing the kinetic energy with a form of nonlocal-in-time kinetic energy. It leads to a hypothetical extension of Newton's second law of motion. In a first stage the obtainable solution form is studied by considering an unknown value for the nonlocality time extent. This is done in relation to higher-order Euler–Lagrange equations and a Hamiltonian framework. In a second stage the free particle case and harmonic oscillator case are studied and compared with quantum mechanical results. For a free particle it is shown that the solution form is a superposition of the classical straight line motion and a Fourier series. We discuss the link with quanta interpretations made in Pais–Uhlenbeck oscillators. The discrete nature emerges from the continuous time setting through application of the least action principle. The harmonic oscillator case leads to energy levels that approximately correspond to the quantum harmonic oscillator levels. The solution to the extended Newton equation also admits a quantization of the nonlocality time extent, which is determined by the classical oscillator frequency. The extended equation suggests a new possible way for understanding the relationship between classical and quantum mechanics.