Abstract
Forl a 1-D conservative system with a position depending mass within a dissipative medium, its effect on the body is to exert a force depending on the squared of its velocity, a constant of motion, Lagrangian, generalized linear momentum, and Hamiltonian are obtained. We apply these new results to the harmonic oscillator and pendulum under the characteristics mentioned about, obtaining their constant of motion, Lagrangian and Hamiltonian for the case when the body is increasing its mass.
Highlights
Variable mass problems without dissipation have a long history and are known as Gylden-Meshcherskii problems [1] [2] [3] [4] [5]
Forl a 1-D conservative system with a position depending mass within a dissipative medium, its effect on the body is to exert a force depending on the squared of its velocity, a constant of motion, Lagrangian, generalized linear momentum, and Hamiltonian are obtained
Newton’s equation with position mass depending is not invariant under Galileo’s transformation [6] [7], and Sommerfeld gave a modification of this equation to overcome this problem [8]. This modification has a fundamental problem when external force is zero, and that is why one considers Newton’s equation of motion as a good equation of motion for these types of problems [9] [10]. This approach was used for 1-D conservative systems with position depending mass [11], binary stars with mass exchanged [12] [13], binary galaxies with mass exchanged [14], and fluid dynamics [15]
Summary
Variable mass problems without dissipation have a long history and are known as Gylden-Meshcherskii problems [1] [2] [3] [4] [5] As it is known, Newton’s equation with position mass depending is not invariant under Galileo’s transformation [6] [7], and Sommerfeld gave a modification of this equation to overcome this problem [8]. Newton’s equation with position mass depending is not invariant under Galileo’s transformation [6] [7], and Sommerfeld gave a modification of this equation to overcome this problem [8] This modification has a fundamental problem when external force is zero, and that is why one considers Newton’s equation of motion as a good equation of motion for these types of problems [9] [10]. The results will be applied to the study on the dynamics of the harmonic oscillator and pendulum systems with this dissipation and with increasing of mass behavior
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