One Dimensional Conservative System with Quadratic Dissipation and Position Depending Mass

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.


Introduction
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]. However, 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]. On the other hand, 1-D systems with constant mass and quadratic dissipation have also been studied [16]. Therefore, in this paper both situations are considered at the same time, position mass de-pending and quadratic dissipation on 1-D conservative systems, and for these systems one will find a Constant of Motion, Lagrangian, Generalized Linear Momentum, and Hamiltonian. 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.

Analytical Approach
where v represents the velocity, d d v x t x = =  , of the body, and α is a constant. One will consider that represents the initial mass of the system at the point 0 x = . Equation (1) can be written as an autonomous dynami- where x m has been defined as d d A constant of motion for this system is a function that is, it must satisfy the following first order partial differential equation which can be solved by the characteristics method [17], where the equations for the characteristics are The last term just tell us that the function K must be an arbitrary function of the characteristic C obtained from the others two terms, where G is arbitrary. From the others two terms, one can write the following equation This expression is of the form where T α is some type of effective kinetic energy of the system, and V α is just the effective potential Then, one can say that represents the effective energy of the system.

Special Cases
Let us note the following: First, one has the following limit which is the expression obtained in reference [11]. Second, assuming the mass as constant, which is the expression obtained in references [16] [18] (for the non relativistic

Lagrangian and Hamiltonian
Now, since ( ) ( ) In this way and considering (13), one gets The generalized linear momentum is With this expression and the Legendre's transformation, , the Hamiltonian of the system is given by If we apply the above observations (11) on the expressions (10), (15), (16), and (17), one gets the corresponding correct expression for these cases.
Let us notice from (1) that the dissipation for 0 v < can be obtained by making the change α α → − on the expressions already found. Therefore, the constant of motion, Lagrangian, generalized linear momentum, and Hamiltonian when 0 v < are given by However, notice from (13)

Mass Linear Dependence on Position
In this case, one has the following dependence of the mass with respect the position of the body where β is a constant. Then, it follows that So, form the expressions (10), (15), (16), and (17), one obtains where the effective potential V α is given by

Harmonic Oscillator
For the harmonic oscillator, one has that ( ) F x kx = − , and using the following the effective potential is where the constant term ( )  . Note that since 0 β ≥ , the system is acquiring mass as the position is increasing. Because of this, and due that one has dissipation in the system, the body will perform a damping spiral behavior on the phase spaces ( ) x p , which is not shown here.
To determine this spiral damping behavior and assuming always and increasing of mass, one would have to divide the phase space ( ) , x v in four regions: 1)

Pendulum
The position on the pendulum is determined by its displacement s respect its , being g the constant acceleration due to gravity. Using the following integration [22] ( ) ( )( where γ is the uncompleted gamma function [22] (page 940). If we select the G. V. López, E. S. Madrigal due to increasing of mass during its motion and the damping factor. The upper lines represent the rotational spiral damping behavior of the body due to the same reason (this spiral damping behavior is not shown on these plots. To get this behavior one would need to proceed similarly as it was explained for the harmonic oscillator part).
The effective potential V α has an oscillatory increasing behavior as a function of the displacement s. Therefore, it does not matter which value of the effective energy K or H takes, due to the increasing of mass and damping factor, the body will perform an oscillatory damping behavior, that is, the origin of the phase space is an attractor of the dynamics of the body (as it happened with the first example). On Figure 2(b) one sees an apparent increasing of the generalized linear momentum as the body is rotating. However, eventually will reach the return point of the potential and the generalized linear momentum will be zero (as the yellow line indicates).

Conclusions and Comments
In general, we have constructed constant of motion, Lagrangian, generalized linear momentum, and Hamiltonian for a 1-D conservative system with position depending mass and embedded in a medium where the body feels a dissipative force which depends quadratically on its velocity. In particular, we made the analysis for the case when the body increases its mass linearly on its displacement, where the dynamics in the phase spaces ( ) , x v and ( ) , x p is plotted on one quadrant of these spaces, which could be very important if one wants to use quantum mechanics for theses system, and we have shown the damping effect on the motion of the body for the harmonic oscillator and pendulum systems due to dissipative force and the increasing of its mass.
We want to comment something for the case of mass lost, we have seen from our model (19) with 0 β < that the motion is limited to a displacement given : Uncompleted gamma function.