On the dynamics of spring-pendulum system: an overview of configuration space and phase space

. The dynamics of the spring-pendulum system with two degrees of freedom were studied. The motion of this system is restricted to be in a vertical plane so that the chosen generalized coordinates are the increased length of the spring 𝑢 and the swing angle of pendulum 𝜃 . Hamiltonian of the system is obtained from the Legendre transformation of Lagrangian. Hamilton’s equation yields four differential equations that represent the dynamic of the system. The obtained results were visualized in configuration space and phase space trajectories. It is shown that generally the greater the initial swing angle, the more complex pattern will occur followed by the appearance of chaotic phenomena.


INTRODUCTION
The review of the physical pendulum system is still evolving even though it is a very familiar case in classical mechanics.One of them is a physical pendulum system driven by a magnetic field through a theoretical and numerical analysis of one-side oscillation [1].The existence of chaotic behavior and multiperiodicity for various values of the frequency of the current signal was shown from bifurcation diagrams obtained numerically and verified by experimental estimates.Previously, observations of magnetic interactions in the double pendulum system had been carried out numerically and experimentally [2].Few chaotic zones have been detected numerically and confirmed experimentally, where the bifurcation diagrams are also used to show the scenarios of transition from regular to chaotic motion and vice versa.The chaotic motions of a double pendulum demonstrate how complicated the motion of a simple dynamic system can be when the system and the motion become nonlinear [3][4][5][6].The next work is a double pendulum case that has been modeled with the fractional dynamics approach to find their equation of motion [7].Moreover, a double pendulum system with magnetic field interaction has been adopted to improve the efficiency of piezoelectric energy harvesters (PEH) to harvest energy from human motions [8].
Apart from the double pendulum, the case that is often discussed is the spring-pendulum system.Wahyuni et al. have derived the equations of motion for this case in their Lagrangian form [9].The equation of motion represented by the second-order differential equation is obtained from the two general coordinates used, i.e. the increase in the length of the spring and the angle of the pendulum.The same system but with a different arrangement was studied by Rini et al., namely by inverting the spring to be at the bottom of the pendulum [10].Both of these systems are solved by the Lagrangian method, and then the dynamics of the generalized coordinates to time are described.In addition, the usual way of describing system dynamics can be done through configuration space and phase space.Configuration space is the space defined by the generalized coordinates, while phase space is defined by the position and momentum of the particle as they change in time.The state of a particle can be represented by a point in phase space, and its movement consequently creates a path or trajectory within that space [11].
This study is a continuation of the work of Wahyuni et al. [9] with a Hamiltonian review.The work only derives the Lagrangian equation and then describes the generalized coordinate dynamics of  and  with respect to time.This study is very different from the studies mentioned above [1][2][3][4][5][6][7][8][9][10][11] because this is only a theoretical study.The important point of this study is the process of deriving the system's equations of motion and then visualizing it from different spaces, namely the configuration space and the phase space.This drawing will be carried out for several samples of initial swing angles.

METHODOLOGY
The Spring-Pendulum System The spring-pendulum system was depicted in Figure 1, consisted of the spring with length  !, increased length of the spring , and mass  ! in the end of spring.In addition, the pendulum with mass  " connected to the mass  ! with the massless rod  " .The angle  represent the swing angle of the pendulum with respect to the vertical line.We assume that all movement occurs in two dimensionals, vertical plane.It is assumed that the spring can only move up and down in oscillations, not deviate with the pendulum, which represents applications in daily tools that springs only move in one dimension.The mass  ! is constrained by the spring constant, and the pendulum is constrained by the constant length of the rod.Starting from two degrees of freedom for each of the two objects, the two constraints reduce the number of degrees of freedom to two, one for each object.

Research Methodology
Based on Figure 1 we can determine that the generalized coordinates for this system are  and .The Hamiltonian () of this system is given by the Legendre transformation of the Lagrangian () and Hamilton's equation of this system as two pairs of first-order differential, i.e.  coordinates representation as and  coordinates representation as The study of such differential equations is crucial for understanding the system's behavior.These two pairs yield four equations of motion were solved using numerical methods.The numerical method chosen is the 4 th order Runge-Kutta which has a good approximation [12][13][14][15].Let %& %' = (, ), we can find the approximation of ( + Δ) as where We use the visualization results only to explain the dynamics of motion.So, the solution of the differential equations was not generated, but are presented in the form of graphs that represent the motion of objects, both in the configuration space and the phase space.To facilitate visualization, several parameters are used as needed.

RESULTS AND DISCUSSION
The Hamilton's Equation The Lagrangian of the system is given by the equation where ̇= / and  ̇= / [9].The Lagrangian was chosen where the zero potential energy at the point  !hanged.
Eliminate  ̇ from equation ( 7) and ( 8) by adding them together, so we get In the end, we obtain the rate of increased length of the spring with time as To get the rate of swing angle with time, substitute equation (10) into equation ( 5), we get Figure 1.Spring-pendulum system ̇= ( The Hamiltonian () of mechanical system in function of the rate of coordinates with time, conjugate momenta, and Lagrangian is obtain using Legendre Transformation, i.e with  1 is the generalized momenta and where  1 is the generalized coordinates [16].We obtain Substitute equations ( 4), (10), and ( 13) together into equation ( 15), we get Hamiltonian of the springpendulum system as equation ( 16) with the set of equations ( 1) and ( 2 (20) where  =  !+  " ,  = 2    ,  =  !+  "  " , and  =  "  .

and ℎ is time step.
There are several representations to describe the equation of motion, such as time evolution, configuration space, phase-space, Poincare section, bifurcation, or the detailed methods to expose the motion.For example, Amer et al. uses evolutionary time, phase space, and Poincare representations to describe system dynamics [17].In this work, we vary the initial swing angle  5 to visualize the configuration space and the threedimensional phase space.The initial coordinates of the motion are (0,0,0) and the parameters used for all the visualization are shown in Table 1.It can be seen that for small values  5 phase trajectories are corresponding to motion on the linear oscillation mode.These shapes gradually begin to distort with increasing  5 and turn into other phase trajectories that are more complex in character.This is same interpretation with the work of Smirnov et.al.[18].

Table 1. Parameters used for all the visualization
The three-dimensional phase-space was shown in Figure 3.The system was plotted for initial values  5 = 0.2 m,  $ =  # = 0 and corresponding initial angles,  5 = 0.6 rad and  5 = 1.2 rad.Those values are chosen for the simulation to distinguish significant states of the system.The simulations have allowed the motion to evolve for a time  = 15 seconds with  = 0.00001.The vertical axis represents time evolution or the flow of the time, while the horizontal axes represent the two phase-space coordinates separately, (,  # ) and (,  $ ).This description follows Semkiv et al. [19] who made an integral curve in the phase space.In this work, coordinates  represent the spring motion, while coordinates  represent the pendulum motion.System leads to periodic motion for small angle  5 = 0.6 rad, while for large angle  5 = 1.2 rad indicates system approach to a chaotic motion.Figure 3(a) shows that the dynamics of the system with small angle condition is predictable for both  and  coordinates.However, a different situation can be seen in Figure 3(b), where  coordinates can still be predicted, but  coordinates already show a chaotic phenomenon.
It can be seen that even though the  coordinates appear irregular but there is no broken path, different from the  coordinates.Therefore, it becomes interesting to investigate the  coordinates further.
In order to investigate the characteristics of  coordinates, we describe a two-dimensional phase space in the (,  # ) plane compared to time evolution of .Each left side of Figure 4 shows a two-dimensional projection of a phase-space for the initial values  5 = 0.2 m,  $ $ = 0,  # $ = 0.2 kg ms -1 with different initial angles.The closed curve of Figure 4(a) and 4(b) is composed of eleven overlapping projected curves, in accordance with eleven patterns of oscillation in right side of the figure.Even though there are slight variations, it can still be said that system move in smooth oscillations.The precise alignment and closure of these curves demonstrate the stability and precise repetition of the oscillation.The presence of repeated oscillation patterns proves that small swing angles lead to steady state oscillations.
The greater the initial angle, more complex trajectories of the systems.It can be seen in Figures 4(c) and 4(d) that the horizontal scale has exceeded the range − ≤  ≤ .Because it has exceeded one period 2, the visualization starts again from - so that several broken patterns appear.The chaotic phenomenon is strengthened by the increasingly irregular oscillation patterns that appear both in the visualization of the phase space and time evolution of .This allows us to visually illustrate one of the defining characteristics of a complex system: unpredictable behavior.The state space is highdimensional making it difficult to analyze and visualize the behavior of the system for varying input conditions [20].It turns out that theoretical studies like this can also be developed and applied in everyday life, for example controlling the kinematics of a spring-pendulum system using an energy harvesting device.This model has become essential in recent times as it uses control sensors in industrial applications, buildings, infrastructure, automobiles, and transportation [21].

CONCLUSION
The Hamiltonian of the spring-pendulum system has been derived with the general coordinates was the increase in the length of the spring  and the swing angle of the pendulum .Visualization results in configuration space and phase space trajectory show that in general the larger the initial swing angle, the more complex patterns will occur with complex characteristics, followed by the appearance of chaotic phenomena.

Figure 2
Figure2demonstrates the configuration space on the planes (, ) where several phase trajectories with constant initial value  5 = 0.2 m,  $ $ =  # $ = 0 and corresponding to different values  5 .The plane area was  !−  to  !+  and − 5 to + 5 .It can be seen that for small values  5 phase trajectories are corresponding to motion on the linear oscillation mode.These shapes gradually begin to distort with increasing  5 and turn into other phase trajectories that are more complex in character.This is same interpretation with the work of Smirnov et.al.[18].
) yields four first-order differential equation, that are.