Simple pendulum phase space

In summary, the conversation discusses the calculation of the phase space volume of a simple pendulum using the total energy equation and the small angle approximation. The potential energy is expressed in terms of elliptic integrals and the use of the small angle approximation allows for the use of the (0, 2(pi)) limits of integration instead of -x_max to x_max.
  • #1
shehry1
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Homework Statement


Pathria 2.6 (2nd Edition): Phase space volume of a simple pendulum.

The total energy can be expressed in the form of the time derivative of the angle + the Sin^2 of that angle.

From this I want to calculate the phase space volume. Mathematica gives the solution in the form of elliptic integrals. So I used the Simple pendulum angle limit ie: Six(x) = x. But because of this I cannot introduce a pi into the equation as by doing so, the limits of the phase space volume should change to x_max and -x_max. Is that correct?

Or can I still use the 0 to 2(Pi) limits of integration. I did a quick calculation and the answer comes correct when I use both the small angle approximation and integrate over the (0, 2(pi)) limit. But it seems wrong somehow.

Homework Equations


Total energy of a simple pendulum



The Attempt at a Solution



Thanks
 
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  • #2
for the help! Yes, I am trying to calculate the phase space volume of a simple pendulum. The total energy of a simple pendulum is given by: E = (1/2)*m*l^2*x_dot^2 + m*g*l*(1-cos(x))Where x is the angle, x_dot is the angular velocity, m is the mass of the pendulum and l is the length of the pendulum. I want to calculate the phase space volume given by:V = Integral from x_min to x_max (dx/sqrt(2(E-V(x)))) where V(x) is the potential energy of the system. Using the small angle approximation, the potential energy reduces to: V(x) = m*g*l*x^2/2. Substituting this into the expression for the phase space volume gives:V = Integral from 0 to 2(pi) (dx/sqrt(2(E - m*g*l*x^2/2))) Which gives the solution in terms of elliptic integrals. Since I am using the small angle approximation, it is not necessary to use the -x_max to x_max limits of integration. Is that correct?
 

Related to Simple pendulum phase space

1. What is a simple pendulum?

A simple pendulum is a weight or bob suspended from a fixed point so that it is able to swing freely back and forth under the influence of gravity.

2. What is the phase space of a simple pendulum?

The phase space of a simple pendulum is a mathematical representation of the possible states of the pendulum, including the position and velocity of the bob, at any given time.

3. How is the phase space of a simple pendulum visualized?

The phase space of a simple pendulum is often visualized as a plot with the position of the bob on the y-axis and the velocity on the x-axis. This plot is known as a phase portrait.

4. What factors affect the phase space of a simple pendulum?

The phase space of a simple pendulum is affected by several factors, including the length of the pendulum, the mass of the bob, the angle at which it is released, and any external forces acting on it.

5. What is the significance of studying the phase space of a simple pendulum?

Studying the phase space of a simple pendulum can help scientists understand the behavior of other systems with oscillatory motion, as well as predict and analyze the behavior of pendulum-like objects in various real-world applications.

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