# Simple pendulum using Lagrange's equation

Defines a LagrangianPendulum class that is used to generate basic pendulum plots from solving Lagrange's equations.

* Last revised 17-Mar-2019 by Dick Furnstahl (furnstahl.1@osu.edu).

## Euler-Lagrange equation

For a simple pendulum, the Lagrangian with generalized coordinate $\phi$ is

$\begin{align}
  \mathcal{L} = \frac12 m L^2 \dot\phi^2 - mgL(1 - \cos\phi)
\end{align}$

The Euler-Lagrange equation is

$\begin{align}
 \frac{d}{dt}\frac{\partial\mathcal{L}}{\partial \dot\phi} = \frac{\partial\mathcal L}{\partial\phi}
 \quad\Longrightarrow\quad
 m L^2 \ddot \phi = -mgL\sin\phi
  \ \mbox{or}\ \ddot\phi = - \omega_0^2\sin\phi = 0
  \;.
\end{align}$

## Hamilton's equations

The generalized momentum corresponding to $\phi$ is

$\begin{align}
   \frac{\partial\mathcal{L}}{\partial \dot\phi} = m L^2 \dot\phi \equiv p_\phi
   \;.
\end{align}$

We can invert this equation to find $\dot\phi = p_\phi / m L^2$.
Constructing the Hamiltonian by Legendre transformation we find 

$\begin{align}
  \mathcal{H} &= \dot\phi p_\phi  - \mathcal{L} \\
              &= \frac{p_\phi^2}{m L^2} - \frac12 m L^2 \dot\phi^2 + mgL(1 - \cos\phi) \\
              &= \frac{p_\phi^2}{2 m L^2} + mgL(1 - \cos\phi)
              \;.
\end{align}$

Thus $\mathcal{H}$ is simply $T + V$.  Hamilton's equations are

$\begin{align}
  \dot\phi &= \frac{\partial\mathcal{H}}{\partial p_\phi} = \frac{p_\phi}{m L^2} \\
  \dot p_\phi &= -\frac{\partial\mathcal{H}}{\partial \phi} = -mgL \sin\phi 
  \;.
\end{align}$


```python
%matplotlib inline
```


```python
import numpy as np
from scipy.integrate import odeint, solve_ivp

import matplotlib.pyplot as plt
```


```python
# The dpi (dots-per-inch) setting will affect the resolution and how large
#  the plots appear on screen and printed.  So you may want/need to adjust 
#  the figsize when creating the figure.
plt.rcParams['figure.dpi'] = 100.    # this is the default for notebook

# Change the common font size (smaller when higher dpi)
font_size = 12
plt.rcParams.update({'font.size': font_size})
```

## Pendulum class and utility functions


```python
class LagrangianPendulum():
    """
    Pendulum class implements the parameters and Lagrange's equations for 
     a simple pendulum (no driving or damping).
     
    Parameters
    ----------
    L : float
        length of the simple pendulum
    g : float
        gravitational acceleration at the earth's surface
    omega_0 : float
        natural frequency of the pendulum (\sqrt{g/l} where l is the 
        pendulum length) 
    mass : float
        mass of pendulum

    Methods
    -------
    dy_dt(t, y)
        Returns the right side of the differential equation in vector y, 
        given time t and the corresponding value of y.
    """
    def __init__(self, L=1., mass=1., g=1.
                ):
        self.L = L
        self.g = g
        self.omega_0 = np.sqrt(g/L)
        self.mass = mass
    
    def dy_dt(self, t, y):
        """
        This function returns the right-hand side of the diffeq: 
        [dphi/dt d^2phi/dt^2]
        
        Parameters
        ----------
        t : float
            time 
        y : float
            A 2-component vector with y[0] = phi(t) and y[1] = dphi/dt
            
        Returns
        -------
        
        """
        return [y[1], -self.omega_0**2 * np.sin(y[0]) ]
    
    def solve_ode(self, t_pts, phi_0, phi_dot_0, 
                  abserr=1.0e-9, relerr=1.0e-9):
        """
        Solve the ODE given initial conditions.
        Specify smaller abserr and relerr to get more precision.
        """
        y = [phi_0, phi_dot_0] 
        solution = solve_ivp(self.dy_dt, (t_pts[0], t_pts[-1]), 
                             y, t_eval=t_pts, 
                             atol=abserr, rtol=relerr)
        phi, phi_dot = solution.y

        return phi, phi_dot
```


```python
def plot_y_vs_x(x, y, axis_labels=None, label=None, title=None, 
                color=None, linestyle=None, semilogy=False, loglog=False,
                ax=None):
    """
    Generic plotting function: return a figure axis with a plot of y vs. x,
    with line color and style, title, axis labels, and line label
    """
    if ax is None:        # if the axis object doesn't exist, make one
        ax = plt.gca()

    if (semilogy):
        line, = ax.semilogy(x, y, label=label, 
                            color=color, linestyle=linestyle)
    elif (loglog):
        line, = ax.loglog(x, y, label=label, 
                          color=color, linestyle=linestyle)
    else:
        line, = ax.plot(x, y, label=label, 
                    color=color, linestyle=linestyle)

    if label is not None:    # if a label if passed, show the legend
        ax.legend()
    if title is not None:    # set a title if one if passed
        ax.set_title(title)
    if axis_labels is not None:  # set x-axis and y-axis labels if passed  
        ax.set_xlabel(axis_labels[0])
        ax.set_ylabel(axis_labels[1])

    return ax, line
```


```python
def start_stop_indices(t_pts, plot_start, plot_stop):
    start_index = (np.fabs(t_pts-plot_start)).argmin()  # index in t_pts array 
    stop_index = (np.fabs(t_pts-plot_stop)).argmin()  # index in t_pts array 
    return start_index, stop_index
```


```python

```

## Make simple pendulum plots 


```python
# Labels for individual plot axes
phi_vs_time_labels = (r'$t$', r'$\phi(t)$')
phi_dot_vs_time_labels = (r'$t$', r'$d\phi/dt(t)$')
state_space_labels = (r'$\phi$', r'$d\phi/dt$')

# Common plotting time (generate the full time then use slices)
t_start = 0.
t_end = 50.
delta_t = 0.001

t_pts = np.arange(t_start, t_end+delta_t, delta_t)  

L = 1.
g = 1.
mass = 1.

# Instantiate a pendulum 
p1 = LagrangianPendulum(L=L, g=g, mass=mass)

```


```python
# both plots: same initial conditions
phi_0 = (3./4.)*np.pi
phi_dot_0 = 0.
phi, phi_dot = p1.solve_ode(t_pts, phi_0, phi_dot_0)


# start the plot!
fig = plt.figure(figsize=(15,5))
overall_title = 'Simple pendulum from Lagrangian:  ' + \
                rf' $\omega_0 = {p1.omega_0:.2f},$' + \
                rf'  $\phi_0 = {phi_0:.2f},$' + \
                rf' $\dot\phi_0 = {phi_dot_0:.2f}$' + \
                '\n'     # \n means a new line (adds some space here)
fig.suptitle(overall_title, va='baseline')
    
# first plot: phi plot 
ax_a = fig.add_subplot(1,3,1)                  

start, stop = start_stop_indices(t_pts, t_start, t_end)    
plot_y_vs_x(t_pts[start : stop], phi[start : stop], 
            axis_labels=phi_vs_time_labels, 
            color='blue',
            label=None, 
            title=r'$\phi(t)$', 
            ax=ax_a)    
                              
# second plot: phi_dot plot 
ax_b = fig.add_subplot(1,3,2)                  

start, stop = start_stop_indices(t_pts, t_start, t_end)    
plot_y_vs_x(t_pts[start : stop], phi_dot[start : stop], 
            axis_labels=phi_dot_vs_time_labels, 
            color='blue',
            label=None, 
            title=r'$\dot\phi(t)$', 
            ax=ax_b)    

# third plot: state space plot from t=30 to t=50   
ax_c = fig.add_subplot(1,3,3)                  

start, stop = start_stop_indices(t_pts, t_start, t_end)    
plot_y_vs_x(phi[start : stop], phi_dot[start : stop], 
            axis_labels=state_space_labels, 
            color='blue',
            label=None, 
            title='State space', 
            ax=ax_c)    

fig.tight_layout()
fig.savefig('simple_pendulum_Lagrange.png', bbox_inches='tight')  
```

Now trying the power spectrum, plotting only positive frequencies and cutting off the lower peaks:


```python
start, stop = start_stop_indices(t_pts, t_start, t_end)    
signal = phi[start:stop]
power_spectrum = np.abs(np.fft.fft(signal))**2
freqs = np.fft.fftfreq(signal.size, delta_t)
idx = np.argsort(freqs)

fig_ps = plt.figure(figsize=(5,5))
ax_ps = fig_ps.add_subplot(1,1,1)
ax_ps.semilogy(freqs[idx], power_spectrum[idx], color='blue')
ax_ps.set_xlim(0, 1.)
ax_ps.set_ylim(1.e5, 1.e11)
ax_ps.set_xlabel('frequency')
ax_ps.set_title('Power Spectrum')

fig_ps.tight_layout()
```


```python

```

44ecc1b739718ad228e8cec3b57448ed149514f3
ipynb
Jupyter Notebook
2020_week_10/Lagrangian_pendulum.ipynb
CLima86/Physics_5300_CDL
d9e8ee0861d408a85b4be3adfc97e98afb4a1149
MIT
2020_week_10/Lagrangian_pendulum.ipynb
CLima86/Physics_5300_CDL
d9e8ee0861d408a85b4be3adfc97e98afb4a1149
MIT
2020_week_10/Lagrangian_pendulum.ipynb
CLima86/Physics_5300_CDL
d9e8ee0861d408a85b4be3adfc97e98afb4a1149
MIT
Qwen/Qwen-72B
1. YES
2. YES