Harmonic Oscillators¶

In this tutorial, we demo how to generate data of harmonic oscillators.

In [1]:

import pandas as pd
import plotly.express as px

from hamilflow.models.harmonic_oscillator import (
    DampedHarmonicOscillator,
    SimpleHarmonicOscillator,
)

import pandas as pd import plotly.express as px from hamilflow.models.harmonic_oscillator import ( DampedHarmonicOscillator, SimpleHarmonicOscillator, )

In [2]:

n_periods = 3
n_samples_per_period = 200

n_periods = 3 n_samples_per_period = 200

Simple Harmonic Oscillator¶

For an simple harmonic oscillator, the action of a simple harmonic oscillator is

$$S_L[x] = \int_{t_0}^{t_1} \mathbb{d}t \left\{\frac{1}{2} m \dot x^2 - \frac{1}{2} m \omega^2 x^2 \right\}\,,$$

where the least action principle leads to the following equation of motion,

$$ \ddot x + \omega^2 x = 0\,. $$

A simple harmonic oscillator is a periodic motion.

In [3]:

sho_omega = 0.5

sho = SimpleHarmonicOscillator(system={"omega": sho_omega})

sho_omega = 0.5 sho = SimpleHarmonicOscillator(system={"omega": sho_omega})

In [4]:

df_sho = sho(n_periods=n_periods, n_samples_per_period=n_samples_per_period)
df_sho.head()

df_sho = sho(n_periods=n_periods, n_samples_per_period=n_samples_per_period) df_sho.head()

Out[4]:

	t	x
0	0.000000	1.000000
1	0.062832	0.999507
2	0.125664	0.998027
3	0.188496	0.995562
4	0.251327	0.992115

In [5]:

px.line(
    df_sho,
    x="t",
    y="x",
    title=rf"Simple Harmonic Oscillator (omega = {sho_omega})",
    labels={
        "x": r"Displacement $x(t)$",
        "t": r"$t$",
    },
)

px.line( df_sho, x="t", y="x", title=rf"Simple Harmonic Oscillator (omega = {sho_omega})", labels={ "x": r"Displacement $x(t)$", "t": r"$t$", }, )

Damped Harmonic Oscillator¶

A damped harmonic oscillator is a simple harmonic oscillator with damping force that is proportional to its velocity,

$$ \ddot x + \omega^2 x = - 2\xi\omega \dot x\,. $$

In this section, we demonstrate three scenarios of a damped harmonic oscillator.

In [6]:

dho_systems = {
    "Underdamped": {"omega": 0.5, "zeta": 0.2},
    "Critical Damped": {"omega": 0.5, "zeta": 1},
    "Overdamped": {
        "omega": 0.5,
        "zeta": 1.2,
    },
}

dfs_dho = []

for s_name, s in dho_systems.items():
    dfs_dho.append(
        DampedHarmonicOscillator(system=s)(
            n_periods=n_periods,
            n_samples_per_period=n_samples_per_period,
        ).assign(
            system=rf"{s_name} (omega = {s.get('omega')}, zeta = {s.get('zeta')})",
        ),
    )

fig = px.line(
    pd.concat(dfs_dho),
    x="t",
    y="x",
    color="system",
    title=r"Damped Harmonic Oscillator",
    labels={
        "x": r"Displacement $x(t)$",
        "t": r"$t$",
    },
)
fig.update_layout(legend={"yanchor": "top", "y": -0.2, "xanchor": "left", "x": 0})

dho_systems = { "Underdamped": {"omega": 0.5, "zeta": 0.2}, "Critical Damped": {"omega": 0.5, "zeta": 1}, "Overdamped": { "omega": 0.5, "zeta": 1.2, }, } dfs_dho = [] for s_name, s in dho_systems.items(): dfs_dho.append( DampedHarmonicOscillator(system=s)( n_periods=n_periods, n_samples_per_period=n_samples_per_period, ).assign( system=rf"{s_name} (omega = {s.get('omega')}, zeta = {s.get('zeta')})", ), ) fig = px.line( pd.concat(dfs_dho), x="t", y="x", color="system", title=r"Damped Harmonic Oscillator", labels={ "x": r"Displacement $x(t)$", "t": r"$t$", }, ) fig.update_layout(legend={"yanchor": "top", "y": -0.2, "xanchor": "left", "x": 0})

In [ ]: