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Add an interface of integrals of motion #35

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cmp0xff opened this issue Mar 23, 2024 · 2 comments · May be fixed by #54
Open

Add an interface of integrals of motion #35

cmp0xff opened this issue Mar 23, 2024 · 2 comments · May be fixed by #54
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documentation Improvements or additions to documentation enhancement New feature or request model Change of models

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cmp0xff commented Mar 23, 2024

In the Lagrangian action of an undamped simple harmonic oscillator (#34), one can easily find the first integral
$$\frac{1}{2} m \dot x^2 + \frac{1}{2} m \omega^2 x^2 \equiv E \eqqcolon \frac{1}{2} m \omega^2 x_0^2\,,$$
where $E$ is the integral or motion (constant), having the interpretation of energy, and $x_0$ can be interpreted as the maximal displacement.

Integrating this first-order differential equation directly gives rise to another constant of motion $t_0$,
$$t-t_0 = \frac{1}{\omega} \arccos \frac{x}{x_0}\,.$$
In other words, the motion of the system is directly given by the two constant $(x_0, t_0)$, in this framework of Lagrangian mechanics.

I propose we add the initial condition $(x_0, t_0)$, in addition to $(x_0, v_0)$, the latter of which is closer to the second-order differential equation of motion, which is less appealing in analytical mechanics.
@cmp0xff cmp0xff added documentation Improvements or additions to documentation enhancement New feature or request model Change of models labels Mar 23, 2024
@cmp0xff cmp0xff mentioned this issue Mar 23, 2024
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cmp0xff added a commit that referenced this issue Jun 15, 2024
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chore(poetry): #35 add pre-commit
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refactor(oscillator): #35 move to subfolder
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refactor(oscillator): #35 factor out initial conditions
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feat(oscillator): #35 parse initial condition for sho
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refactor(oscillator): #35 move tester
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@cmp0xff cmp0xff changed the title Add an alternative initial condition for undamped simple harmonic oscillator Add an alternative initial condition for simple harmonic oscillator Jun 15, 2024
@cmp0xff cmp0xff linked a pull request Jun 15, 2024 that will close this issue
feat(model): #35 add an interface of integrals of motion #54
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@cmp0xff cmp0xff self-assigned this Jun 17, 2024
@cmp0xff cmp0xff closed this as not planned Won't fix, can't repro, duplicate, stale Jul 23, 2024
@cmp0xff cmp0xff reopened this Aug 18, 2024
@cmp0xff cmp0xff changed the title Add an alternative initial condition for simple harmonic oscillator Add an interface of integrals of motion Aug 18, 2024
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cmp0xff commented Sep 2, 2024 •
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We have a few options to specify the system:

  1. initial conditions
  2. integrals of motion
  3. mathematical / geometric specifications

If we focus on closed systems with exact solutions, I suggest we focus on 2 and 3, making 1 optional.

Initial conditions

These are the King in numeric simulations. For exact solutions, however, they are often not the best specifications. For examples, in Kepler problem, we are likely to be forced to convert initial conditions to the energy and angular momentum, before we can proceed.

Integrals of motion

By definition, all exactly solvable conserved systems have integrals of motion. These quantities have special importance in analytical mechanics.

Typical examples include

  1. Energy and initial time (reaching maximum amplitude) for oscillators
  2. Energy and angular momentum for Kepler problem

Mathematical / geometric specifications

Good examples include

  1. Amplitudes and initial phase for oscillators
  2. Parameter and eccentricity for the Kepler problem

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cmp0xff commented Sep 17, 2024

Currently we provide users intermediate variables which are used for analytical or numerical purposes. Maybe we should remove them.

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feat(model): #35 add an interface of integrals of motion kausalflow/hamilflow
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