Harmonic Osillator and Least Action

[tgupta2000@gmail.com]

Could someone elaborate on the steps to obtain the equation of motion
of a harmonic oscillator using the principle of least action, and then
extending it to a damped oscillator?
So, if L = T - V,
L = 0.5 m [ (xdot)^2 - (omega)^2 * (x)^2 ]

S = Integrate [ L dt ] ....

Is it essential to use trial functions?

How does one extend this to a damped oscillator?

Thanks in advance

[mpalenik (He-Jutsu)]

Are you posting homework questions here? It's just that this sounds a
little bit like a homework question. . .

Anyway, it's a little bit tricky to a lagrangian for a damped
oscillator, because you can't define a potential for the damping
force. You have to add it in as a non-conservative force after coming
up with the equations of motion for an undamped oscillator.

For an undamped oscillator:
U = 1/2*k*x^2
T = 1/2m*v^2 = 1/2m*xdot^2

L = T - U = 1/2m*xdot^2 - 1/2*k*x^2

So, use the same procedure that you do for everything else--use
Euler's formula to find the path that minimizes L: dL/dx - d/dt(dL/
dxdot) = 0
and you should get:
mxdouble_dot + k*x = 0

For a damped oscillator, you also have a force proportional to v (or
xdot), so just add in a term -B*xdot to the right hand side

mxdouble_dot + k*x = -B*xdot

The fact is, for an oscillator, it's actually easier to just write out
the equations of motion based on the forces you know are present,
rather than deriving them from the lagrangian.

Mark

On Feb 2, 3:50 pm, tgupta2...@gmail.com wrote:
> Could someone elaborate on the steps to obtain the equation of motion
> of a harmonic oscillator using the principle of least action, and then
> extending it to a damped oscillator?
> So, if L = T - V,
> L = 0.5 m [ (xdot)^2 - (omega)^2 * (x)^2 ]
>
> S = Integrate [ L dt ] ....
>
> Is it essential to use trial functions?
>
> How does one extend this to a damped oscillator?
>
> Thanks in advance

[tgupta2000@gmail.com]

Thanks for the reply, especially Mark.
For a change...it is not a homework question. I am 40 years old with a
PhD degree working in the industry trying to get back into Physics and
toying with problems that can later be extended into research
problems.
It is true, it sounds like homework. There is no sure way to verify.

On Feb 3, 3:22*pm, "mpalenik (He-Jutsu)" <markpale...@gmail.com>
wrote:
> Are you posting homework questions here? *It's just that this sounds a
> little bit like a homework question. . .
>
> Anyway, it's a little bit tricky to a lagrangian for a damped
> oscillator, because you can't define a potential for the damping
> force. *You have to add it in as a non-conservative force after coming
> up with the equations of motion for an undamped oscillator.
>
> For an undamped oscillator:
> U = 1/2*k*x^2
> T = 1/2m*v^2 = 1/2m*xdot^2
>
> L = T - U = 1/2m*xdot^2 - 1/2*k*x^2
>
> So, use the same procedure that you do for everything else--use
> Euler's formula to find the path that minimizes L: dL/dx - d/dt(dL/
> dxdot) = 0
> and you should get:
> mxdouble_dot + k*x = 0
>
> For a damped oscillator, you also have a force proportional to v (or
> xdot), so just add in a term -B*xdot to the right hand side
>
> mxdouble_dot + k*x = -B*xdot
>
> The fact is, for an oscillator, it's actually easier to just write out
> the equations of motion based on the forces you know are present,
> rather than deriving them from the lagrangian.
>
> Mark
>
> On Feb 2, 3:50 pm, tgupta2...@gmail.com wrote:
>
>
>
> > Could someone elaborate on the steps to obtain the equation of motion
> > of a harmonic oscillator using the principle of least action, and then
> > extending it to a damped oscillator?
> > So, if L = T - V,
> > L = 0.5 m [ (xdot)^2 - (omega)^2 * (x)^2 ]
>
> > S = Integrate [ L dt ] ....
>
> > Is it essential to use trial functions?
>
> > How does one extend this to a damped oscillator?
>
> > Thanks in advance- Hide quoted text -
>
> - Show quoted text -