Solve Lagrangian Oscillator: Damped, Driven System

[tburke2]

Homework Statement

I'm given a driven, dampened harmonic oscillator (can it be thought of as a spring-mass system with linear friction?) Is it possible to solve the equation of motion using Lagrangian mechanics? I could solve it with the usual differential equation x''+βx'+ωₒ²x=fₒcos(ωt) but as we have just started learning Lagrangian in class I'd like to do it that way.

Homework Equations

x''+βx'+ωₒ²x=fₒcos(ωt)

The Attempt at a Solution

I know how to do it with an undampened, undriven spring-mass system but am unsure how to include the energies for the driving force and damping force.

For undampended and undriven:
L= 1/2mx'² - 1/2kx²

 

[Orodruin]

There are ways of including half derivatives in time which will let you do this but it is significantly more advanced than your typical basic course in Lagrangian mechanics. The usual treatment cannot deal with dissipative systems.

 

[PhysyCola]

Hi tburke2,

I believe that you can solve the system by considering the displacement x of the particle as the displacement from some support that oscillates in the same way as your forcing.

This would lead to a lagrangian of:
L =1/2mx'^2 - 1/2kx^2
were x = x_o - z, where x_o = F_o cos(wt) and z is the actual displacement of your forcing. Of course, this is only valid for some kind of mechanical forcing. I would recommend reading Morin Classical Mechanics as it covers Lagrangian Mechanics is a good level of detail.

 

[Orodruin]

This would lead to a lagrangian of:
L =1/2mx'^2 - 1/2kx^2
were x = x_o - z, where x_o = F_o cos(wt) and z is the actual displacement of your forcing. Of course, this is only valid for some kind of mechanical forcing. I would recommend reading Morin Classical Mechanics as it covers Lagrangian Mechanics is a good level of detail.

This does not involve any damping, which is a dissipative effect and what the OP was asking for.