1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Time-dependent mass and the Lagrangian

  1. Oct 7, 2017 #1
    I was talking to a friend about Lagrangian mechanics and this question came out. Suppose a particle under a potential ##U(r)## and whose mass is ##m=m(t)##. So the question is: the Lagrangian of the particle can be expressed by

    ##L = \frac{1}{2} m(t) \dot{\vec{r}} ^2 -U(r)##

    or I need to re-write the kinetic energy? Maybe this way

    ## \displaystyle T = \int \vec{F} \cdot d\vec{r} = \int \frac{d\vec{p}}{dt} \cdot \vec{v} \: dt = \int \vec{v} \cdot d\vec{p} = \int \vec{v} \cdot (\vec{v} \: dm + m \: d \vec{v}) = \int v^2 \: dm + \int m \: \vec{v} \cdot d \vec{v} ##
     
  2. jcsd
  3. Oct 7, 2017 #2

    hilbert2

    User Avatar
    Science Advisor
    Gold Member

    It's as in your first equation. If ##r## is a single 1d coordinate, the equation of motion will be

    ##0 = \frac{\partial L}{\partial r} - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{r}}\right)##
    ## = -\frac{\partial U}{\partial r} - \frac{d}{dt}( m(t)\dot{r})##
    ## = -\frac{\partial U}{\partial r} - m(t)\ddot{r} - \dot{m}(t)\dot{r}##
     
  4. Nov 5, 2017 #3

    Jano L.

    User Avatar
    Gold Member

    The first method is appropriate when the process that changes the mass does not result in additional force on the remaining body. For example, when the body ejects mass in two opposite directions with the same rate. It is not valid for a rocket in flight, because mass is thrown away in a preferred direction and as a result, there is a strong force acting on the rocket. The second method is not a valid derivation, since for variable mass systems, external force does not in general equal d(mv)/dt.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted