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    Hi, trying to do some revision and came across this question, any ideas how to tackle it? im findin the topic quite hard! theres no rush on replying cos it doesnt have to be handed in or anything but any thoughts would be appriciated!

    A particle of mass m moves in 2-dimensions under the action of a force due to a potential V (x, y). Write down an expression for the Lagrangian in terms of the Cartesian coordinates x and y and hence show that the equations of motion are the same as newton’s equations.
    The same particle is observed by someone on a roundabout rotating with angular velocity w. This observer uses ˜x and ˜y coordinates that are realted to the Cartesian coordiantes by

    x = ˜x cos(wt) − ˜y sin(wt)
    y = ˜x sin(wt) + ˜y cos(wt)

    Differentiate these equations to ˙x and ˙y in terms of ˙˜x and ˙˜y. Hence find the kinetic energy and Lagrangian as measured by the rotating observer. Calculate the Lagrange equations of motion for ˜x and ˜y and show that it looks to such an observer that there
    are additional forces acting on the particle. Note: These correpond to the cetripetal accelerartion and Coriolis acceleration for rotating frames in Newtonian mechanics.
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    Well the Lagrangian for a particle moving in a 2d potential is:

     \mathcal{L} = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2) - V(x,y)
    and the Euler-Lagrange equations are:

     \frac{d}{dt} (\frac{\partial \mathcal{L}}{\partial \dot{q}_i}) - \frac{\partial \mathcal{L}}{\partial q_i} = 0

    as for the second part, sub in your expressions into the lagrangian and do the same procedure as above, you should get a "radial" expression and an "angular" expression
 
 
 
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