- #1
Peeter
- 305
- 3
In the solution of a pendulum attached to a wheel problem, I was initially suprised to see that a term of the form:
[tex]
\frac{df}{dt}
[/tex]
"can be removed from the Lagrangian since it will have no effect on the equations of motion".
ie: [itex]L' = L \pm df/dt[/itex] gives identical results.
f in this case was cos(\omega t + \theta) where theta was the generalized coordinate.
I confirmed this for the cosine function in this example by taking derivatives, and then confirmed that this is in fact a pretty general condition, given two conditions:
1) equality of mixed partials:
[tex]
\frac{\partial^2 f}{\partial t \partial q^i} = \frac{\partial^2 f}{\partial q^i \partial t}
[/tex]
2) no dependence on velocity coordinates for time partial derivative:
[tex]
\frac{\partial^2 f}{\partial \dot{q}^i \partial t} = 0
[/tex]
Does this ability to add/remove time derivatives of functions from the Lagrangian that aren't velocity dependent have a name?
[tex]
\frac{df}{dt}
[/tex]
"can be removed from the Lagrangian since it will have no effect on the equations of motion".
ie: [itex]L' = L \pm df/dt[/itex] gives identical results.
f in this case was cos(\omega t + \theta) where theta was the generalized coordinate.
I confirmed this for the cosine function in this example by taking derivatives, and then confirmed that this is in fact a pretty general condition, given two conditions:
1) equality of mixed partials:
[tex]
\frac{\partial^2 f}{\partial t \partial q^i} = \frac{\partial^2 f}{\partial q^i \partial t}
[/tex]
2) no dependence on velocity coordinates for time partial derivative:
[tex]
\frac{\partial^2 f}{\partial \dot{q}^i \partial t} = 0
[/tex]
Does this ability to add/remove time derivatives of functions from the Lagrangian that aren't velocity dependent have a name?