- #1

atomqwerty

- 94

- 0

## Homework Statement

Let be the lagrangian given by

[itex]L(x,y,\dot{x},\dot{y})=\frac{m}{2}(\dot{x^2} +\dot{y^2})-V(x^{2}+y^{2})[/itex]

and

[itex]L(x,y,\dot{x},\dot{y})=\frac{m}{2}(\dot{x^2} + \dot{y^2})-V(x^{2}+y^{2}) - \frac{k}{2}x^{2}[/itex]

and the transformation

[itex]x'=\cos\alpha x - \sin\alpha y[/itex]

[itex]y'=\sin\alpha x + \cos\alpha y[/itex]

Show the invariant observable with respect to the symmetry of this transformation (Noether's Theorem) for both Lagrangians.

## Homework Equations

Noether Theorem says that for each variable x such as [itex]\partial L/\partial x = 0[/itex] then x is said to be invariant and [itex]\partial L/\partial \dot{x} = 0[/itex] is constant of motion for that system.

## The Attempt at a Solution

First, the transformation from (x,y) to (x',y') doesn't change the Lagrangian, and since L does depend on [itex]x, y, \dot{x}, \dot{y}[/itex] on both Lagrangians, then the only non explicit variable is time t, and in that case the conserved quantity would be the Energy E = T + V. I must be wrong since this is a problem of an exam of Theoretical Mechanics in University...

Thanks a lot!

Last edited by a moderator: