- #1

koil_

- 5

- 1

- Homework Statement
- Find the conserved quantity of the Lagrangian $$L = (\frac {\dot{x}}{x})^2$$ associated with the invariance given by the transformation $$ x \to sx$$

- Relevant Equations
- $$ \frac {\partial L}{\partial \dot{Q}} \frac {\partial Q}{\partial s}$$

So Noether's Theorem states that for any invarience that there is an associated conserved quantity being:

$$ \frac {\partial L}{\partial \dot{Q}} \frac {\partial Q}{\partial s}$$

Let $$ X \to sx $$

$$\frac {\partial Q}{\partial s} = \frac {\partial X}{\partial s} = \frac {\partial (sx)}{\partial s}= x $$

This is the part that I'm now unsure about:

$$\frac {\partial L}{\partial \dot{Q}}=\frac {\partial (\frac {\dot{X}}{X})^2} {\partial \dot{X}} = \frac {2\dot{X}} {X^2} $$

This would make the conserved quantity therefore:

$$ \frac {2\dot{X}} {X^2} * x $$

However I'm not sure where to go from here as there X and x is present and 'reversing' the transformation produces an 's' in the quantity which wouldn't make sense as this quantity is supposed to be conserved.

I'm sure it is quite evident that this is the first piece we have been given on this so I may have the method completely wrong as, in some examples I have seen, the derivatives are not fully expanded. e.g:

$$\frac {\partial L}{\partial \dot{X}} * x$$ is some conserved quantity.

Many thanks in advance.

$$ \frac {\partial L}{\partial \dot{Q}} \frac {\partial Q}{\partial s}$$

Let $$ X \to sx $$

$$\frac {\partial Q}{\partial s} = \frac {\partial X}{\partial s} = \frac {\partial (sx)}{\partial s}= x $$

This is the part that I'm now unsure about:

$$\frac {\partial L}{\partial \dot{Q}}=\frac {\partial (\frac {\dot{X}}{X})^2} {\partial \dot{X}} = \frac {2\dot{X}} {X^2} $$

This would make the conserved quantity therefore:

$$ \frac {2\dot{X}} {X^2} * x $$

However I'm not sure where to go from here as there X and x is present and 'reversing' the transformation produces an 's' in the quantity which wouldn't make sense as this quantity is supposed to be conserved.

I'm sure it is quite evident that this is the first piece we have been given on this so I may have the method completely wrong as, in some examples I have seen, the derivatives are not fully expanded. e.g:

$$\frac {\partial L}{\partial \dot{X}} * x$$ is some conserved quantity.

Many thanks in advance.