Interpretation of the Lagrangian

[Logarythmic]

I have showed that if the integrand $$f$$ in the variational problem
$$\delta (\int f dx ) = 0$$
does not depend explicitly on the independent variable x, i.e. satisfies $$f = f(y, \dot{y})$$, then the Euler equation can be integrated to

$$\dot{y} \frac{\partial f}{\partial \dot{y}} - f = const.$$

How can I give an interpretation of this constant for the case that $$f = L$$ is the Lagrangian and $$x = t$$ is the time?

 

[dextercioby]

It's the energy of the system.

Daniel.

 

[Logarythmic]

So I can interpret this as that the Energy of the system is constant?

 

[dextercioby]

Yes, of course. It's a symmetry of the action.

Daniel.