How to tell if Energy is Conserved from the Lagrangian?

[penguin46]

Homework Statement

When attempting to determine if energy is conserved by looking at the Lagrangian for a system, one must differentiate wrt time to determine if the Lagrangian is constant in time. When doing this, does one take a partial time derivative or a total time derivative?

Relevant Equations

Lagrangian, Lagrangian equations of motion, multivariable chain rule

I am fairly certain that the answer here is to differentiate partially with respect to time rather than fully. In Landau and Lifshitz' proof of energy conservation one of the hypotheses is that the partial of L wrt time is zero. Am I on the right track?

 

[ergospherical]

That’s right, yes. It’s an example of the Beltrami identity, whereby independence of the Lagrangian with respect to the independent variable implies a first integral (in this case the energy).

 

[PeroK]

Homework Statement:: When attempting to determine if energy is conserved by looking at the Lagrangian for a system, one must differentiate wrt time to determine if the Lagrangian is constant in time. When doing this, does one take a partial time derivative or a total time derivative?
Relevant Equations:: Lagrangian, Lagrangian equations of motion, multivariable chain rule

I am fairly certain that the answer here is to differentiate partially with respect to time rather than fully. In Landau and Lifshitz' proof of energy conservation one of the hypotheses is that the partial of L wrt time is zero. Am I on the right track?

To analyse the Lagrangian itself, we consider it an abstract function of independent variables. If (as an abstract function) it is independent of $t$, then energy is conserved. That means taking the "partial" derivative.

To perform a "total" derivative, we would have to take the other (no-longer-independent) variables as functions of $t$. To do that we would have to consider the Lagrangian as a time-dependent quantity that we calculate along a curve, say.

There's an important distinction between the roles of the Lagrangian in these two aspects of Lagrangian mechanics.