How to tell if Energy is Conserved from the Lagrangian?

In summary: It's quite subtle.In summary, when determining energy conservation through the Lagrangian, one must differentiate partially with respect to time. This is because the Lagrangian must be independent of time in order for energy to be conserved. Taking a total derivative would require considering the Lagrangian as a time-dependent quantity, which is a subtle distinction from its role as an abstract function of independent variables.
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penguin46
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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?
 
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  • #2
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).
 
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  • #3
penguin46 said:
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.
 
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FAQ: How to tell if Energy is Conserved from the Lagrangian?

How is energy conserved in a system described by the Lagrangian?

The Lagrangian is a mathematical function that describes the dynamics of a system in terms of its generalized coordinates and velocities. The conservation of energy in such a system can be determined by examining the Lagrangian and applying the principle of least action, which states that the path taken by a system between two points in time is the one that minimizes the action integral. If the Lagrangian does not explicitly depend on time, then the energy of the system is conserved.

Can energy be conserved even if the Lagrangian explicitly depends on time?

Yes, in some cases, energy can still be conserved even if the Lagrangian explicitly depends on time. This can occur if the system has a symmetry that leads to a conserved quantity, such as angular momentum or linear momentum. In these cases, the energy may change over time, but the total energy of the system remains constant.

How can we tell if a system has a symmetry that leads to energy conservation?

To determine if a system has a symmetry that leads to energy conservation, we can examine the Lagrangian and look for any terms that are invariant under certain transformations, such as translations, rotations, or time shifts. If the Lagrangian remains unchanged under these transformations, then the system has a corresponding conserved quantity, and energy may be conserved as a result.

Is energy always conserved in systems described by the Lagrangian?

No, energy is not always conserved in systems described by the Lagrangian. In some cases, the Lagrangian may explicitly depend on time, or the system may not have any symmetries that lead to conserved quantities. In these cases, energy may not be conserved, and the total energy of the system may change over time.

How can we experimentally verify if energy is conserved in a system described by the Lagrangian?

To experimentally verify if energy is conserved in a system described by the Lagrangian, we can measure the energy of the system at different points in time and compare it to the predicted energy from the Lagrangian. If the measured energy remains constant or follows the predictions from the Lagrangian, then we can conclude that energy is conserved in the system. However, if there are discrepancies, it may indicate that energy is not conserved, and further investigation is needed.

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