Free energy is a generalization of energy when the system exchanges heat with the environment. Energy, in its turn, can be extracted from lagrangian under the symmetry of time $(\frac{\partial L}{\partial t}=0)$. However, I can't find any sources that links the free energy directly from lagrangian. Is there any way to do that? If it is, what symmetry is it based on?
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3$\begingroup$ If you like this Phys.SE question, you may also enjoy reading this Phys.SE post. $\endgroup$– Qmechanic ♦Commented Jun 16, 2014 at 16:12
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