In mechanics we perform the Legendre transform to go from the Lagrangian $L(q, \dot{q})$ to the Hamiltonian $H(q, p)$. This seems to be describing the same physical system. $L$ and $H$ both describe an object(s) moving under certain constraints. In contrast, in stat mech we transform internal energy $U(S,V)$ to enthalpy $H(S, P)$. This seems to be describing different physical systems. $U(S,V)$ holds pressure constant. An example system would be a closed rigid container. On the other hand $H(S,P)$ holds volume constant. An example system would be a beaker in a lab. Is this understanding correct and if so, why does Legendre transform describe 2 views on the same physical system in mechanics but 2 different physical systems in stat mech?
Asked
Active
Viewed 60 times
1
Kevin
- 131
- 3
-
1Related: https://physics.stackexchange.com/q/4384/2451 and links therein. – Qmechanic Mar 15 '22 at 09:49
-
Agreed that post explains Legendre transformations well. But after reading all the answers, I still couldn't figure out why in mechanics going from Lagrangian to Hamiltonian seems to be different ways of describing the same physics while going from internal energy to enthalpy in stat mech seems to be talking about different physics... – Kevin Mar 15 '22 at 13:32
-
1Suggestion to the post (v1): Consider to make the title more nuanced, reflecting the actual question. – Qmechanic Mar 15 '22 at 13:47
-
The question is the last sentence... why isn't it in the title? – Cosmas Zachos Mar 15 '22 at 14:13
-
Not Strictly Related : A mathematically illogical argument in the derivation of Hamilton's equation in Goldstein. – Frobenius Mar 15 '22 at 14:18