First Law of Thermodynamics
Symmetry of physical laws under variation of the path in time
Principle of stationary action → Euler–Lagrange equation
\begin{align}
t &\rightarrow t + \varepsilon \\[6pt]
L' &= L + \varepsilon \frac{dL}{dt} \\[6pt]
\frac{dL}{dt} &= \frac{\partial L}{\partial x} \frac{dx}{dt} + \frac{\partial L}{\partial v} \frac{dv}{dt} \\[6pt]
\frac{\partial L}{\partial x} &= \frac{d}{dt}\left( \frac{\partial L}{\partial v} \right) \\[6pt]
\frac{dL}{dt} &= \frac{d}{dt}\left( \frac{\partial L}{\partial v} \right)\frac{dx}{dt}
+ \frac{\partial L}{\partial v}\frac{dv}{dt} \\[6pt]
\frac{dL}{dt} &= \frac{d}{dt}\left( \frac{\partial L}{\partial v} \right)v
+ \frac{\partial L}{\partial v}\frac{dv}{dt}
\end{align}Conservation of Energy
For a static empty universe
\begin{align}
L &= T - V = \frac{1}{2}mv^2 - V \\[6pt]
\frac{d}{dt}\left(mv^2 - L\right) &= 0 \\[6pt]
\frac{d}{dt}\left(mv^2 - \frac{1}{2}mv^2 + V\right) &= 0 \\[6pt]
\frac{d}{dt}\left(\frac{1}{2}mv^2 + V\right) &= 0 \\[6pt]
\frac{dE}{dt} &= 0
\end{align}However, our universe is not globally symmetric with respect to time since spacetime itself evolves. Global energy conservation does not hold, even though local conservation () still applies.
