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Then we have this Lagrangian:

$$L= \frac{1}{2}(-f(x)\dot{t}^2 + g(x)\dot{x}^2 + 2l(x)\dot{x}\dot{t}).$$

The Euler-Lagrange equation for $$t$$ is:

since $$t$$ is not there in the Lagrangian then $$\partial L/ \partial t=0$$

This implies that $$\frac{d}{d\tau}\frac{\partial L }{\partial \dot{t}}= 0$$

so $$\frac{\partial L }{\partial \dot{t}}$$ is a conserved quantity we call energy and I got it equal to $$-f\dot{t} + l\dot{x}$$ where my professor only got it $$ E= -f\dot{t}$$

Am I mistaken somewhere?