Time invariance and energy conservation
In this blog post, I explain how energy conservation in classical mechanics results from time invariance in the Lagrangian description. By using the formalism of generalized coordinates, I introduce the Hamiltonian \mathcal{H}:
\mathcal{H} = \sum_j p_j \dot{q}_j - \mathcal{L}
where p_j = \frac{\partial \mathcal{L}}{\partial \dot{q}_j}. If the Lagrangian \mathcal{L} does not explicitly depend on time, I show that the total time derivative of \mathcal{H} is:
\frac{\mathrm{d}\mathcal{H}}{\mathrm{dt}} = -\frac{\partial \mathcal{L}}{\partial t}
which means \mathcal{H} is conserved whenever \mathcal{L} is time independent. For systems where the generalized coordinates are defined via time-independent constraints, the kinetic energy T is a homogeneous quadratic function of velocities. In this case, the Hamiltonian simplifies to:
\mathcal{H} = T + V
where V is the potential energy, and T is the kinetic energy. This is the familiar total mechanical energy of the system. Energy conservation follows directly from the invariance of the Lagrangian under time translations.
For more insights into this topic, you can find the details here.