\(\S1\) Objectives |

To show that energy conservation law holds for systems if the Lagrangian is independent of time.

\(\S2\) Let's Recall and Discuss |

- Give an example of application of energy conservation.
- What do you understand by conservation law? When we say something \(X(q,p)\) is a constant of motion what exactly it means mathematically?
- Give examples of a system for which energy is not conserved.
- Consider an example of a particle moving in one dimension in a potential \(V(x)\). Taking total time derivative of total energy, \(E= \frac{1}{2}m\dot{x}^2 + V(x)\) we would get \begin{equation}\label{EQ1A} \frac{dE}{dt} = m \dot{x} \ddot{x} + \frac{dV(x)}{dx} \frac{dx}{dt}. \end{equation}

The right hand side does not seem to become zero !

What is missing?

How do we see that the right hand side of \eqref{EQ1A} is zero?

\(\S4\) EndNotes |

- The canonical momentum is in general not the same as momentum of a particle.
- The Hamiltonian is defined as \begin{equation} H = \sum_k p_k \dot{q}_k -L(q, \dot{q}, t) \end{equation} where \(p_k\) is {\it canonical momentum} conjugate to \(q_k\).
- For conservative mechanical systems the Hamiltonian coincides the energy of the system.
- The Hamiltonian is just the total energy of a system. It is conserved when the Lagrangian does not have explicit time dependence.

Every book on classical mechanics will have a discussion of energy conservation, see for Landau Lifshitz \cite{LandMe} and Calkin \cite{Calkin}.

References |

- Landau, L. D. and Lifshitz E. M., "
*Mechanics*, Volume 1 of Course of Theoretical Physics",

Butterworth-Heinenann Linacre House, Jordan Hill, Oxford 3rd Ed.(1976).

We have closely followed the treatment by Landau Lifshitz in \(\S6\). -
Calkin, M.G. "

*Lagrangian and Hamiltonian Mechanics"*, World Scientific Publishing Co.Pte. Ltd. (1996);

Here the energy conservation is discussed as an application of Noether's theorem to time translations.

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