Lesson/CM02-02 Conservation of Energy

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

1. Give an example of application of energy conservation.
2. What do you understand by conservation law? When we say something \(X(q,p)\) is a constant of motion what exactly it means mathematically?
3. Give examples of a system for which energy is not conserved.
4. 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?

\(\S3\)Energy conservation

1. The canonical momentum is in general not the same as momentum of a particle.
2. 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 \(q_k\).
3. For conservative mechanical systems the Hamiltonian coincides the energy of the system.
4. 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}.

1. 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 \(\S6\). 

2. 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.