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# Lesson/CM02-02 Conservation of Energy

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 $$\S1$$ Objectives

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

 $$\S2$$ Let's Recall and Discuss
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 $$\label{EQ1A} \frac{dE}{dt} = m \dot{x} \ddot{x} + \frac{dV(x)}{dx} \frac{dx}{dt}.$$
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

 $$\S4$$ EndNotes
1. The canonical momentum is in general not the same as momentum of a particle.
2. The Hamiltonian is defined as $$H = \sum_k p_k \dot{q}_k -L(q, \dot{q}, t)$$ 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}.
 References
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.

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