### Lecture 4 Time Evolution of Physical Quantities Csir Net

### Time Evolution of Physical Quantities in Classical Mechanics

Understanding the time evolution of physical quantities is fundamental in classical mechanics. This concept involves analyzing how different properties of a system change over time under the influence of forces. The Hamiltonian formulation provides a robust framework for this analysis.

This reflects the classical analogue of Ehrenberg’s theorem in quantum mechanics, which relates the time evolution of the expectation value of an operator to its commutator with the Hamiltonian.

#### Applications in Various Systems

**Rotating Systems:**- In rotating systems, angular velocity and torque can be analyzed using the time evolution of angular momentum.

**Electromagnetic Fields:**- Charged particles in electromagnetic fields follow time evolution governed by the Lorentz force, which can be framed using Hamiltonian mechanics.

**Coupled Oscillators:**- For systems of coupled oscillators, the time evolution of energy exchange between oscillators can be studied using their Hamiltonian.

#### Conclusion

The time evolution of physical quantities in classical mechanics is elegantly described by the Hamiltonian formulation. Using Poisson brackets and Hamilton’s equations, we can systematically analyze how systems evolve over time. This framework not only simplifies the process of solving dynamic problems but also lays the groundwork for understanding more complex systems in both classical and quantum mechanics.

Students must download the study material in pdf form and save the pdf according to topic, chapter, and sub-topic respectively. It is advisable for the students to download one chapter at a time, complete that and then go for the next chapter.

Those students doing self-study should consult the appropriate textbook as well as they can take help from our video lectures.

Students should solve the examples on their own after understanding the concepts given in the theory. They may use the solutions only as a last option.