The Most Beautiful Result in Classical Mechanics

Noether's theorem says that a symmetry of a Lagrangian implies a conservation law. But to fully appreciate the connection we need to go to Hamiltonian mechanics and see how symmetries act on phase space! Get the notes for free here: https://courses.physicswithelliot.com...

The connection between symmetries and conservation laws is one of the deepest relationships in physics. Noether's theorem says that for every continuous symmetry of a Lagrangian, you'll find a corresponding conserved quantity. But to fully understand the connection between the two, we need to investigate their relationship in Hamiltonian mechanics. Any function on phase space generates a "flow," similar to how the Hamiltonian generates time evolution. Then the rate of change of any other function along the flow is given by its Poisson bracket with the generator. A quantity will be conserved if and only if the flow that it generates leaves the Hamiltonian invariant!

Get all the links here: https://www.physicswithelliot.com/ham...

Watch these first!:

The principle of least action:    • Explaining the Principle of Least Action: ...  

Noether's theorem:    • Symmetries & Conservation Laws: A (Physics...  

Poisson brackets:    • Before You Start On Quantum Mechanics, Lea...  

Also check out:

A 20 minute intro to Lagrangian and Hamiltonian mechanics:    • Lagrangian and Hamiltonian Mechanics in Un...  

The relativistic action:    • The Special Relativistic Action, Explained  

Tutoring inquiries: https://www.physicswithelliot.com/tut...

If you find the content I’m creating valuable and would like to help make it possible for me to continue sharing more, please consider supporting me! You can make a recurring contribution at   / physicswithelliot  , or make a one time contribution at https://www.physicswithelliot.com/sup.... Thank you so much!

About physics mini lessons:
In these intermediate-level physics lessons, I'll try to give you a self-contained introduction to some fascinating physics topics. If you're just getting started on your physics journey, you might not understand every single detail in every video---that's totally fine! What I'm really hoping is that you'll be inspired to go off and keep learning more on your own.

About me:
I’m Dr. Elliot Schneider. I love physics, and I want to help others learn (and learn to love) physics, too. Whether you’re a beginner just starting out with your physics studies, a more advanced student, or a lifelong learner, I hope you’ll find resources here that enable you to deepen your understanding of the laws of nature. For more cool physics stuff, visit me at https://www.physicswithelliot.com.