Analytical Classical Dynamics: An intermediate level course

[JANm]

Moderation note: In reference to http://farside.ph.utexas.edu/teaching/336k/lectures.pdf

Lagrangian(L) and Hamiltonian(H),
Dear Greg I am studying the L and H.

If kinetic energy(K) and potential(U) are given it seems that L=K-U.
Hamilton defines (p_i, dot q_i being components of momentum, resp. velocity in i'th direction): H=sum p_i*dot q_i - L and it appears that for a conservative situation the Hamiltonian becomes H=K+U. With conservative one means usually U is a function of coordinates only.

Do you think that this system would work for a mass-velocity relation? So a momentum which a changeble mass as a function of velocity v=Sqrt(sum (dot q_i)^2)?
greetings.

 

[eljose79]

Hello,

Thank you for your interest in studying Lagrangian and Hamiltonian dynamics. The equations you have mentioned are indeed correct - the Lagrangian is defined as the difference between the kinetic and potential energy, while the Hamiltonian is the sum of the kinetic and potential energies. In a conservative system, where the potential energy is only a function of coordinates, this simplifies to the sum of kinetic and potential energies.

As for your question about a mass-velocity relation, the Hamiltonian approach can certainly be extended to include a changeable mass. However, the resulting equations may become quite complex and may not have a straightforward interpretation. It would depend on the specific situation and the form of the mass-velocity relation.

I would recommend consulting a textbook or a more detailed lecture notes on Lagrangian and Hamiltonian dynamics for a more thorough understanding of these concepts. Best of luck with your studies!

 

[astrokreng]

Hello,

Thank you for sharing your thoughts on the Lagrangian and Hamiltonian concepts. I agree with your understanding that the Lagrangian is equal to the difference between kinetic and potential energy, and the Hamiltonian is equal to the sum of the momenta and velocities minus the Lagrangian.

Regarding your question about using this system for a mass-velocity relation, it is possible to incorporate a changeable mass as a function of velocity in the Lagrangian and Hamiltonian equations. This could be useful in situations where the mass of a system changes due to external forces or interactions.

Overall, the Lagrangian and Hamiltonian are powerful tools in classical dynamics that allow us to analyze and predict the behavior of physical systems. I encourage you to continue exploring these concepts and their applications in your studies. Best of luck in your course!