Thank for this suggestion. I have some other videos to make first, but would be happy to make a video on Kane's Method in the future.

Go for it. That's what they're for.

My pleasure!

I'm not sure what the problem is here. For some reason KZfaq seems unable to auto-generate the transcript for this video like it has done for all the other videos, which is strange because the audio seems fine. I have now used a 3rd party service to successfully produce a transcript for this video and I uploaded that, but it is still not showing up. I'll keep working at this an let you know if I manage to fix it.

I’m not familiar with this. Do you perhaps have some reference material?

@@Freeball99 Yes, I have. I will send some links and screenshots to your e mail

Yes, the kinetic energy can depend on both the velocity AND position, in general. This comes up often in rotating system, for example the elastic pendulum kzfaq.info/get/bejne/n7t8lJxylZauoHU.htmlsi=CAZUfbyWsu-joQsY&t=153

@@Freeball99 Thank you

γ is a coefficient that quantifies the extent of damping relative to the internal elastic forces. This is a quantity that is typically determined in the lab and is a function of the material of the beam, its geometry and the boundary conditions and the type of excitation. Typical values range from about 0.01 to 0.1. The specific reference I used for the is "Dynamics of Structures" by Hurty & Rubinstein. The book is long since out of print, but you can find a copy at archive.org. Not sure how helpful you'll find it though. For a reference on variational principles, my go-to reference is "Structural Dynamics: A Variational Approach" by Dym & Shames which is an excellent book! You'll likely also find an archived copy somewhere online.

@@Freeball99 thank you very much sir! i'll keep watching

The app is called "Paper" by WeTransfer. Running on an iPad Pro 13 inch and using an Apple Pencil.

hello Sir, please help me out of this query as soon as possible!!!!!

Sorry for the delayed response, but I missed your comments until now... Both methods will yield the same equations of motion for systems where they are applicable, but they offer different perspectives and may be more convenient for different kinds of problems. Hamilton's Principle was discovered about 100 years after the concept of the Lagrangian. Originally, d'Alembert's Principle was used to derive the Euler-Lagrange equation, but both lead you to the same result? Hamilton's Principle is generally more abstract and is based on the concept of minimizing action. It often applies in a broader context, including fields beyond classical mechanics. d'Alembert's Principle is more intuitive and directly connected to the forces in the problem. It is particularly useful when dealing with non-conservative forces.

The app is called "Paper" by WeTransfer. Running on an iPad Pro 13 inch and using an Apple Pencil.

It depends on the type of damping that you have, but usually you can model a damping force as external if you're unsure. Viscous damping can be treated as either.

The time derivative of the potential energy with respect to q_dot will just be zero, so only the kinetic energy will contribute to the first term of equation 13. Also, in many problems, the kinetic energy will not be depend directly on the generalized coordinate, q, and so, in that case, the kinetic energy would have zero contribution to the second term of equation 13. Therefore, this is not a problem and it allows us to write Lagrange's equation is the compact form shown in 13.

No, these are not Laplacian operators. The Laplacian operator represents the divergence squared which is not what appears in either of these equations. Also, Laplace's equation typically deals with an equilibrium state (so no time dependency) while these equations are time-dependent.

A virtual displacement means displacing the object/surface by an infinitesimal amount - it's the same idea as taking the variation of the path. This is similar to considering differential slices of objects when studying differential calculus.

This goes back to an earlier video on Variational Calculus, this is the equivalent of the boundary condition ...the idea is that because we know the state of the system at times t1 and t2, therefore the variation at each of the end-points of the path is zero. In other words, this conditions says that when considering an optimal path from t1 to t2, all trial paths must be constrained to have the same beginning and ending points as the original path.