How did we arrive to Hamilton's principle ?

[Nikitin]

Hi. The Lagrange equations can be derived simply by using Newton's laws and defining potential, work and kinetic energy. So that's just a mathematical reformulation of known results.

Is it the same with Hamilton's principle? Is the concept of "action" and it being stationary just another mathematical reformulation from Newton's laws, or is it something else? Because Hamilton's principle leads to the Langrange equations just the same as Newton's laws.

My professor did not discuss this. He simply presented Hamilton's principle as a postulate and left it at that.

PS: What is up with this "generalized force" thing? I don't exactly understand what that is all about. Would appreciate it if somebody could explain briefly.

 

[dextercioby]

The principle of stationary action (in the Lagrangian or Hamiltonian form) is usually taken as an axiom. You can only constantly check its validity by producing the correct equations of motion.

 

[Incand]

Hamiltons equations follow from the definition of the hamiltonian as a Legendre transformation $H = \dot qp - L$
If you then write down the differential $dH$ you instantly get hamiltons out of it (try it!).

Generalised forces work just the same as coordinates, velocities, moment etc the force along your coordinate vector. It's just that the only forces we care about the non-conservative ones that you can't account for as energy so in a lot of cases you just don't care about the force part at all.
So
$$\frac{d}{dt}(\frac{\partial L}{d\dot q_i}) - \frac{\partial L}{dq_i} = F_i$$
would reduce to a 0 in the right part if we only have conservative forces.

 

[Jano L.]

Hi. The Lagrange equations can be derived simply by using Newton's laws and defining potential, work and kinetic energy. So that's just a mathematical reformulation of known results.

Is it the same with Hamilton's principle? Is the concept of "action" and it being stationary just another mathematical reformulation from Newton's laws, or is it something else? Because Hamilton's principle leads to the Langrange equations just the same as Newton's laws.

My professor did not discuss this. He simply presented Hamilton's principle as a postulate and left it at that.

PS: What is up with this "generalized force" thing? I don't exactly understand what that is all about. Would appreciate it if somebody could explain briefly.

Hamilton's principle (principle of stationary action) can be derived from Newton's laws and some additional assumptions about the mechanical system. The important one that comes to mind is that all constraints have to be holonomic. If the mechanical problem has non-holonomic constraints (e.g. sphere rolling on ground with no slipping), usual Hamilton's principle does not work. Also if friction dissipates mechanical energy Hamilton's principle is hard to formulate and I think it is little used there.

Check the book H. Goldstein, Classical Mechanics for exposé of Hamilton's principle. There is also book by C. Lanczos, The Variational Principles of Mechanics which might help.

 

[Nikitin]

allright. thanks guys