Difference hamiltonian and energy

[Gavroy]

Hi

I am looking for the MOST GENERAL statement that determines, when the Hamilton function and the energy are equal to each other in classical mechanics.

 

[Popper]

Hi

I am looking for the MOST GENERAL statement that determines, when the Hamilton function and the energy are equal to each other in classical mechanics.

I've been doing a lot of digging lately on this subject to determine exactly this. Mind you, what I'm posting may not be exactly precise because I'm only looking at the statements and haven't followed the derivations if many years.

From Classical Mechanics - Third Edition by Goldstein, Safko and Poole page 345

Further, it was proved in Section 2.7 that if the equations of transformation that define the generalized coordinates (1.38),

r_m = r_m(q₁, …, q_n; t)

do not depend explicitly upon time, and if the potential is velocity independent, then H is the total energy T + V. The identification of H as a constant of the motion and as total energy are two separate matters, and the conditions sufficient for one are not enough for the other. It can happen that Eqs. (1.38) do involve time explicitly but that H does not. In this case H is a constant of motion but is not the total energy/ etc.

I also found the following comment regarding Jacobi's integral, aka the energy function, in Classical Dynamics, by Donald T. Greenwood, Dover Pub. (1977). When the energy function is expressed in terms of generalized coordinates and canonical momentum it's called the Hamiltonian. On page 73 the author defines a natural system as a conservative system which has the additional properties

(1) it is described by the standard holonomic form of Lagrange's equations
(2) the kinetic energy is expressed as a homogeneous quadratic function of the generalized velocities.

He then states that under these circumstances Jacobi's integal, aka the energy function, is the total mechanical energy of the system and is an integral of motion, i.e. constant.

Nice! :)

I assume that you know what a conservative system is, yes? I think the term has much to be desired. It makes one think of a quantity that is constant in time. However it's defined as a system in which the force can be expressed as the gradient of a function. If the function does not contain the time explicitly then it's called a conserved system. But such a function can depend on time. E.g. this happens when a charged particle is moving through an electromagnetic field which is time dependent.