On the basis of the concept, idea and knowledge I till have developed, I hereby attempt to answer as possibly best:
The inter-relation of the Hamiltonian and the energy of a system seems to be very simple, whereas it is not. As of now, I only discuss this topic in the context of classical mechanics. In the Hamiltonian formalism, we mainly rely on two equations of motion only and they are:
1. explicit time variation of generalised momentum Pi = -( variation of the Hamiltonian explicitly with generalised coordinate Qi )
2. explicit time variation of generalised coordinate Qi = variation of the Hamiltonian explicitly with generalised momentum Pi
Whereas, there is another one equation which relates the explicit time dependency of the Hamiltonian with that (negatively) of the Lagrangian of a particular classically dynamical system. If put into words then, it is meant by the equation: the summation of explicit time dependency of both the Lagrangian and the Hamiltonian of a particular system is zero. And the eqn is:
Explicit time dependency of the Hamiltonian of a system = -( Explicit time dependency of the Lagrangian of that system )
or,
[ ( Explicit time dependency of the Hamiltonian of a system ) + ( Explicit time dependency of the Lagrangian of that system )] = 0
Now, the above-mentioned phenomenon or result further leads to conservation of energy of a system. Now, there is a twist & it is: the Hamiltonian of a system refers to the energy of that system and this very property of a classically dynamical system is solely based upon the notion of the principle of conservation of energy. Now as has been stated above, the conservation of energy of a dynamical system is resulted out of the sum of the explicit time dependency of both the Hamiltonian and the Lagrangian of that system to be zero or the Hamiltonian being not time-dependent explicitly. And only then, the Hamiltonian of a ( classical ) system can be referred to be the energy of that system.
Further more:
this phenomenon or the property of a system is conditional and the conditions being:
1. The kinetic energy of the system must have to be a quadratic function of velocity.
2. The potential the concerned system is driven by must not be dependent upon the velocity or the explicit change in generalised coordinate of the system.
3. The system has to be holonomic and scleronomic .
4. The system must not be subjected to such a force which has no potential, which is called pseudo-force; for example the coriolis force is a pseudo-force.
So, at the last: the Hamiltonian of a system does not correspond to the energy of the system always.
To be honest the answer is non-statisfactoty even to me, the explanation seems to be hollow and inflated to me.
Again may I have a help, how to drop question ??
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On the basis of the concept, idea and knowledge I till have developed, I hereby attempt to answer as possibly best:
The inter-relation of the Hamiltonian and the energy of a system seems to be very simple, whereas it is not. As of now, I only discuss this topic in the context of classical mechanics. In the Hamiltonian formalism, we mainly rely on two equations of motion only and they are:
1. explicit time variation of generalised momentum Pi = -( variation of the Hamiltonian explicitly with generalised coordinate Qi )
2. explicit time variation of generalised coordinate Qi = variation of the Hamiltonian explicitly with generalised momentum Pi
Whereas, there is another one equation which relates the explicit time dependency of the Hamiltonian with that (negatively) of the Lagrangian of a particular classically dynamical system. If put into words then, it is meant by the equation: the summation of explicit time dependency of both the Lagrangian and the Hamiltonian of a particular system is zero. And the eqn is:
Explicit time dependency of the Hamiltonian of a system = -( Explicit time dependency of the Lagrangian of that system )
or,
[ ( Explicit time dependency of the Hamiltonian of a system ) + ( Explicit time dependency of the Lagrangian of that system )] = 0
Now, the above-mentioned phenomenon or result further leads to conservation of energy of a system. Now, there is a twist & it is: the Hamiltonian of a system refers to the energy of that system and this very property of a classically dynamical system is solely based upon the notion of the principle of conservation of energy. Now as has been stated above, the conservation of energy of a dynamical system is resulted out of the sum of the explicit time dependency of both the Hamiltonian and the Lagrangian of that system to be zero or the Hamiltonian being not time-dependent explicitly. And only then, the Hamiltonian of a ( classical ) system can be referred to be the energy of that system.
Further more:
this phenomenon or the property of a system is conditional and the conditions being:
1. The kinetic energy of the system must have to be a quadratic function of velocity.
2. The potential the concerned system is driven by must not be dependent upon the velocity or the explicit change in generalised coordinate of the system.
3. The system has to be holonomic and scleronomic .
4. The system must not be subjected to such a force which has no potential, which is called pseudo-force; for example the coriolis force is a pseudo-force.
So, at the last: the Hamiltonian of a system does not correspond to the energy of the system always.
To be honest the answer is non-statisfactoty even to me, the explanation seems to be hollow and inflated to me.
Reference:
1. Classical Mechanics; H. Goldstein; Pearson.
2. Classical Mechanics; N.C. Rana, P.S. Joag; McGraw Hill; pp: 63-69; Sub-topic: 2.5 & 2.6.
- অর্ণব বিশ্বাস ( Arnab Biswas )
May I have a help, actually I can't paste the Images of Hamiltonian Ian eqn of motion which are instrumental to answer this question.
You can copy paste the equations. eg. q˙i=∂H/∂pi and −p˙i=∂H/∂qi