Physical Significance of Hamiltonian

The Hamiltonian in classical and quantum mechanics is a mathematical object that encodes the total energy of a physical system. It plays a central role in determining the time evolution of a system and is used to formulate the laws of motion in terms of equations of motion for the system's variables. The Hamiltonian is therefore a key quantity in understanding the dynamics and behaviour of physical systems, making it of great physical significance.

ELI5: Imagine the Hamiltonian is like a pot of energy for a system, like a toy car or a ball. It tells us how much energy the system has and how it is being used. By knowing the Hamiltonian, we can understand how the system will move and change over time, just like how you can know how your toy car will move when you push it, or how your ball will roll. The Hamiltonian helps us explain why things happen in the way they do.

Like Lagrangian, H also posses the dimensions of energy but in all circumstances it is not equal to the total energy E. Restriction for this equality, i.e., E= H are:

1. The system be conservative one, i.e. P.E is coordinate dependent and not velocity dependent and

2. Coordinate transformation equations be independent of time, so that `∑pjqj = 2T`

These are conditions necessary for Hamiltonian H to represent the total energy E. It may happen that though coordinate transformation equation involve time explicitly yet L be still independent of time, then

`H = ∑pjq̇j - L(qj, q̇j)` will be a constant i.e H will continue to be a constant of motion but not total energy. It can be shown that it does not occur in Lagrangian L then Hamiltonian H will not involve t. Since

H = H(qj,pj,t)

Also from canonical equation,

∂H/ ∂qj = -ṗj and ∂H/ ∂pj = q̇j

Therefore dH/dt = - ∑ṗjq̇j + ∑ṗjq̇j + ∂H/ ∂t

But ∂H/ ∂t = - ∂L/ ∂t

If L does not involve time t, then ∂L/ ∂t =0 shows that

∂H/ ∂t = 0

So `dH/dt =0`

Which mean t will not appear in H. Here , Cyclic Hamiltonian is constant of motion. A cyclic Hamiltonian, also known as an integral of motion or constant of motion, refers to a quantity that remains constant throughout the motion of a physical system. In other words, it is a function of the system's coordinates and momenta that does not change with time.

When a Hamiltonian is cyclic, it implies that the corresponding physical quantity associated with that cyclic coordinate or momentum is conserved.

The significance of a cyclic Hamiltonian lies in the fact that it provides insights into the behavior and properties of the system. By identifying and analyzing cyclic Hamiltonians, one can determine conserved quantities and constants of motion, which greatly simplify the description of the system's dynamics. Cyclic Hamiltonians often lead to symmetries in the system and allow for the application of powerful mathematical techniques, such as integrability and separation of variables.

In general Hamiltonian physical significance lies in several key aspects:

1. Conservation of Energy: The Hamiltonian represents the total energy of a system, including both kinetic and potential energy. In classical mechanics, the time evolution of a system is governed by Hamilton's equations, which ensure that the total energy, given by the Hamiltonian, is conserved. This conservation of energy is a fundamental principle in physics and has profound implications for the behavior of physical systems.

2. System Dynamics: The Hamiltonian provides a mathematical framework to describe the dynamics of a physical system. It defines the equations of motion and determines how the system evolves over time. By analyzing the Hamiltonian, one can obtain information about the stability, equilibrium points, and long-term behaviour of the system.

3. Symmetry and Conservation Laws: The Hamiltonian is intimately linked to symmetries and conservation laws in physics. Noether's theorem establishes a connection between symmetries and conserved quantities. For example, if the Hamiltonian of a system is invariant under spatial translations, it implies the conservation of linear momentum. Similarly, time translation symmetry leads to the conservation of energy. Therefore, the Hamiltonian provides insight into the symmetries and associated conserved quantities of a physical system.

4. Quantization in Quantum Mechanics: In quantum mechanics, the Hamiltonian operator plays a central role. It represents the observable associated with the total energy of a quantum system. The eigenvalues and eigenstates of the Hamiltonian operator provide information about the allowed energy levels and corresponding wave functions of the system. Thus, the Hamiltonian allows for the quantization of energy and forms the basis for understanding the behaviour of quantum systems.

5. Control and Manipulation of Systems: The Hamiltonian is used in control theory and optimal control applications to design strategies for manipulating physical systems. By formulating the Hamiltonian and employing mathematical optimization techniques, one can determine the most efficient control inputs or paths to achieve desired outcomes.

Overall, the Hamiltonian is a fundamental concept in physics that encapsulates the total energy, dynamics, symmetries, conservation laws, and quantization of a physical system. Its physical significance extends across various branches of physics, providing deep insights into the behaviour and manipulation of systems.

This note is taken from Classical Mechanics, Msc physics, Nepal.

This note is a part of the Physics Repository.