In classical mechanics, the Hamiltonian of a particle in one dimension is defined as the sum of its kinetic energy and potential energy. It is denoted by the symbol H.
The Hamiltonian, H, is given by:
H = T + V,
where T represents the kinetic energy and V represents the potential energy of the particle.
For a particle with mass m and velocity v, the kinetic energy, T, is given by:
T = (1/2)mv^2.
The potential energy, V, depends on the specific system in which the particle is located. It can take various forms depending on the forces acting on the particle. Some common examples include gravitational potential energy, spring potential energy, or electric potential energy.
The Hamiltonian captures the total energy of the particle and provides a convenient way to describe its dynamics using Hamilton's equations of motion. In quantum mechanics, the Hamiltonian takes on a different form, where it represents the operator associated with the total energy of a quantum system.