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Hamiltonian as total energy
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The Hamiltonian is the energy expressed in coordinates and momenta.
H = p²/2m + UThe Hamiltonian H(q, p) = p²/2m + U(q) is the total energy written in terms of position and momentum. Hamilton's equations q̇ = ∂H/∂p, ṗ = −∂H/∂q give the motion.
For a spring H = p²/2m + ½kx².
H equals energy only if the constraints are time independent.
In more detail
In classical mechanics, the Hamiltonian is the total energy expressed in terms of position and momentum. Its equations of motion describe how the system evolves in phase space.
Real-world uses
- Physicists use Hamiltonians to model planetary orbits and molecular vibrations.
- Quantum mechanics builds on the Hamiltonian as the operator for total energy.
- Quantum and molecular simulations use the Hamiltonian to compute the total energy of each state of a system.
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Frequently asked questions
Is the Hamiltonian always energy?
In many systems yes, specifically when the potential is time-independent.
Why use a Hamiltonian?
It gives a unified framework for mechanics, statistical physics and quantum theory.
Why is the Hamiltonian useful?
It gives the time evolution of a system in a form that works for quantum mechanics as well as classical physics.
Related topics in Work, Energy & Power
Virial theorem · Principle of virtual work · Work–energy with variable mass · Lagrangian mechanics intro · Euler–Lagrange equations · Principle of least action