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Hamiltonian momentum
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Generalised momentum is derived from the Lagrangian.
p = ∂L/∂q̇In advanced mechanics the generalised momentum is pᵢ = ∂L/∂q̇ᵢ. It reduces to mv for simple cases but can include extra terms, such as qA in a magnetic field.
For a charge in a magnetic field p = mv + qA.
Canonical momentum is not always mv.
In more detail
In Hamiltonian mechanics, momentum is one of the coordinates in phase space. Conserved momenta correspond to cyclic coordinates in the Hamiltonian, which simplifies the equations.
Real-world uses
- Physicists use canonical momentum to analyse charged particles in magnetic fields.
- Computational physics uses Hamiltonian methods for long-term simulations.
- In advanced mechanics, the generalised momentum for each coordinate is found from the Lagrangian, which can differ from mv.
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Frequently asked questions
What is canonical momentum?
The momentum conjugate to a coordinate, which may differ from mv in electromagnetic fields.
Why is it useful?
It reveals conserved quantities directly.
Is generalised momentum always mv?
No. In a magnetic field or with unusual coordinates, the generalised momentum includes extra terms.
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