6.2 Hamiltonian mechanics
Returning to the general case, with LagrangianL(q,q ̇;t), one defines the momentum pi
canonically conjugate toqias follows,
pi=
∂L
∂q ̇i
(6.13)
For the simplest systems, like the LagrangianLV, the canonical momentum coincides with the
mechanical momentum, given by mass times velocity,pi=miq ̇i. But for more complicated
Lagrangians, likeLA, the two quantities differ, and one has in this casep=mr ̇+eA.
We shall assume that the relationpi=pi(q,q ̇;t) is invertible,^5 so that the generalized
velocities can be obtained uniquely as a function of the generalized coordinates and momenta,
q ̇i= ̇qi(q,p;t) (6.14)
The Hamiltonian is now defined as the Legendre transform of the Lagrangian,
H(q,p;t) =
∑Ni=1piq ̇i−L(q,q ̇;t) (6.15)
with the understanding that the velocities are eliminated in favor ofq andpon the rhs
using (6.14). The space of all allowed positionsqiand momentapiis referred to asphase
space. Hamiltonian mechanics is formulated in terms of time evolution equations (or flows)
on phase space.
The Hamilton equations are obtained as follows. Under a general variation ofqandp,
the rhs above transforms as,
δH=
∑i(δpiq ̇i+piδq ̇i−
∂L
∂qi
δqi−
∂L
∂q ̇i
δq ̇i
)(6.16)
Notice that the second and fourth terms on the rhs cancel by the very definition of pi.
Alternatively, the variation ofHin termspandqonly gives,
δH=
∑
i(∂H
∂qi
δqi+
∂H
∂pi
δpi
)(6.17)
Comparing the two expressions forδH and using the Euler-Lagrange equation gives the
Hamilton equations,
∂H
∂qi
=−p ̇i
∂H
∂pi
= ̇qi (6.18)
(^5) There are many physically interesting Lagrangians where this relation is not invertible. These systems
are referred to as havingconstraints.