Last Updated：2022/12/24
A Legendre transformation looks like this: H=∑ᵢ₌₁ⁿp_i̇q_i-L=∑ᵢ₌₁ⁿ∂L/∂̇q_i̇q_i-L, where the q_i are generalized coordinates, their dotted versions ̇q_i are their time derivatives, the p_i=∂L/∂̇q_i are generalized momenta or conjugate momenta, L=L(q_1,...,q_n;̇q_1,...,̇q_n) is a Lagrangian function and H=H(q_1,...,q_n;p_1,...,p_n) is a Hamiltonian function.
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A
Legendre
transformation
looks
like
this:
H=∑ᵢ₌₁ⁿp_i̇q_i-L=∑ᵢ₌₁ⁿ∂L/∂̇q_i̇q_i-L,
where
the
q_i
are
generalized
coordinates,
their
dotted
versions
̇q_i
are
their
time
derivatives,
the
p_i=∂L/∂̇q_i
are
generalized
momenta
or
conjugate
momenta,
L=L(q_1,...,q_n;̇q_1,...,̇q_n)
is
a
Lagrangian
function
and
H=H(q_1,...,q_n;p_1,...,p_n)
is
a
Hamiltonian
function.
