Hamiltonian - General Approach

Notes to a video lecture on UNIZOR.COM

General Approach
to Hamiltonian

In the previous introductory lecture we have examined the way to simplify the system of n Euler-Lagrange differential equations of second order into a system of 2n equations of the first order using coordinates q={q_i} and momenta p={p_i}.

For simple classical systems the Lagrangian Mechanics uses Euler-Lagrange equations
∂L/∂q_i = d/dt ∂L/∂q̇_i
where Lagrangian L=T−U,
T is the full kinetic energy of a system (independent of coordinates),
U is its potential energy (independent of velocities).

Instead, we constructed mathematically equivalent system of Hamiltonian equations
q̇_i = ∂(T+U)/∂p_i
−ṗ_i = ∂(T+U)/∂q_i
and introduced a new function called Hamiltonian
H(q,p) = T(p) + U(q)
where q=(q₁,...,q_n) are coordinates
and p=(p₁,...,p_n) are momenta.

This transformation increased the number of variables from n to 2n, but reduced the complexity of differential equations to the first order and made them symmetric and esthetically more appealing.

However, this construction relied on special properties of the system:
kinetic energy T depended only on momenta,
potential energy U depended only on coordinates.

In general mechanical systems, the Lagrangian L(q,q̇,t) may depend on coordinates and velocities in a more complicated way, kinetic and potential energies might not be separable and the Lagrangian might not be written as a sum of a function of p and a function of q.
Therefore, the construction H=T+U is not applicable to the general case.

The Hamiltonian approach is attractive for these simple mechanical systems, so it is natural to ask whether it can be generalized.
More specifically, given a Lagrangian L(q,q̇,t), we'll construct a function H(q,p,t) whose partial derivatives generate the differential equations similar to above and equivalent to Euler-Lagrange equations of motion.

To accomplish this, we will reformulate the expression H=T+U to avoid explicitly using energies and, instead, express H in terms of coordinates, momenta and a Lagrangian.

Recall the calculations of the Hamiltonian for simple mechanical system in the previous introductory lecture:
T = Σ_i½m_i·q̇_i² = Σ_i½p_i²/m_i

Partial derivative of T by p_i produces
∂T/∂p_i = p_i/m_i = q̇_i
Therefore,
q̇_i = ∂T/∂p_i
Our expression for H=T+U in that case can be represented as
H = 2T − (T−U) = 2T − L =
= Σ_im_i·q̇_i² − L =
= Σ_i(m_i·q̇_i)·q̇_i − L =
= Σ_ip_i·q̇_i − L

The above expression
H(q,p,t) = Σ_ip_i·q̇_i − L(q,q̇,t)
formally seems to correspond to our task of constructing a function H(q,p,t) from coordinates, momenta and a given Lagrangian L(q,q̇,t).
It does not explicitly rely on kinetic or potential energy. Since it is written only in terms of the Lagrangian, generalized coordinates, generalized velocities, and generalized momenta, it is a natural candidate for the Hamiltonian in the general case.

It's very important to understand the change of a viewpoint on relationship between arguments, which is a key idea of the Hamiltonian Mechanics.
While in the Lagrangian formulation the independent variables are q, q̇ and t, in the Hamiltonian approach the independent variables are q, p and t with generalized velocities q̇ become dependent on them:
q̇_i = q̇_i(q,p,t)

Let's check if this expression corresponds to differential equations involving the Hamiltonian.
∂H(q,p,t)/∂p_k =
[take into consideration that now q_i and p_i are independent variables, while generalized velocities q̇_i are functions of these variables: q̇_i=q̇_i(q,p,t)]
= ∂/∂p_k [Σ_ip_i·q̇_i − L] =
[use chain rule]
= q̇_k + Σ_ip_i·(∂q̇_i/∂p_k) −
− Σ_i(∂L/∂q̇_i)·(∂q̇_i/∂p_k) =
= q̇_k+Σ_i[p_i−∂L/∂q̇_i]·(∂q̇_i/∂p_k) =
[recall that p_i=∂L/∂q̇_i by definition]
= q̇_k
as required.

∂H(q,p,t)/∂q_k =
= Σ_ip_i·(∂q̇_i/∂q_k) − ∂L/∂q_k −
− Σ_i(∂L/∂q̇_i)·(∂q̇_i/∂q_k) =
= Σ_i[p_i−∂L/∂q̇_i]·(∂q̇_i/∂q_k) −
− ∂L/∂q_k =
[recall that p_i=∂L/∂q̇_i by definition]
= − ∂L/∂q_k =
[using Euler-Lagrange equation]
= − d/dt ∂L/∂q̇_k =
[using the definition of generalized momentum]
= − d/dt p_k = −ṗ_k
as required.

So, from the definitions of momenta as
p_i = ∂L/∂q̇_i
the Hamiltonian as
H(q,p,t) = Σ_ip_i·q̇_i − L(q,q̇,t)
and Euler-Lagrange equation for L
∂L/∂q_i = d/dt ∂L/∂q̇_i
follows that both differential equations for H are satisfied:
∂H/∂p_i = q̇_i
∂H/∂q_i = −ṗ

In reverse, assuming Hamiltonian equations are satisfied, we can derive Euler-Lagrang equations.
∂H(q,p,t)/∂q_k =
= ∂/∂q_k [Σ_ip_i·q̇_i − L] =
= Σ_ip_i·(∂q̇_i/∂q_k) − ∂L/∂q_k −
− Σ_i(∂L/∂q̇_i)·(∂q̇_i/∂q_k) =
= Σ_i[p_i−∂L/∂q̇_i]·(∂q̇_i/∂q_k) −
− ∂L/∂q_k =
[recall that p_i=∂L/∂q̇_i by definition]
= − ∂L/∂q_k
Since we assumed that Hamiltonian differential equations are true,
∂H(q,p,t)/∂q_k = −ṗ_k
Therefore,
− ∂L/∂q_k = −ṗ_k

From the definition
ṗ_k = dp_k/dt = d/dt ∂L/∂q̇_k
follows:
∂L/∂q_k = d/dt ∂L/∂q̇_k
which is the Euler-Lagrange equation.
That completes the proof that a system of n second order Euler-Lagrange differential equations is equivalent to 2n first order Hamiltonian equations.

The construction of the Hamiltonian
H = Σ_ip_i·q̇_i − L
from the Lagrangian L is called the Legendre transformation.