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The Extension of the Methodology of
Lagrangian and Hamiltonian Mechanics
to a Continuum of Variables

Summary of Lagrangian and Hamiltonian
Dynamics for the Finite Variable Case

Let Q be the vector of generalized coordinates (q1, q2, …, qn) and V be the vector of their time derivatives (dq1/dt, dq2/dt, …, dqn/dt). The number of generalized coordinates n is also known as the number of degrees of freedom.

The Lagrangian L of a system is the difference between the kinetic energy K of the system and its potential energy U:

L = K − U

For the physically relevant cases K is a function of V and U is a function of Q and possibly time t. Thus L is a function of Q, V and possibly t. The integral of L over time is called the action S of the system.

S = ∫abL(Q, V, t)dt


Physical systems take on values of Q and hence V over time such that S takes on a stationary value. Originally the stationary value was presumed to be a mininum and the principle was called the Principle of Least Action. Later it was found that the proper statement was that the variation in the action, δS, had to be zero and that could occur for maxima and inflection points as well as minima.

A variation in Q(t) of δQ leads to a variation in velocities δV=d(δQ)/dt and consequently to a variation in the action, expressed here symbolically as,

δS = ∫ab[(∂L/∂Q)δQ + (∂L/∂V)δV]dt
or, equivalently
δS = ∫ab[(∂L/∂Q)δQ + (∂L/∂V)(δQ/dt)]dt

where δQ(a)=0 and δQ(b)=0.

The second term in the integral can be integrated by parts and when the end point conditions are taken into account the above equation reduces to

δS = ∫ab[(∂L/∂Q)δQ − d(∂L/∂V)/dt)(δQ]dt
and equivalently as
δS = ∫ab[(∂L/∂Q) − d(∂L/∂V)/dt)](δQdt

If δS=0 then

d(∂L/∂V)/dt = (∂L/∂Q)
which is a symbolical expression of
d(∂L/∂vi)/dt = (∂L/∂qi)
for i=1 to n

These are known as the Lagrangian equations.

The momentum pi conjugate to qi is defined as

pi = (∂L/∂vi)

From the Lagrangian equation it follows that

(dpi/dt) = (∂L/∂qi)

Let P denote the vector of momenta.

The Hamiltonian H of the system is defined as

H = VTP −L

If L is not explicitly a function of time then

H = 2K − (K − U)
and hence
H = K + U

The time derivative of Q and P can be expressed as

(dqi/dt) = (∂H/∂pi)
(dpi/dt) = (∂H/∂qi)

Extension of the Analysis to Fields

A field is merely a function over all of space. The value of the function may be a scalar, a vector, tensor or any other mathematical entity such as a spinor. The set of points in space takes the place of the indices 1 to n. Let X be point in space (a 3 dimensional vector) and G(X) be the field variable at that point. At a point there is the Lagrangian density L(G, t) so the Langrangian for the field is given by

L(G, t) = ∫d³xL(G(X), t)
and the action by
S = ∫abdt[ ∫d³xL(G(X), t)]

Let ∂(∂tG) denote the vector ∂(∂G/∂t)/∂X). Then the vector of the momenta conjugate to X is a function of Lagrangian density given by

Γ(X) = (∂L/∂[∂(∂tG)])

(To be continued.)

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