Hamilton’s principle - Classical Mechanics, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

1. Start with a single particle of mass m, with position

q(t) = (q₁ (t), q₂ (t), q₃ (t)),                        (1)

velocity    and acceleration  at time t. Suppose that the force F = (F₁ , F₂ , F₃ ) acting the on the particle is conservative, with potential U (q). Then

(2)
and Newton’s second law is

 (3)

Think of the particle as a mechanical “system.” The state, or configuration of the system at time t is q(t). The set C of possible configurations is called the configuration space. In this case, C = R³ . The time-evolution of the configuration is governed by the system of equations (3).

2. Suppose that between times a and b, the particle traces a tra jectory q in C with endpoints

q(a) = α and q(b) = β . (4)

The potential energy of the particle is U (q), and its kinetic energy 

(5)
Consider the action functional

 (6)
defined on a domain  of smooth curves q satisfying the boundary conditions (4). The class  of admissible variations consists of smooth curves h(t) = (h₁ (t), h₂ (t), h₃ (t)) that vanish at a and b. By the usual steps, the extremizing condition

            (7)

leads to the Euler-Lagrange equations

          (8)

where the Lagranigian is L = T − U . Note that L_qk = −U_qk and . Plug these into (8) to get

 (9)

which are Newton’s equations. Thus, you could say that

a. The particle’s trajectory q(t) through R³ is determined by Newton’s equations.
Though you might prefer to say that

b. The mechanical system’s trajectory q(t) through the configuration space C is an extremal of the action functional.
This second characterization is a crude version of Hamilton’s principle, a variational generalization of Newton’s second law.

3. It is important to note that q₁ , . . . , q_n are not necessarily the rectangular coordinates of a particle, but rather a set of quantities that describe the state of a mechanical system.
Thus the qk are called generalized coordinates, and the  the generalized velocities.
As pointed out above, for the Lagrangian L in (6),  . For this reason, the quantities

      (10)

are called the generalized momenta. And since L_qk = −U_qk , the L_qk are called the generalized forces. The number n of generalized coordinated is called the number of degrees of freedom of the system.

4. Example: Consider a pendulum with string of length Λ and negligible mass and and a bob of mass m. As usual, θ is the angle made by the string and the pendulum. The kinetic energy is

and the potential,

U = mgΛ(1 − cos θ).

Hence the Lagrangian is

  (11)

By Hamilton’s principle, the equation of motion is

which reduces to

In this example the generalized coordinate is θ, the generalized velocitythe generalized momentum

and the generalized force

We take θ to be dimensionless. Note that the generalized velocity, momentum and force do not have the dimensions of their standard counterparts.

5. Since the Lagrangian in the previous example does not depend explicitly on time, the Hamiltonian

= T + U.            (12)

is a first integral. Thus the total energy T + U is constant, i.e. the energy is conserved.

6. Consider the planar motion of a body of mass m sub ject to the gravitational attraction F created by a body of mass M fixed at the origin. Let x = (x₁ , x₂ ) be the position of former body. Then

The potential for the force field F is

(13)
and the kinetic energy

 (14)

We can make things easier by introducing polar coordinates:

x₁ = r cos θ and x₂ = r sin θ.

In terms of r and θ, the potential energy is

       (15)

and the kinetic,

     (16)

The Lagrangian is

      (17)

According to Hamilton’s principle, the equations of motion are

or

In this example, the system has two degrees of freedom. The generalized coordinates are r and θ, the generalized velocities , and the generalized momenta  and

7. Since the Lagrangian does not depend explicitly on time, the Hamiltonian

= T + U. (18)

is a first integral. Thus the total energy T + U is constant, i.e. the energy is conserved.