Lagrangian and Hamiltonian Formalism and equations of motion - 2 | Physics for IIT JAM, UGC - NET, CSIR NET PDF Download

2.3 Hamilton’s Principle

The configuration of a system at any moment is speci ed by the value of the generalized coordinates q_j (t), and the space coordinatized by these q₁,......,q_N is the configuration space. The time evolution of the system is given by the tra jectory, or motion of the point in conguration space as a function of time, which can be specied by the functions q_i(t).
One can imagine the system taking many paths, whether they obey Newton’s Laws or not. We consider only paths for which the q_i(t) are differentiable. Along any such path, we dene the action as

(2.16)

The action depends on the starting and ending points q(t₁) and q(t₂), but beyond that, the value of the action depends on the path, unlike the work done by a conservative force on a point moving in ordinary space. In fact, it is exactly this dependence on the path which makes this concept useful | Hamilton’s principle states that the actual motion of the particle from is along a path q(t) for which the action is stationary.
That means that for any small deviation of the path from the actual one, keeping the initial and nal congurations xed, the variation of the action vanishes to rst order in the deviation.

To find out where a differentiable function of one variable has a stationary point, we differentiate and solve the equation found by setting the derivative to zero. If we have a diff
erentiable function f of several variables x_i, the first-order variation of the function is  so unless     for all i, there is some variation of the {x_i} which causes a first order variation of f, and then x₀ is not a stationary point.

But our action is a functional, a function of functions, which represent an in nite number of variables, even for a path in only one dimension. Intuitively, at each time q(t) is a separate variable, though varying q at only one point makes   hard to interpret. A rigorous mathematician might want to describe the path q(t) on t ∈ [0, 1] in terms of Fourier series, for which   Then the functional S(f) given by

becomes a function of the in nitely many variables q₀,q₁,a₁,.....The endpoints x q₀ and q₁, but the stationary condition gives an in nite number of equations 

It is not really necessary to be so rigorous, however. Under a change

functional S will vary by

where we integrated the second term by parts. The boundary terms each have a factor of δq at the initial or nal point, which vanish because Hamilton tells us to hold the qi and qf xed, and therefore the functional is stationary if and only if

(2.17)

We see that if f is the Lagrangian, we get exactly Lagrange’s equation. The above derivation is essentially unaltered if we have many degrees of freedom qi instead of just one.

2.3.1 Examples of functional variation

In this section we will work through some examples of functional variations both in the context of the action and for other examples not directly related to mechanics.

The falling particle

As a rst example of functional variation, consider a particle thrown up in a uniform gravitional eld at t = 0, which lands at the same spot at t = T .

We make no assumptions about this path other than that it is differentiable and meets the boundary conditions

The action is

The fourth term can be integrated by parts,

The boundary term vanishes becamugs Δz = 0 where it is evaluated, and the other term cancels the sixth term in S ,so

The rst integral is independent of the path, so the minimum action requires the second integral to be as small as possible. But it is an integral of a nonnegative quantity, so its minimum is zero, requiringat all times, and the path which minimizes the action is the one we expect from elementary mechanics.

Is the shortest path a straight line?

The calculus of variations occurs in other contexts, some of which are more intuitive. The classic example is to nd the shortest path between two points in the plane. The length ‘ of a path y(x) from (x₁,y₁)to(x₂,y₂) is given⁵ by

and the path is a straight line.

2.4 Conserved Quantities 

2.4.1 Ignorable Coordinates 

If the Lagrangian does not depend on one coordinate, say q_k , then we say it is an ignorable coordinate. Of course, we still want to solve for it, as its derivative may still enter the Lagrangian and effect the evolution of other coordinates. By Lagrange’s equation

so if in general we de ne

as the generalized momentum, then in the case that L is independent of q_k , P_k is conserved, dP_k /dt =0.

Linear Momentum 

As a very elementary example, consider a particle under a force given by a potential which depends only on y and z, but not x. Then

is independent of x, x is an ignorable coordinate and

is conserved. This is no surprize, of course, because the force is and

Note that, using the definition of the generalized momenta

Lagrange’s equation can be written as

Only the last term enters the de nition of the generalized force, so if the kinetic energy depends on the coordinates, as will often be the case, it is not true that  dP_k =dt = Q_k . In that sense we might say that the generalized momentum and the generalized force have not been de ned consistently.

Angular Momentum

As a second example of a system with an ignorable coordinate, consider an axially symmetric system described with inertial polar coordinates (r, θ, z ), with z along the symmetry axis. Extending the form of the kinetic energy we found in sec (1.3.4) to include the z coordinate, we haveThe potential is independent of θ, because otherwise the system would not be symmetric about the z-axis, so the Lagrangian

does not depend on θ, which is therefore an ignorable coordinate, and

We see that the conserved momentum P_θ is in fact the z-component of the angular momentum, and is conserved because the axially symmetric potential can exert no torque in the z -direction: 

Finally, consider a particle in a spherically symmetric potential in spherical coordinates. In section (3.1.2) we will show that the kinetic energy in

Again, φ is an ignorable coordinate and the conjugate momentum  is conserved. Note, however, that even though the potential is independent of θ as well, θ does appear undifferentiated in the Lagrangian, and it is not an ignorable coordinate, nor is P_θ conserved.

If qj is an ignorable coordinate, not appearing undifferentiated in the Lagrangian, any possible motion q_j (t) is related to a different tra jectory in the sense that they have the same action, and if one is an extremal path, so will the other be. Thus there is a symmetry of the system under q_j → q_j + c, a continuous symmetry in the sense that c can take on any value. As we shall see in Section 8.3, such symmetries generally lead to conserved quantities. The symmetries can be less transparent than an ignorable coordinate, however, as in the case just considered, of angular momentum for a spherically symmetric potential, in which the conservation of L_z follows from an ignorable coordinate φ, but the conservation of L_x and L_y follow from symmetry under rotation about the x and y axes respectively, and these are less apparent in the form of the Lagrangian.

2.4.2 Energy Conservation 

We may ask what happens to the Lagrangian along the path of the motion.

In the first term the first factor is

 by the equations of motion, so

We expect energy conservation when the potential is time invariant and there is not time dependence in the constraints, i.e. when , so we rewrite this in terms of

Then for the actual motion of the system,

If , H is conserved.

H is essentially the Hamiltonian, although strictly speaking that name is reserved for the function H (q, p, t) on extended phase space rather than the function with arguments . What is H physically? In the case of Newtonian mechanics with a potential function, L is an inhomogeneous quadratic function of the velocities . If we write the Lagrangian L = L₂ + L₁ + L₀ as a sum of pieces purely quadratic, purely linear, and independent of the velocities respectively, then

is an operator which multiplies each term by its order in velocities,

and

H = L₂ − L₀,
For a system of particles described by their cartesian coordinates, L₂ is just the kinetic energy T , while L₀ is the negative of the potential energy L₀ = −U ,so H = T + U is the ordinary energy. There are, however, constrained systems, such as the bead on a spoke of Section 2.2.1, for which the Hamiltonian is conserved but is not the ordinary energy.

Hamilton’s Equations 

We have written the Lagrangian as a function of q_i, , and t, soitisa function of N + N + 1 variables. For a free particle we can write the kinetic energy either as More generally, we can⁷ reexpress the dynamics in terms of the 2N + 1 variables q_k , P_k , and t.

The motion of the system sweeps out a path in the space ora path in (q, P, t). Along this line, the variation of L is

where for the rst term we used the de nition of the generalized momentum and in the second we have used the equations of motionThen examining the change in the Hamiltonian along this actual

2.5. HAMILTON’S EQUATIONS

Note this is just the sum of the kinetic and potential energies, or the total energy.

Hamilton’s equations give

These two equations verify the usual connection of the momentum and velocity and give Newton’s second law.
The identication of H with the total energy is more general than our particular example. If T is purely quadratic in velocities, we can write T =

so we see that the Hamiltonian is indeed the total energy under these circumstances.