Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

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2.3 Hamilton’s Principle

The configuration of a system at any moment is speci ed by the value of the generalized coordinates q(t), and the space coordinatized by these q1,......,qN 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 qi(t).
One can imagine the system taking many paths, whether they obey Newton’s Laws or not. We consider only paths for which the qi(t) are differentiable. Along any such path, we de ne the action as

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev(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 Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRevis 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 xi, the first-order variation of the function is Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev so unless  Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev   for all i, there is some variation of the {xi} 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 Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev  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 Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev  Then the functional S(f) given by

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

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  Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

It is not really necessary to be so rigorous, however. Under a changeLagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

functional S will vary by

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

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

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev(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 .

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

We make no assumptions about this path other than that it is differentiable and meets the boundary conditions Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

The action is

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

The fourth term can be integrated by parts,

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

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

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

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, requiringLagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRevLagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRevat 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

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

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 qk , 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

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

so if in general we de ne

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

as the generalized momentum, then in the case that L is independent of qk , Pk is conserved, dPk /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

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

is independent of x, x is an ignorable coordinate and

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

is conserved. This is no surprize, of course, because the force is Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRevandLagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

Note that, using the definition of the generalized momenta

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

Lagrange’s equation can be written as

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

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  dPk =dt = Q. 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 haveLagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRevLagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRevThe potential is independent of θ, because otherwise the system would not be symmetric about the z-axis, so the Lagrangian

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

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

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

We see that the conserved momentum Pθ 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: 

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

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

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

Again, φ is an ignorable coordinate and the conjugate momentum  Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRevis 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 Pθ conserved.

If qj is an ignorable coordinate, not appearing undifferentiated in the Lagrangian, any possible motion qj (t) is related to a different tra jectoryLagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev 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 qj → q+ 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 Lz follows from an ignorable coordinate φ, but the conservation of Lx and Ly 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.

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

In the first term the first factor is

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

 by the equations of motion, so

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

We expect energy conservation when the potential is time invariant and there is not time dependence in the constraints, i.e. when Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev, so we rewrite this in terms of

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

Then for the actual motion of the system,

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

If Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev, 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 Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev. What is H physically? In the case of Newtonian mechanics with a potential function, L is an inhomogeneous quadratic function of the velocities Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev. 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

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

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

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

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 qi, Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev, and t, soitisa function of N + N + 1 variables. For a free particle we can write the kinetic energy either as Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRevMore generally, we can7 reexpress the dynamics in terms of the 2N + 1 variables qk , P, and t.

The motion of the system sweeps out a path in the space Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRevora path in (q, P, t). Along this line, the variation of L is

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

where for the rst term we used the de nition of the generalized momentum and in the second we have used the equations of motionLagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRevThen examining the change in the Hamiltonian Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRevalong this actual

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev


2.5. HAMILTON’S EQUATIONS

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

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

Hamilton’s equations give

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

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 =

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev  

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

Lagrangian and Hamiltonian Formalism and equations of motion (Part - 2) - CSIR-NET Physical Sciences Physics Notes | EduRev

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

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