Associate Professor

Mathematics Department

D.B.S. (P.G.) College

 Dehradun, Uttarakhand, India  

Abstract

In modern physics, the Hamiltonian plays a very important role. Apart from being a powerful tool for solving classical-mechanical problems, it is crucial for bridging classical and quantum mechanics. This chapter considers dynamical systems described by Hamilton’s equations. It deals with the development of the explicit equations of motion for such systems when constraints are imposed on them. Such explicit equations do not appear to have been obtained hereto. The holonomic or non- holonomic constraints imposed can be non-linear functions of canonical momenta, the coordinates, and time, and they can be functionally dependent. These explicit equations of motion for constrained system are obtained through the development of the connection between the Lagrangian concept of virtual displacements and Hamiltonian dynamics. A simple three step approach for obtaining the explicit equations of motion of constrained Hamiltonian system is presented.

Keywords: Constrained, Hamiltonian system, explicit equations of motion, canonical equations

Introduction:

Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces(generalized) velocities  used in Lagrangian mechanics with (generalized) momenta. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry(notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics. In reality, most of the classical system are holonomic conservative system for which potential energy is only function of generalised coordinates but not generalised velocity. For system with this kind of potential energy, Hamilton’s canonical equations are very usual and with these equations one can show that for N particle holonomic system, if the instantaneous position of any constituent particle is not explicit function of time then Hamiltonian of that system i.e. the total energy of that system will be conserved. In this mathematical study I have tried to construct the form of Hamilton’s canonical equations for the classical system with generalised potential energy.

Hamilton’s Equations:

Hamiltonian: The Hamiltonian is a function of momentum and position. For a single particle the one-dimensional Hamiltonian is defined as:

where p stands for the generalized coordinate. Note that the Lagrangian is a function of position, velocity, and time, L= L (q,q,t). The Hamiltonian is a function of momentum, position, and time,    

H = H(p,q,t). If you are asked to determine a Hamiltonian, make sure your final expression does not  explicitly contain any velocities.

For more than one dimension and /or more than one particle the Hamiltonian is :

More explicitly, for a system of N particles in three dimensions,

Having obtained the Hamiltonian you can use it to obtain the equations of motion of the system. Recall that the equation of motion as expressed by Newton’s second law is a second order differential equation for the position. On the other hand the Hamiltonian formulation yields two first order equations for the momentum and position. They are:

These are Hamilton’s equations of motion and they are completely equivalent to Newton’s second law and to the Lagrange equations of motion.

There are a number of different ways to drive Hamilton’s equations. One way is to start with equation (2) and write the differential of H as

Using the definition of generalized momentum to cancel the first and last term in parentheses, and using Lagrange’s equation to write

 we obtain

But the differential of H as obtained from

is

 A term -by-term comparison of the two expressions for  yields Equations (3) as well as

For the sake of a specific example, consider the Hamiltonian of a particle of mass m moving vertically in a uniform gravitational field g. (For example, you might want to determine the Hamiltonian and Hamilton’s equations for a ball of mass m thrown vertically upward.) Begin with the Lagrangian. It is

By the definition of generalized momentum,

Therefore,

Equation (1) then gives

Note that I was careful to replace  by p/m. inserting this Hamiltonian into Hamilton’s equations yields the following equations of motion:

 The second equation reads , which is just the definition of momentum. The first equation states that

That is, , as expected.

Now

and

V = mgz

so

Therefore, in this case, H = E. in fact, the Hamiltonian is nearly always the total energy expressed in terms of generalized momentum and position. Students frequently assume that H is always equal to E. this is not true. The following conditions determine when H is constant and when it is equal to E.

1. If the Lagrangian does not depend explicitly on time, the Hamiltonian is constant but not necessarily equal to the energy.

2. If the transformation equations do not depend on time and the potential energy is conservative (that is, the work is independent of the path), then the Hamiltonian is equal to the total energy but it may not be constant.

3. If the constraints and the transformation equations are time independent and the potential energy is conservative and time independent, and does not depend on velocity, then the Hamiltonian is equal to the total energy and is constant.

The third condition is satisfied by many systems, and frequently when requested to write the Hamiltonian (especially in quantum mechanical problems) a physics student simply write H = p² / 2m +V. However, this can be dangerous because there are problems in which the Hamiltonian is not equal to the total energy.

Students are often asked to determine the Hamiltonian for a particular system. A common error is to write an expression for H that involves generalized velocities. This is not the way a Hamiltonian should be written. The final expression for H must only contain momenta and positions (and possibly the time) because H = H(p,q,t).

Finally, I would like to mention that if you transform to a new coordinate system, you will have a new set of generalized coordinates, say P_i and Q_i. If these coordinates maintain the form of Hamilton’s equations unchanged, then they are called “canonical conjugates”.

Conclusion: Hamilton’s canonical equations for velocity dependent potential energy are constructed which will be very helpful for analysis for similar classical system.

References:

1.  R. G. Takwale and P. S. Puranik, ‘Introduction to Classical Mechanics’, Tata McGraw – Hill Publishing Company Limited, Ch. 8, 226-227 (2006)

2.  H. Goldstien ‘Classical Mechanics’, Narosa Publishing House, Ch. 1, 20-21 (1986)

3.  H. Goldstien ‘Classical Mechanics’, Narosa Publishing House, Ch. 1, 22-23 (1986)

4.  J. C. Upadhyaya ‘Classical Mechanics’, Himalaya Publishing House, Ch. 11, 324-325 (2009)

5. M. D.Semon and G. M. Schmieg, “Note on the analogy between inertial and electromagnetic forces”, Am. J. Phys. 49, 689-690 (1981)

6.  F.E. Udwadia, R. E. Kalaba,: A new perspective on constrained motion. Proc. R. Soc. Lond. Ser. A 439, 407-410 (1992)

7.  F.E. Udwadia, A.D. Schutte,: Equations of motion for general constrained systems in Lagrangian mechanics. Acta Mech. 213, 111-129 (2010)