5 Hamilton’s Principle and Lagrange’s Equation

Ashok Goyal

1. Introduction:

Non-relativistic dynamics in an inertial frame are described by Newton’s equation  . In actual physical situation the dynamical system in general, is constrained by a prior unknown forces. A particle may be constrained to move on a given surface, the motion may be restricted within certain boundaries and so on. The constraint forces may be quite complicated in which case one may have to look for alternative formalism. This alternative formalism of course, cannot go beyond Newton’s Laws but may result in the simplification of the problem and may even have wider application. Historically a minimum principle based on the notion that nature always act in a way that during the development of a dynamical system, certain quantities are minimized, has been used in an alternative formulation of mechanics. We will discuss in this unit a very powerful principle due to Hamilton for the formulation of what is known Hamiltonian dynamics.

2. Hamilton’s Principle.

The principle states that “of all the possible paths consistent with the constraints along which a dynamical system can evolve from one point to another, the actual path followed is the one which minimizes the action”. Where action is defined as the time integral of the Langrangian of the system i.e.

Lagrange’s Equation of motion

Consider a multiparticle system characterized by a Lagrangian function.

Where  and  are the sets of generalized coordinate and velocities. The action for this Lagrangian is given as

Hamilton’s Principle states that of the various paths given by

The path followed by the system is the one for which the corresponding action is minimum.Consider the path  specified by the set  and a nearby path  characterized by the set

The change  in action is then given by

The Lagrangian as defined is not unique, we can add to it a term which is a total time derivative of any function  with no change in the equations of motion. The new action we get is

The last term being constant has no effect on    and therefore, on Lagrange’ equations.

Lagrange’s Equation with Undetermined Multipliers:

In the above derivation we had assumed that the constraints are holonomic and can be expressed in terms of algebraic relations. If same of the constraints are expressed in terms of velocities and are in the form of non-integrable equations like

For arbitrary ’s, ’s and ’s , it is possible to incorporate them in the Lagrange’s eqations by means of the Lagrangian undermined multipliers. Take for example, non-holonomic constraints of the force

For constraints expressible as

Where ’s are the undetermined multipliers.

1. Solved Examples:

1.Disc rolling down an inclined plane

A disc of mass M and radius R is rolling down an inclined planed without slipping write down the Lagrangian of the system and obtain the equation of motion.

The kinetic energy of the disc is the sum of translational and rotational energy

(Note that if instead of disc, we had a cylinder, a sphere, a ring or a spherical shell, the acceleration can be obtained by putting the corresponding expression for the moment of inertia).

2. Double Pendulum:

For a double pendulum consisting of the masses  and  tied by string of length  and  supported at a point in the horizontal plane, obtain the Lagrangian and the equations of motion. Solve them for small amplitude.

Ans.: Set up the coordinate system as shown

The coordinate of the pendulum can be expressed in terms of angles  and  as

The caxpled set of differential equations can be solved by assuming the solution to be

Thus we get the two frequencies.

3. Particle on a cycloid :

A particle is moving on a cycloid ,  under the action of gravity set up the Lagrange’s equation of motion and find the frequency for Solution: An infinitesimal distance element on the surface of the cycloid is

4. Summary 

• Principle of least action is a very powerful method used for an alternative formulation of mechanics.
• Hamilton’s Principle states that dynamical system evolves along a path that minimizes.