Hamilton's Principle and Lagrangians: An Alternative Method to Newtonian Mechanics

Hamilton’s Principle provides an alternate method to Newtonian mechanics when solving problems and determining a system’s equations of motion.  The idea behind this method is to determine the system’s path of motion, otherwise known as its “action”, which minimizes the difference between its kinetic and potential energies.  To do this, we begin by formulating equations defining the kinetic energy T and potential energy U for each degree of freedom of the system.  Next, we formulate our Lagrangian equation L using the relationship

Once our Lagrangian is established, we can apply the Euler-Lagrange equation

to determine our equation of motion with minimized action.  In this equation, x_i represents the function with respect to time of one of the system’s degrees of freedom, λ is essentially the force of constraint of our system, and g_i is the function establishing our system’s constraint equations.  Solving the resulting system of equations after applying the Euler-Lagrange equation will provide the equation of motion for the system at hand.

Example:

(Ex 7.10 from Classical Dynamics of Particles and Systems by Thornton and Marion)

Consider a particle of mass m starting at rest on top of a smooth, fixed hemisphere of radius a.  Find the force of constraint and determine the angle at which the particle leaves the hemisphere.

Figure 1: This is the system.

Image Credit: Classical Dynamics of Particles and Systems by Thornton and Marion

To begin, we will find the constraint equation.  As the surface of the hemisphere can be described as

our constraint equation is

Next, we must formulate our kinetic and potential energies and establish our Lagrangian equation.  To determine our kinetic energy T, we must consider the kinetic energy of both degrees of freedom: r and θ.  Thus,

The potential energy of the system is dependent on the height of the mass and can be represented by

Thus, our Lagrangian is

Now, to determine our equations of motion with minimized action, we must apply the Euler-Lagrange equation for each degree of freedom.  Applying this process for the r degree of freedom, we see

and

Thus, the Euler-Lagrange equation for r is

Likewise, when applying this process for θ, we see

and

Thus, the Euler-Lagrange equation for θ is

Finally, we can determine our force of constraint by solving this system of equations, including the two Euler-Langrange equations and the equation of constraint, for λ.  This provides us with the relation

As the mass will leave the hemisphere at the angle θ in which the force of constraint λ becomes 0, we determine that the particle will depart at

The approximate point of departure is illustrated on the figure below.

Figure 2:  The red dot represents the particle approximately at the angle of departure from the hemisphere.