Lagrangian Mechanics

The Principle of Stationary Action

The principle that particles follow the path of least action in mechanical systems, known as Hamilton’s principle or the principle of stationary action (sometimes also inaccurately referred to as the principle of least action), is a fundamental idea in classical mechanics. The “action” refers to a specific quantity (called the action integral) that depends on the path a particle takes through space and time. The principle states that out of all possible paths a particle could take between two points, the actual path is the one that minimizes (or more generally, extremizes) the action. The principle of stationary action was first developed in the 18th century by Pierre-Louis Moreau de Maupertuis and later refined by Euler, Lagrange, and Hamilton. The principle of stationary action encapsulates Newton’s laws of motion in a more general and elegant form. In nature, systems tend to follow paths that extremize (minimize or maximize) certain quantities. For example, light follows the path of extremal time (Fermat’s principle).

Mechanical systems tend to follow the path that extremizes the action, which can be understood as an optimization principle in nature. Just as systems naturally move toward states of lower energy, the action minimizes in a way that balances kinetic and potential energy. This balance ensures that the system evolves in the most “economical” way, subject to the constraints of the physical forces acting on it. In this sense, the path of least action, for example, represents the most efficient way for the system to move from one state to another. The principle of stationary action provides a unifying framework for understanding the dynamics of systems and plays a crucial role in all modern physical theories.

Action and Lagrangian Mechanics

In classical mechanics, the action is defined as the integral of the Lagrangian over time. The Lagrangian is the difference between the kinetic energy and potential energy of the system:

where:

• is the Lagrangian (always a scalar function),
• represents the generalized coordinates of the system (such as position),
• is the generalized velocity (the time derivative of ).

Generalized positions and velocities can be vectors. In systems with multiple particles, more than one generalized coordinate vector may be needed, with the exact number determined by the constraints.

The principle of least action states that the actual path taken by the particle minimizes (or extremizes) the action . The action is a functional — a “function of functions.” When demand the action be stationary under variations in the paths , i.e., , we arrive - after some algebra - at the Euler-Lagrage equations.

Euler–Lagrange Equations

The principle of least action is mathematically related to the Euler–Lagrange equations, which describe the equations of motion for a system.

These equations are equivalent to Newton’s second law of motion. Thus, the path of least action corresponds to the same path that satisfies the classical equations of motion derived from Newton’s laws.

Example: Marble in a Hemispherical Bowl

A marble of mass is rolling in a hemispherical bowl of radius . The bowl is stationary, and friction is negligible. The bead is free to move in all directions and it is subject to gravity, but it is small enough to be treated as a point mass. Its position from the center of the (hemi)sphere that constrains the motion is denoted by the variables .

Step 1: Determine Constraints

As long as the bead remains in the bowl, this system is equivalent to the spherical pendulum. Our constraint reads:

as long as , where is the height of the bowl. In this case . Strictly speaking, this constitutes a non-holonomic constraint, but if we separate cases between and and focus on the former, we can use the standard methodology.

Step 2: Find Generalized Coordinates

Spherical coordinates with polar angle and azimuth angle are a natural choice for the generalized coordiantes and :

Note that the distance of the marble to the center is fixed while in the bowl, i.e. . Hence, we do not need to consider as a generalized coordinate since its evolution is already known. For the next step it is often convenient to express x,y,z coordinates of the marble in generalized coordinates

Step 3: Calculate Kinetic and Potential Energy

First, compute the time derivatives to obtain velocities:

The squared velocity reads

Thus, the kinetic and potential energies are

and

Step 4: Euler–Lagrange Equations

Applying the Euler–Lagrange equations to derive equations of motion from the Lagrangian. Since we have two generalized coordinates, we end up with two Euler-Lagrange equations.

Inserting the Lagrangian we obtain two ordinatry differential equations

Applying the differential operators leads to the final equations of motion

Final Equations of Motion

The resulting equations of motion in our generalized coordinates and are

Note, that the equations of motion are coupled and non-linear in the generalized coordinates!

What if the Marble Leaves the Bowl?

Starting the marble with sufficient initial velocity can make it leave the bowl. If that happens, the constraint no longer holds, but in our model the marble is then only subject to gravity. Since is no longer constant, spherical coordinates offer no advantage. The equations of motion become:

These correspond to simple projectile motion. The solution will depend on the initial conditions when the marble leaves the bowl.

Generalized Forces

In many mechanical systems, external forces (such as friction, tension, or control inputs) cannot be easily expressed in from of a potential energy function. To include such non-conservative forces in Lagrangian mechanics, we introduce the concept of generalized forces. For a system with generalized coordinates , the generalized force associated with each coordinate appears on the right-hand side of the extended Euler-Lagrange equation:

If a system is subjected to external forces acting at points with positions , then the generalized force is

This formula projects the applied force onto the direction of virtual displacement associated with the coordinate . In other words, represents more or less how much work each applied force does along the direction of the generalized coordinate . If, on the other hand, is conservative (i.e., ), then , and the force can be absorbed into the potential .

Example: Damped Pendulum

Consider a simple pendulum of length , mass , and angular coordinate . Suppose we add viscous damping (e.g., air resistance) proportional to the velocity of the pendulum ().

The Lagrangian without damping reads

Since damping is not conservative we need first calculate

and then add the generalized force to the Euler-Lagrange equation:

This is the equation of motion for the damped pendulum, incorporating both conservative and non-conservative effects.