11.2: Principle of Least Action

Define the action \(\mathbb{S}\) as the magnitude of the integral of the Lagrangian along the path.

\[\mathbb{S}= \left| \int_{t_{0}}^{t_{1}} \mathcal{L}\left(t, y, \frac{d y}{d t}\right) dt \right| \nonumber \]

Assuming the independent variable \(t\) represents time in seconds, the action will have the units joule seconds. For energy conversion processes, the path found in nature experimentally is the path that minimizes the action. This idea is known as the Principle of Least Action or sometimes as Hamilton's principle [163, p. 11]. The idea of conservation of energy is contained in this principle.

To find a minimum or maximum of a function, find where the derivative of the function is zero. Here, \(\mathcal{L}\) and \(H\) are not quite functions. Instead, they are functionals. A function takes a scalar quantity as an input and returns a scalar quantity. A functional takes a function as an input and returns a scalar quantity. Both \(\mathcal{L}\) and \(H\) take the function \(y(t)\) as input and return a scalar quantity in joules. The idea of taking a derivative and setting it to zero to find a minimum is still useful, but we have to take the derivative with respect to the function \(y(t)\). The process of finding the maximum or minimum of a functional described by an integral relationship is known as calculus of variations.

It is often easier to work with differential relationships than integral relationships. We can express the Principle of Least Action as differential equation, and it is called the Euler-Lagrange equation.

\[\frac{\partial \mathcal{L}}{\partial y}-\frac{d}{d t} \frac{\partial \mathcal{L}}{\partial\left(\frac{d y}{d t}\right)}=0 \label{11.2.2} \]

If the Lagrangian \(\mathcal{L}\) is known, we can simplify the Euler-Lagrange equation to an equation involving only the unknown path. The resulting equation in terms of path \(y(t)\) is called the equation of motion.

The Lagrangian provides a ton of information about an energy conversion process. If we can describe the difference between two forms of energy by a Lagrangian \(\mathcal{L} \left(t, y, \frac{dy}{dt} \right)\), we can set up the Euler-Lagrange equation. From the Euler-Lagrange equation, we may be able to find the equation of motion and solve it. The resulting path minimizes the action and describes how the energy conversion process evolves with time. We can find the generalized potential of the system as a function of time too. The Euler-Lagrange equation is a conservation law for the generalized potential. The symmetries of the equation of motion may lead to further conservation laws and invariants. These last two ideas, and the math behind them, are often known as Noether's theorem. Noether's theorem says that there is a very close relationship between symmetries of either the path or the equation of motion and conservation laws [165] [166]. These ideas are discussed further in Sec. 14.5.

Notice the mix of partial and total derivative symbols in Equation \ref{11.2.2}. Since \(y(t)\) depends on only one independent variable, there is no need to use partial derivatives in expressing \(\frac{dy}{dt}\). The derivative \(\frac{dy}{dt}\) is written in shorthand notation as \(\dot y\), and \(\ddot y\) may be used in place of \(\frac{d^2y}{dt^2}\). The Lagrangian \(\mathcal{L}\) depends on three independent-like variables: \(t\), \(y\), and \(\frac{dy}{dt}\). Thus, the partial derivative symbols are used to indicate which partial derivative of \(\mathcal{L}\) is being considered.

The first term of the Euler-Lagrange equation, \(\frac{\partial \mathcal{L}}{\partial y}\), is the generalized potential defined above. The units of the generalized potential are joules over units of path, \(\frac{J}{\text{units of path}}\). Each term of the Euler-Lagrange equation has these units. For example, if \(y(t)\) is in the units of meters, the generalized potential is in \(\frac{J}{m}\) or newtons. Each term of the Euler-Lagrange equation represents a force, and the Euler-Lagrange equation is a conservation relationship about forces. As another example, if the path \(y(t)\) represents charge in coulombs, then the generalized potential has the units \(\frac{J}{C}\) which is volts. The Euler-Lagrange equation in this case is a conservation relationship about voltages.