Direct Energy, 2018a

248 11.3 Principle of Least Action
The Lagrangian provides a ton of information about an energy conversion
process. If we can describe the dierence between two forms of energy
by a Lagrangian L ( )
t, y, dy
dt , we can set up the Euler-Lagrange equation.
From the Euler-Lagrange equation, we may be able to nd 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 nd 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 Eq. 11.8.
Since y(t) depends on only one independent variable, there is no need
to use partial derivatives in expressing dy
dy
. The derivative is written in
dt dt
shorthand notation as ẏ, and ÿ may be used in place of d2 y
. The Lagrangian
dt 2
L depends on three independent-like variables: t, y, and dy . Thus, the
dt
partial derivative symbols are used to indicate which partial derivative of
L is being considered.
The rst term of the Euler-Lagrange equation, ∂L , is the generalized potential
dened above. The units of the generalized potential are joules over
∂y
units of path, J
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 m J 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 J C which
is volts. The Euler-Lagrange equation in this case is a conservation relationship
about voltages.