Notes to a video lecture on UNIZOR.COM

Lagrangian for
Nonconservative System

All the advantages of Lagrangian Mechanics are best visible in its approach to the analysis of conservative mechanical system, where the forces acting on system's components are conservative and, as such, can be represented as gradients of a potential - a scalar field with values at each point in space depending only on the location.

Attempts to extend this approach to nonconservative systems encounter technical difficulties but are nevertheless worthwhile.
This lecture exemplifies this case.

Assume a simple system with forces depending not only on position but on velocity of an object as well.
Let a stone of mass m fall vertically into the water with no initial velocity.
The force of gravity will pull it down, but the water exerts a resistive force opposite to the direction of motion.
Assume the resistive force is proportional to the stone's speed v. Then the drag force has magnitude k·v, where k is the coefficient of resistance, and it always acts opposite to the direction of motion.


Our first task is to analyze the movement using Newtonian Second law.
Newton's Second Law produces the following differential equation of motion m·ẍ(t) = m·g − k·ẋ(t)
that can be simplified to a standard linear form
ẍ(t) + (k/m)·ẋ(t) − g = 0

The solution to this equation with initial conditions
x(0)=0 and x'(0)=0
is
x(t) =
= (mg/k)·t+(m²g/k²)·(e^−kt/m−1)
Condition x(0)=0 is satisfied.
The speed (time-derivative) is
x'(t) = (mg/k)·(1−e^−kt/m)
Condition x'(0)=0 is satisfied.
The acceleration is
x"(t) = g·e^−kt/m
Newton's differential equation of motion above is satisfied.

Two important conclusions:
1. Since the positive direction is downward, the acceleration remains positive but decreases exponentially toward zero.
2. Therefore, the speed is always increasing but asymptotically approaches its maximum mg/k.

Now we will try to approach this task from the Lagrangian viewpoint.
A simple Lagrangian in terms of a difference between kinetic and potential energy is not a good choice since the force of resistance to a movement is dependent on velocity and, therefore, is not conservative.

However, one possible Lagrangian, first proposed by Bateman and later developed by Caldirola and Kanai, is obtained as follows.
Consider again Newton's Second law equation for our system
ẍ(t) + (k/m)·ẋ(t) − g = 0
Suppose there exists a Lagrangian L(x(t),ẋ(t),t) suitable for this nonconservative system.
Then its Euler Lagrange equation
d/dt [∂L/∂ẋ] = ∂L/∂x
must be equivalent to Newton's Second equation above.
In other words, both equations must be transformable into each other.

1. Multiply Newton's Second law equation by e^kt/m.
e^kt/m[ẍ(t)+(k/m)·ẋ(t)−g] = 0
or
e^kt/m[ẍ(t)+(k/m)·ẋ(t)] = g·e^kt/m
2. Notice that
e^kt/m[ẍ(t)+(k/m)·ẋ(t)] =
= d/dt [e^kt/m·ẋ(t)]
3. Notice that an expression in [...] on the right side of the equation above can be written as a derivative by ẋ
e^kt/m·ẋ = ∂/∂ẋ[e^kt/m·½ẋ²]
4. Therefore, we have an equation
d/dt ∂/∂ẋ[e^kt/m·½ẋ²] = g·e^kt/m
5. The right side of the equation in item 4 equals to
∂/∂x [g·x·e^kt/m]
6. Our transformed Newton's Second law equation looks now as
d/dt ∂/∂ẋ[e^kt/m·½ẋ²] =
=∂/∂x [g·x·e^kt/m]
7. Adding to the expression in [...] on the left side of the equation any function of x won't change the partial derivative by ẋ. Adding to the expression in [...] on the right side of the equation any function of ẋ won't change the partial derivative by x.
Hence, we may replace both bracketed expressions by the single function
e^kt/m·½ẋ² + g·x·e^kt/m
8. Multiplying the function by the constant m and writing it in a more compact form does not change the Euler-Lagrange equation.
Hence, both expressions in [...] can be replaced by a function
L = e^kt/m·(½m·ẋ² + mg·x)
and we have the Euler-Lagrange equation
d/dt ∂L/∂ẋ = ∂L/∂x

Surprisingly or not, the Lagrangian in this case differs from the Lagrangian for a conservative system only by a multiplier e^kt/m since
½m·ẋ² is kinetic energy T of a falling stone,
−mg·x is its potential energy U and
½m·ẋ² + mg·x = T − U
and the Lagrangian for this nonconservative system is
L = e^kt/m·(T−U)

Unlike the Lagrangian of a conservative system, this Lagrangian depends explicitly on time through the exponential factor e^kt/m.
This explicit dependence reflects the continuous loss of mechanical energy caused by the resistive force. Therefore, although the motion can still be described by Euler-Lagrange equations, many of the elegant conservation properties of conservative systems no longer hold.