# 10: Nonconservative Systems


• 10.1: Introduction to Nonconservative Systems
A conservative force has the property that the total work done moving between two points is independent of the taken path. That is, a conservative force is time symmetric and can be expressed in terms of the gradient of a scalar potential V. The focus of this chapter is to discuss the origins of nonconservative motion and how it can be handled in algebraic mechanics.
• 10.2: Origins of Nonconservative Motion
Nonconservative degrees of freedom involve irreversible processes, such as dissipation, damping, and also can result from course-graining, or ignoring coupling to active degrees of freedom.
• 10.3: Algebraic Mechanics for Nonconservative Systems
Since Lagrangian and Hamiltonian formulations are invalid for the nonconservative degrees of freedom, there are three primary approaches used to include nonconservative degrees of freedom directly in the Lagrangian and Hamiltonian formulations of mechanics.
• 10.4: Rayleigh’s Dissipation Function
As mentioned above, nonconservative systems involving viscous or frictional dissipation, typically result from weak thermal interactions with many nearby atoms, making it impractical to include a complete set of active degrees of freedom. In addition, dissipative systems usually involve complicated dependences on the velocity and surface properties that are best handled by including the dissipative drag force explicitly as a generalized drag force in the Euler-Lagrange equations.
• 10.5: Dissipative Lagrangians
Dissipative Lagrangians can be proposed that introduce non-conservative forces.
• 10.S: Nonconservative systems (Summary)
Dissipative drag forces are non-conservative and usually are velocity dependent. The motion of non-linear dissipative dynamical systems can be highly sensitive to the initial conditions and can lead to chaotic motion.

Thumbnail: This low-pressure system over Iceland spins counterclockwise due to balance between the Coriolis force and the pressure gradient force. (Public Domain; NASA’s Aqua/MODIS satellite).

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