Conservative system
Consider a mechanical system moving under forces. The system is conservative if the total energy of the system stays constant as the system moves.
More precisely, for a system with total energy
\[E = U + K\]where $U$ is the potential energy and $K$ is the kinetic energy, the value of $E$ is constant over time:
\[\frac{d}{dt} E = 0.\]Consider a mechanical system with generalized coordinates $q = (q_1, \dots, q_n)$. The system is conservative if every force on the system comes from a potential energy $V(q)$ that depends only on the positions:
\[F = -\nabla V(q).\]A dynamical system $\dot x = f(x)$ is called conservative if there exists a real-valued function $x \mapsto V(x)$ such that
\[f(x) = -\nabla V(x)\]Examples
Examples
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A mass $m$ on a frictionless spring with stiffness $k$, at position $q$
Undergraduate
A mass on a frictionless spring is a conservative system, because the spring force $-k q$ is the negative derivative of the potential energy $V(q) = \frac{1}{2} k q^2$.
Properties
- In a conservative system, the total energy $E = U + K$, the potential energy plus the kinetic energy, is constant along every motion.
- Friction, air resistance, and other damping forces make a system non-conservative, because these forces turn mechanical energy into heat.
- A conservative system has the Lagrangian $L = T - V$.
Notes
In dynamical systems, a system $\dot{x} = f(x)$ is sometimes called conservative if the flow preserves volume, which is a separate definition of the word.