General Equation of Dynamics

The general equation of dynamics combines D'Alembert's principle with the principle of virtual work. It provides equations of motion for systems with ideal constraints without explicitly introducing their reactions.

Starting idea

By D'Alembert's principle, inertia forces $\vec F_i^{in}=-m_i\vec a_i$ are added to active forces and constraint reactions. The principle of virtual work is then applied to the formally balanced system.

General equation

For a system with ideal constraints:

$$\sum_i(\vec F_i^{a}-m_i\vec a_i)\cdot\delta\vec r_i=0.$$

Reactions of ideal constraints do not appear because their total virtual work is zero.

Physical meaning

The equation does not mean that the system is in static equilibrium. The inertial terms represent actual accelerations, while virtual displacements are a mathematical device for eliminating constraint reactions.

Generalized coordinates

If a system has $s$ degrees of freedom described by $q_j$, admissible displacements can be expressed through independent variations $\delta q_j$. Grouping coefficients of each $\delta q_j$ yields $s$ independent equations of motion.

Advantages

The general equation is particularly useful for constrained systems in which direct application of Newton's laws introduces many unknown reactions.

Connection with Lagrange's equations

Passing to independent generalized coordinates and expressing the inertial terms through kinetic energy leads to Lagrange's equations of the second kind.

Example

For a one-degree-of-freedom system, all admissible $\delta\vec r_i$ are expressed through one $\delta q$. Substitution gives $B(q,\dot q,\ddot q,t)\delta q=0$. Since $\delta q$ is arbitrary, $B=0$ is the equation of motion.

Common mistakes

• retaining reactions of ideal constraints without using their zero virtual work;
• treating inertia forces as ordinary active forces;
• using dependent coordinate variations as if they were independent;
• confusing the general equation of dynamics with static equilibrium.