## 13.4.2 General Lagrangian Expressions

As more complicated mechanics problems are considered, it is convenient to express the differential constraints in a general form. For example, evaluating (13.130) for a kinematic chain of bodies leads to very complicated expressions. The terms of these expressions, however, can be organized into standard forms that appear simpler and give some intuitive meanings to the components.

Suppose that the kinetic energy is expressed using (13.126), and let denote an entry of . Suppose that the potential energy is . By performing the derivatives expressed in (13.136), the Euler-Lagrange equation can be expressed as scalar equations of the form (13.140)

in which (13.141)

There is one equation for each from to . The components of (13.140) have physical interpretations. The coefficients represent the inertia with respect to . The represent the affect on of accelerating . The terms represent the centrifugal effect induced on by the velocity of . The terms represent the Coriolis effect induced on by the velocities of and . The term usually arises from gravity.

An alternative to (13.140) is often given in terms of matrices. It can be shown that the Euler-Lagrange equation reduces to (13.142)

which represents scalar equations. This introduces , which is an Coriolis matrix. It turns out that many possible Coriolis matrices may produce equivalent different constraints. With respect to (13.140), the Coriolis matrix must be chosen so that (13.143)

Using (13.141), (13.144)

A standard way to determine is by computing Christoffel symbols. By subtracting from the inside of the nested sums in (13.144), the equation can be rewritten as (13.145)

This enables an element of to be written as (13.146)

in which (13.147)

This is called a Christoffel symbol, and it is obtained from (13.145). Note that . Christoffel symbols arise in the study of affine connections in differential geometry and are usually denoted as . Affine connections provide a way to express acceleration without coordinates, in the same way that the tangent space was expressed without coordinates in Section 8.3.2. For affine connections in differential geometry, see ; for their application to mechanics, see .

Subsections
Steven M LaValle 2012-04-20