Abstract
In this chapter, we introduce the principle of virtual work, which is the foundation for all other variational principles of mechanics. We discuss d’Alembert’s principle, which is a simple application of the principle of virtual work to the dynamical equations of Newtonian mechanics, and then derive Lagrange’s equations of the 1st and 2nd kind. We shall see that the Lagrangian formulation of mechanics allows us to analyze problems that are much harder to solve when approached with a direct application of Newton’s laws.
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Notes
- 1.
Our definition of non-holonomic constraints follows the convention used by Hertz (2004) and by Lanczos (1949). Other authors, e.g. Fetter and Walecka (1980) and Flannery (2005), use the term non-integrable for these constraints, and reserve non-holonomic for the more general class of constraints that may not even be expressible in analytic form.
- 2.
The 3-dimensional Levi-Civita symbol is defined in (A.7).
- 3.
In two dimensions, (2.13) is automatically satisfied, since anti-symmetrizing over three indices in a two-dimensional space identically gives zero. Said another way, there is no non-zero 3-index Levi-Civita symbol \(\varepsilon _{\alpha \beta \gamma }\) in two-dimensions. The practical consequence of this is that any differential constraint in two-dimensions is integrable (Exercise B.6).
- 4.
The proof for holonomic constraints would be exactly the same; one simply replaces \(C^A_\alpha \) with the partial derivatives \(\partial \varphi ^A/\partial x^\alpha \).
- 5.
The result for non-holonomic constraints is the same as (2.43) with \(\mathbf {\nabla }_I\varphi ^A\) replaced by \(\mathbf {C}^A_I\).
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Benacquista, M.J., Romano, J.D. (2018). Principle of Virtual Work and Lagrange’s Equations. In: Classical Mechanics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-68780-3_2
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DOI: https://doi.org/10.1007/978-3-319-68780-3_2
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