Abstract
This chapter introduces the variational principles of mechanics in the case of unilateral constraints and impacts. We start with virtual displacements and then proceed with variational inequalities formalisms (equivalently inclusions into normal cones to tangent cones and convex sets), Fourier and Jourdain’s principles. The second part is dedicated to the Lagrange dynamics. The case with exogenous impulsive forces is obtained from the material of Chap. 1. Hamilton’s principle, which is far more involved, is treated in the last part of the chapter. The chapter ends with some comments about the link with optimal control under state inequality constraints.
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Notes
- 1.
Often called the impressed forces.
- 2.
According to our notation, \(\sigma _{f}(t)\mathop {=}\limits ^{\varDelta }f(t^{+})-f(t^{-})\), and \(Q_{imp}\) is a vector of impulsive forces, R is the contact percussion vector (i.e., the density of P considered as a Dirac measure at the impact time).
- 3.
For a vector \(x=(x_{1},x_{2},x_{3},\ldots )^{T}\), \(x \succeq 0\) means that the first nonzero element \(x_{i}\) is nonnegative. Thus either all elements are zero, or the first nonzero element is positive.
- 4.
It is preferable to take I as an open interval, in order to encompass the right limit at possible impact times.
- 5.
J.J. Moreau [891] points out that Fourier’s inequality in (3.2) is not always equivalent to \(Q \in N_{\varPhi }(q)\). One has to assure that \(T_{\varPhi }(q) \not = \emptyset \) to secure this.
- 6.
It is reasonable to define such a virtual displacement vector, since a virtual displacement is an infinitesimal change of coordinates.
- 7.
- 8.
“Every problem of the calculus of variations has a solution, provided the word “solution” is suitably understood.” (D. Hilbert), so that ... in variational problems the original setting must be modified in accordance with the needs of an existence theory. [1306, p. 218].
- 9.
see Definition B.12 in Appendix B.
- 10.
\(\bar{\mathbb {R}}=\mathbb {R}+\{-\infty ,+\infty \}\).
- 11.
Recall that if the level sets are compact, the function is said to be proper [1129, Definition 4.6.1]. Hence properness implies coerciveness.
- 12.
Let us define \(n=\frac{1}{\varepsilon }\). The saw-toothed functions \(x_{n}(t)\) converge uniformly toward the function \(x \equiv 0\) [397, p. 64], indeed \(\sup _{t \in \mathbb {R}}|x_{n}(t)|=\frac{1}{n} \rightarrow 0\) when \(n \rightarrow +\infty \). However, the sequence \(\{\dot{x}_{n}\}\) does not converge toward \(\dot{x} \equiv 0\), not even pointwisely since \(|\dot{x}_{n}(t)|=1\) for almost all t. Note that this is reassuring for a mechanician: if \(\dot{x}_{n} \rightarrow 0\) then \(\ddot{x}_{n} \rightarrow 0\) in the distributional sense so that no impacts occur in the limit. Another point of view is that the infinitesimal zigzag curve can be described by assigning the pair of slope \(+1\) and \(-1\) at each point with a probability \(\frac{1}{2}\) [1306, p. 160]. This makes it clear that the saw-toothed function does not converge to the function \(x\equiv \dot{x}\equiv \ddot{x} \equiv ...\equiv 0\) but to something else in a space of “generalized curves”.
- 13.
i.e., nondifferentiable points of the curves.
- 14.
Obviously if the endpoint conditions do not satisfy the constraints, this problem possesses no solution.
- 15.
Hamilton’s principle with fixed \(t_{0}\), \(t_{1}\), \(q(t_{0})\), and \(\dot{q}(t_{0})\) has been studied in [30]. It is shown that the Lagrangian has to be modified to \(K=-\frac{1}{2}\dot{q}^{T}M(q)\dot{q}(t-t_{1})-q^{T}Kq(t-t_{1})+2F^{T}q(t- t_{1})\) in the action integral, where the last two terms account for the potential energy.
- 16.
In (a) and (b), one may replace min I(q) by extr I(q). Indeed searching for the minimizing curve is very hard even in the nonconstrained case. In general one finds the Euler–Lagrange equations which are only necessary conditions to be satisfied by the extremalizing curve. Whether or not these curves define a minimum point of the action is another problem. Additional assumptions about convexity (i.e., forces derived from a convex potential) permit to derive Hamilton’s principle as a minimum principle [958].
- 17.
Notice that the notation \(\cdot \, (0)\) means that the considered function of \(\alpha \) is evaluated at \(\alpha =0\). It is clear that since we analyze the action on the interval \([t_{1},t_{\alpha })\), then \(t_{\alpha }(0)=t_{0}^{-}\).
- 18.
In Chap. 1 we indicated that distributional derivatives of a function \(f(\cdot )\) are sometimes denoted as Df. The notation df is also used in nonsmooth dynamics to denote the measure associated with a function RCLBV [894] and is called the differential measure of \(f(\cdot )\), see Sect. A.3.2.
- 19.
There is no minus sign before \(\partial \psi _{\tilde{\varPhi }}(\tilde{x}(t))\) because of the presence of a minus sign in \(\left( \begin{array}{c} 0 \\ -C^{T}\end{array}\right) \) in (3.70(a)).
- 20.
By virtual we mean that the control input has to be such that the state will not escape from a certain given set. However, the system’s model itself does not contain unilateral constraints like in mechanics with physical obstacles, or circuits with ideal diodes.
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Brogliato, B. (2016). Variational Principles. In: Nonsmooth Mechanics. Communications and Control Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-28664-8_3
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DOI: https://doi.org/10.1007/978-3-319-28664-8_3
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