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In the previous chapter, we have derived the Euler-Lagrange equations for a particular case of interacting particles. As we have seen, the Lagrangian depends on the “essential variables” \(\mathbf {q}\) and \(\dot {\mathbf {q}}\) . We are now ready to formalize these concepts. We introduce configuration manifolds of the system to treat the “essential coordinates” and tangent bundles of configuration manifolds as the arguments of the Lagrangian. We then introduce the concept of action and formulate the critical action principle as a foundation of a variational approach to mechanics. After analyzing several particular examples to illustrate derived concepts, we focus on systems with holonomic constraints. We finish the chapter by discussing dissipative systems and introducing Rayleigh’s dissipation function.

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Configuration Manifolds, Variational Principle, and Euler-Lagrange Equations

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  • 发表日期：

    2025年

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  • Vakhtang Putkaradze

摘要

In the previous chapter, we have derived the Euler-Lagrange equations for a particular case of interacting particles. As we have seen, the Lagrangian depends on the “essential variables” \(\mathbf {q}\) and \(\dot {\mathbf {q}}\) . We are now ready to formalize these concepts. We introduce configuration manifolds of the system to treat the “essential coordinates” and tangent bundles of configuration manifolds as the arguments of the Lagrangian. We then introduce the concept of action and formulate the critical action principle as a foundation of a variational approach to mechanics. After analyzing several particular examples to illustrate derived concepts, we focus on systems with holonomic constraints. We finish the chapter by discussing dissipative systems and introducing Rayleigh’s dissipation function.