Virtual Work and d'Alembert Principle

February 03, 2024

Hamilton's Principle

In the d'Alembert Principle, we were introduced to the notion of virtual displacement and virtual work. The Lagrange's equation emerged as a consequence of this principle, and it proved to be a powerful tool in deriving equations of motion of any physical system. Here, we derive Lagrange's Equation using Hamilton's Principle. In introductory classical mechanics, we are used to think in terms of forces and vectors while solving physical systems. They were quite intuitive, albeit a little tedious when complex systems were involved. Another way of approaching these problems is to begin with the assumption that " Nature is lazy ". In any kind of dynamical system, she always ensures that a particular quantity takes on the least possible value. We call this quantity, the Action of the system. So, instead of keeping track of all the force vectors that are acting on the system, we ask ourselves, what quantity is being minimized here? Essentially, we try to formulate ...

Previously, we learnt the advantage of using generalized coordinates over traditional coordinate spaces while deriving equations of motion. Here, we build on the idea by introducing Virtual Displacements and Virtual Work. We will use these ideas to arrive at d'Alembert Principle, named after the French physicist and mathematician Jean le Rond d'Alembert.

Virtual Displacement

A virtual displacement of a system is an instantaneous, infinitesimal variation in the positions of the constituents of the system in accordance with the constraints. Virtual displacement is denoted by

δ q_{i}

. This is by no means a real displacement, but an imaginary one. We "test" the system by nudging it a bit and see how it behaves. Since the constraints and the external forces may change with time, we keep it a constant while we virtually displace it.

Virtual Displacement has much more nuances to it, but the definition above is good enough for all intents and purposes. But you can learn more about it from [1].

Virtual Work

d'Alembert Principle

References

1. On Virtual Displacement and Virtual Work in Lagrangian Dynamics - Ray, Shamanna

DOI: http://dx.doi.org/10.1088/0143-0807/27/2/014

arXiv

2. A Student's Guide to Lagrangian and Hamiltonian - Hamill

3. Classical Mechanics - Goldstein, Poole, Safko

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