dorsal/arxiv
View SchemaOn Virtual Displacement and Virtual Work in Lagrangian Dynamics
| Authors | Subhankar Ray, J. Shamanna |
|---|---|
| Categories | |
| ArXiv ID | physics/0510204 |
| URL | https://arxiv.org/abs/physics/0510204 |
| DOI | 10.1088/0143-0807/27/2/014 |
| Journal | European Journal of Physics, 27 (2006) 311-329 |
Abstract
The confusion and ambiguity encountered by students, in understanding virtual displacement and virtual work, is discussed in this article. A definition of virtual displacement is presented that allows one to express them explicitly for holonomic (velocity independent), non-holonomic (velocity dependent), scleronomous (time independent) and rheonomous (time dependent) constraints. It is observed that for holonomic, scleronomous constraints, the virtual displacements are the displacements allowed by the constraints. However, this is not so for a general class of constraints. For simple physical systems, it is shown that, the work done by the constraint forces on virtual displacements is zero. This motivates Lagrange's extension of d'Alembert's principle to system of particles in constrained motion. However a similar zero work principle does not hold for the allowed displacements. It is also demonstrated that d'Alembert's principle of zero virtual work is necessary for the solvability of a constrained mechanical problem. We identify this special class of constraints, physically realized and solvable, as {\it the ideal constraints}. The concept of virtual displacement and the principle of zero virtual work by constraint forces are central to both Lagrange's method of undetermined multipliers, and Lagrange's equations in generalized coordinates.
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"abstract": "The confusion and ambiguity encountered by students, in understanding virtual\ndisplacement and virtual work, is discussed in this article. A definition of\nvirtual displacement is presented that allows one to express them explicitly\nfor holonomic (velocity independent), non-holonomic (velocity dependent),\nscleronomous (time independent) and rheonomous (time dependent) constraints. It\nis observed that for holonomic, scleronomous constraints, the virtual\ndisplacements are the displacements allowed by the constraints. However, this\nis not so for a general class of constraints. For simple physical systems, it\nis shown that, the work done by the constraint forces on virtual displacements\nis zero. This motivates Lagrange\u0027s extension of d\u0027Alembert\u0027s principle to\nsystem of particles in constrained motion. However a similar zero work\nprinciple does not hold for the allowed displacements. It is also demonstrated\nthat d\u0027Alembert\u0027s principle of zero virtual work is necessary for the\nsolvability of a constrained mechanical problem. We identify this special class\nof constraints, physically realized and solvable, as {\\it the ideal\nconstraints}. The concept of virtual displacement and the principle of zero\nvirtual work by constraint forces are central to both Lagrange\u0027s method of\nundetermined multipliers, and Lagrange\u0027s equations in generalized coordinates.",
"arxiv_id": "physics/0510204",
"authors": [
"Subhankar Ray",
"J. Shamanna"
],
"categories": [
"physics.ed-ph",
"physics.class-ph"
],
"doi": "10.1088/0143-0807/27/2/014",
"journal_ref": "European Journal of Physics, 27 (2006) 311-329",
"title": "On Virtual Displacement and Virtual Work in Lagrangian Dynamics",
"url": "https://arxiv.org/abs/physics/0510204"
},
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