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Cyclic Coordinates - Lagrangian and Hamiltonian Equations, Classical Mechanics, CSIR-NET Physical Sc | Physics for IIT JAM, UGC - NET, CSIR NET PDF Download
Lagrangian mechanics:
Lagrangian: 
Cyclic coordinate 
Conserved quantity:
Eliminate
as independent variable.
Do not substitute
into Lagrangian.
Substitute 
into Routhian instead.
Routhian:
Equations of motion: 
Supplement:
Hamiltonian mechanics:
Hamiltonian: 
Cyclic coordinate
Conserved quantity: 
Reduced Hamiltonian:
Angular frequency: 
Equations of motion:
Supplement:
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FAQs on Cyclic Coordinates - Lagrangian and Hamiltonian Equations, Classical Mechanics, CSIR-NET Physical Sc - Physics for IIT JAM, UGC - NET, CSIR NET
| 1. What are cyclic coordinates in classical mechanics? |  |
Ans. Cyclic coordinates are variables in a mechanical system that do not appear explicitly in the Lagrangian or Hamiltonian equations. These coordinates are characterized by the fact that their conjugate momenta are conserved quantities.
| 2. How are Lagrangian and Hamiltonian equations related to cyclic coordinates? | |
Ans. Lagrangian and Hamiltonian equations describe the dynamics of a mechanical system. In the presence of cyclic coordinates, the Lagrangian equation simplifies as the corresponding generalized momentum is conserved. In the Hamiltonian framework, cyclic coordinates lead to a constant of motion in the Hamiltonian equation.
| 3. What is the significance of cyclic coordinates in classical mechanics? | |
Ans. Cyclic coordinates play a crucial role in simplifying the equations of motion in classical mechanics. They allow for the reduction of the number of variables needed to describe a system, making it easier to analyze and solve complex problems.
| 4. How can one identify cyclic coordinates in a given mechanical system? | |
Ans. Cyclic coordinates can be identified by examining the Lagrangian or Hamiltonian equations of a mechanical system. If a coordinate does not appear explicitly in these equations, it is likely to be a cyclic coordinate. Additionally, cyclic coordinates are associated with conserved quantities or constants of motion.
| 5. Can cyclic coordinates be used to simplify the study of complex physical systems? | |
Ans. Yes, cyclic coordinates can greatly simplify the analysis of complex physical systems. By identifying and utilizing cyclic coordinates, one can reduce the number of variables and equations involved, leading to a more manageable and insightful study of the system's dynamics.
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