Explore the comparison between Lagrangian and Hamiltonian approaches in dynamics, covering natural mechanical systems, time-irreversible systems, cyclic variables, integrability, and theorems. Dive into examples like Sokolov's case of a rigid body in a liquid and understand the impact of gauge terms on equations. Discover the essence of Hamilton's equations in transforming potentials and more.
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Presentation Transcript
LAGRAGIAN VERSUS HAMILTONIAN APPROACHES IN CERTAIN PROBLEMS OF DYNAMICS YEHIAHamad Mansoura University Egypt hyehia@mans.edu.eg 1 7/3/2025
For a generalized natural (time-irreversible) mechanical system where b expresses velocity-dependent forces like: - Coriolis forces in rotating frames, - Lorentz forces due to magnetic effect on moving electric charges, - Fictitious forces resulting from ignoring some cyclic coordinates. The corresponding Hamiltonian is ??=? 7/3/2025 3 ? ??????? ???????+ [? +? ? ???????].
1) 1) When one adds a gauge term Lagrange s equations and their integrals do not change, while for the corresponding Hamiltonian ??=? ? ??????? (??+ ???)?????+ [? +? ? (??+ ???)???(??+ ???)]. Hamilton s equations and their integrals change and potential is disguised. 7/3/2025 4
Example: Sokolov s case of rigid body in a liquid: (Sokolov) (Borisov and Mamaev) 7/3/20255
2) Cyclic variable change 2) Cyclic variable change Let q be a cyclic variable in the Lagrangian of an integrable system. Replacing q by q = Q -nt ( ?= ?-n), preserves integrability and conservativity. The Hamiltonian equivalent is ? = ? + ??, where ? is the cyclic integral conjugate to q. 6 7/3/2025
The Lagrangian (1) admits the cyclic integrals Consider another system with the Lagrangian constants (2) i are cyclic variables and let , ,?? ?1 7/3/2025 7
Theorem Theorem: If the system with the Lagrangian (1) is integrable for all initial conditions, then (2) is integrable for arbitrary functions (conditionally) on the level 8 7/3/2025
The general problem of motion of a rigid body about a fixed point under the action of a combination of skew conservative potential and gyroscopic forces, described by the Lagrangian Scalar and vector potentials V, l depend only on the Eulerian angles through the nine direction cosines 14 7/3/2025
The equations of motion in the Euler The equations of motion in the Euler Poisson variables ( (Celest Celest. Mech. 1988) . Mech. 1988) Poisson variables 7/3/2025 15
Cyclic coordinates in rigid body dynamics: 1- Axi-symmetric fields - precession angle cyclic. 2- Axi-symmetric body rotation angle cyclic. 3- Dynamically axi-symmetric body and quaternion symmetry ( cyclic). 16 7/3/2025
Conditional generalizations Transformation with arbitrary function and arbitrary constant takes the Rubanovsky (1968)-Kharlamov (1963) Steklov (1896) case 31 7/3/2025
The second integrable case Asymmetric equivalent of generalized Lyapunov case of a body in liquid (Celest. Mech. 1988). OR The body is heavy, magnetized and electrically charged, acted upon by skew non-uniform gravity, magnetic and electric fields. 36 7/3/2025
