In a conservative system where the Hamiltonian H and Lagrangian L formulations are used, consider a particle moving in a potential field. The system is described by the Hamiltonian \(H = \frac{p^2}{2m} + V(q)\), where p is the generalized momentum, q is the generalized coordinate, m is the mass, and V(q) is the potential energy. Which of the following statements correctly describes the application of Hamilton's and Lagrange's equations to this system?
1
Hamilton's equations can be used to directly derive the force acting on the particle as \(F = -\nabla V(q)\), while Lagrange's equations do not provide information about forces.
2
Lagrange's equations are more general as they can be applied to non-conservative systems, unlike Hamilton's equations which only apply to conservative systems.
3
In Hamilton's formulation, the time evolution of the generalized coordinates and momenta is completely determined by the partial derivatives of the Hamiltonian, whereas in Lagrange's formulation, the equations of motion are derived from the Euler-Lagrange equation based on the stationary action principle.
4
Both Hamilton's and Lagrange's formulations are incapable of handling time-dependent potentials within the system.
5
Question Not Attempted