Generalized coordinates, Euler-Lagrange equations, constraints, conservation laws, central forces and Hamilton equations.
40 cards · basic cards · AI-written, checked twice. Edit anything.
- What is a generalized coordinate in classical mechanics?
Show answer
Any independent variable describing a system's configuration, not necessarily Cartesian position (can be angles, arc lengths, or any parameter).- Define a holonomic constraint.
Show answer
A constraint relating coordinates (and possibly time) that reduces to a restriction on configuration space, e.g., a particle constrained to a surface.- What is a non-holonomic constraint?
Show answer
A constraint involving velocities that cannot be reduced to a coordinate restriction alone, such as a rolling sphere without slipping.- How many degrees of freedom does a system have?
Show answer
The number of independent generalized coordinates needed to fully specify the configuration.- What is a virtual displacement?
Show answer
An infinitesimal displacement respecting instantaneous constraints but not necessarily the actual displacement the system undergoes.- Distinguish constraint forces from applied forces.
Show answer
Constraint forces maintain constraints (often perpendicular to motion); applied forces are external or internal forces independent of constraints.- Why can constraint forces be ignored in Lagrangian mechanics?
Show answer
Constraint forces do no virtual work for holonomic systems, so they vanish from equations of motion derived from the action principle.- What is configuration space?
Show answer
An abstract space with coordinates being the generalized coordinates of a system; each point represents one complete configuration.- Define the Lagrangian.
Show answer
L = T - V, the difference between kinetic energy and potential energy.- State the principle of stationary action (Hamilton's principle).
Show answer
The action integral S = integral L dt is stationary for the actual path compared to nearby virtual paths.- Write the Euler-Lagrange equation.
Show answer
d/dt(dL/dq_dot) - dL/dq = 0 for each generalized coordinate q.- If the Lagrangian does not depend on coordinate q_k, what is conserved?
Show answer
The generalized momentum p_k = dL/dq_dot_k is conserved (constant in time).- What is a cyclic coordinate?
Show answer
A generalized coordinate that does not appear in the Lagrangian (only its time derivative may appear, or neither).- What is the generalized momentum conjugate to q_k?
Show answer
p_k = dL/dq_dot_k, the derivative of the Lagrangian with respect to the velocity of that coordinate.- For a free particle in 1D, write the Lagrangian and deduce the equation of motion.
Show answer
L = (1/2) m (dq/dt)^2; Euler-Lagrange equation gives d^2q/dt^2 = 0, so constant velocity.
FlashKeepers