Analytical Mechanics
Lagrangian and Hamiltonian analytical mechanics formulated via the principle of stationary action.
General idea of the topic
Analytical mechanics reformulates Newtonian mechanics with energies instead of forces. A set of coordinates describing the configuration of the system is chosen and the equations of motion are derived from a single scalar function, without needing to compute the constraint forces.
The block starts from generalized coordinates and constraints, develops the Lagrangian formulation, based on kinetic and potential energy, and the Hamiltonian one, which works with positions and momenta, and ends by applying them to the oscillator, the pendulum and problems with constraints.
What you will truly understand
By the end you will be able to choose coordinates adapted to a problem, write the Lagrangian, obtain the equations of motion and relate symmetries to conserved quantities.
You will also know when this formulation simplifies a problem compared with the vector method.
Recommended learning route
- Generalized Coordinates: coordinates that describe the configuration without redundancy.
- Constraints and Degrees of Freedom: geometric restrictions and the number of degrees of freedom.
- D'Alembert's Principle and Virtual Work: equilibrium and dynamics with virtual work, without ideal constraint forces.
- Lagrangian Formulation: Lagrange function L = T − U and Lagrange's equations.
- Hamiltonian Formulation: conjugate momenta, Hamiltonian and Hamilton's equations.
- Advanced Constraints: nonholonomic constraints and their treatment.
- Applications of Analytical Mechanics: harmonic oscillator, pendulum, small oscillations and systems with constraints solved with these tools.
Physical criterion you must acquire
The essential criterion is to count the degrees of freedom and choose as many independent coordinates as there are. If the constraints are holonomic and ideal, the constraint forces do not appear in the equations. A result is correct if it reproduces Newton's for the same problem.
Connection with real problems
- robotics and articulated mechanisms.
- dynamics of satellites and rigid bodies.
- quantum mechanics.
- optimization and control.
Final goal of the topic
The final goal is to set up and solve the dynamics of systems with constraints through the Lagrangian and Hamiltonian formulations and relate symmetries to conservation laws.
Topics in this section
- Generalized Coordinates
Generalized coordinates and degrees of freedom for constrained mechanical systems.
- Constraints and Degrees of Freedom
Constraints and degrees of freedom in mechanical systems with independent restrictions.
- D'Alembert's Principle and Virtual Work
Reformulates dynamics with ideal constraints through dynamic equilibrium and virtual work.
- Advanced Constraints
Differential constraints and advanced restrictions. Covers: Nonholonomic Constraints.
- Lagrangian Formulation
Generalized coordinates, Lagrangian function, and Euler-Lagrange equations for constrained systems.
- Hamiltonian Formulation
Canonical conjugate momenta, Hamiltonian function, and Hamilton's canonical equations.
- Applications of Analytical Mechanics
Analytical solutions for double pendulums, central force problems, and coupled systems.