5.4 Applications to particle dynamics and rigid body motion

Lagrangian mechanics revolutionizes how we analyze particle and rigid body motion. It simplifies complex systems by using generalized coordinates and the principle of least action, leading to powerful equations of motion and conservation laws.

Advanced applications like the double pendulum, symmetric top, and charged particles in fields showcase Lagrangian mechanics' versatility. These examples demonstrate how to tackle intricate systems, revealing the method's strength in various physics scenarios.

Lagrangian Mechanics for Particles and Rigid Bodies

Lagrangian mechanics for particle motion

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• Lagrangian function describes system dynamics $L = T - V$, kinetic minus potential energy
• Generalized coordinates independently describe system configuration (angles, distances)
• Generalized velocities time derivatives of generalized coordinates (angular velocity)
• Principle of least action system path minimizes action integral over time
• Euler-Lagrange equations $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0$ govern system evolution
• Constraints limit system motion holonomic (pendulum length) or non-holonomic (rolling without slipping)
• Conservation laws emerge from symmetries via Noether's theorem (energy, momentum)
• Rigid body motion involves rotational kinetic energy and moment of inertia tensor

Double pendulum equations of motion

• System consists of two masses connected by massless rigid rods
• Generalized coordinates $\theta_1$ and $\theta_2$ angles from vertical for each pendulum
• Kinetic energy expressions for each mass use $\theta_1$, $\theta_2$, $\dot{\theta_1}$, and $\dot{\theta_2}$
• Potential energy accounts for gravitational potential of each mass
• Lagrangian formulation $L = T - V$ expressed in generalized coordinates
• Euler-Lagrange equations applied to $\theta_1$ and $\theta_2$ yield equations of motion
• Resulting coupled non-linear differential equations describe complex motion

Advanced Applications of Lagrangian Mechanics

Symmetric top motion analysis

• Euler angles describe top orientation $\phi$ (precession), $\theta$ (nutation), $\psi$ (spin)
• Rotational kinetic energy expressed using Euler angles and time derivatives
• Potential energy accounts for gravitational potential of the top
• Lagrangian constructed with Euler angles as generalized coordinates
• Euler-Lagrange equations applied to each Euler angle yield equations of motion
• Cyclic coordinates identify conserved quantities (angular momentum)
• Precession and nutation analyzed from resulting equations

Charged particle motion in fields

• Electromagnetic potentials scalar $\phi$ and vector $\mathbf{A}$ describe fields
• Generalized coordinates use particle's Cartesian position
• Kinetic energy $T = \frac{1}{2}mv^2$ for particle of mass m
• Potential energy $V = q\phi$ for particle of charge q
• Lagrangian for charged particle $L = \frac{1}{2}mv^2 - q\phi + q\mathbf{v} \cdot \mathbf{A}$
• Euler-Lagrange equations applied to each coordinate yield equations of motion
• Lorentz force derived from Lagrangian formulation
• Particle trajectory solved for given electromagnetic fields