Analytics Mechanics (Physics)

Classify mechanical constraints as holonomic or nonholonomic and as scleronomous or rheonomous.

Determine the number of degrees of freedom of a particle system subject to independent holonomic constraints.

Choose generalized coordinates that automatically satisfy a given set of holonomic constraints.

Express Cartesian velocities in terms of generalized coordinates, generalized velocities, and time.

Compute the first variation of a functional that depends on a path and its first derivative.

Use integration by parts and fixed-endpoint conditions in a variational calculation.

Derive the Euler–Lagrange equation from the principle of stationary action.

Derive the equation of motion of a simple pendulum using the Euler–Lagrange equation.

Compute generalized forces from the virtual work of applied Cartesian forces.

Derive Lagrange’s equations with generalized forces from d’Alembert’s principle.

Construct kinetic energy in curvilinear generalized coordinates.

Form a Lagrangian of the form L = T − V for a conservative constrained system.

Derive the coupled equations of motion of a double pendulum from its Lagrangian.

Include linear damping in Lagrange’s equations using a Rayleigh dissipation function.

Show that adding a total time derivative to a Lagrangian leaves the equations of motion unchanged.

Derive the Lorentz-force equation from a velocity-dependent electromagnetic Lagrangian.

Identify a cyclic coordinate and derive its conserved canonical momentum.

Derive the Lagrangian energy function and state the conditions under which it is conserved.

Calculate the infinitesimal change of a Lagrangian under a continuous coordinate transformation.

Derive the Noether conserved quantity associated with a one-parameter point transformation.

Obtain conservation of linear momentum from spatial translation symmetry.

Obtain conservation of angular momentum from rotational symmetry.

Separate center-of-mass motion from internal motion in a translation-invariant particle system.

Distinguish canonical momentum from mechanical momentum for a charged particle.

Reduce a two-body central-force problem to an equivalent one-body problem using the reduced mass.

Use angular-momentum conservation to show that central-force motion is planar.

Derive the one-dimensional radial equation using the effective potential.

Determine the existence and stability of circular orbits from an effective potential.

Derive Binet’s orbit equation for a central force.

Solve Binet’s equation for the inverse-square force and classify the resulting conic-section orbits.

Derive Kepler’s area law from angular-momentum conservation.

Calculate a classical scattering angle from the radial effective-potential integral.

Linearize Lagrange’s equations about a stable equilibrium configuration.

Construct the mass and stiffness matrices for a small-oscillation system.

Derive the generalized eigenvalue equation for normal-mode frequencies.

Compute the normal frequencies and mode shapes of two coupled oscillators.

Establish orthogonality of distinct normal modes with respect to the mass matrix.

Transform a quadratic Lagrangian to normal coordinates.

Solve an initial-value problem by expanding the motion in normal modes.

Identify a zero mode arising from a continuous symmetry.

Represent an infinitesimal rigid-body rotation using an angular-velocity vector.

Construct the inertia tensor of a discrete mass distribution.

Find the principal axes and principal moments by diagonalizing an inertia tensor.

Express rigid-body rotational kinetic energy in principal-axis coordinates.

Relate angular velocity to Euler-angle rates for a chosen Euler-angle convention.

Derive Euler’s equations for a torque-free rigid body.

Determine the stability of steady rotation about each principal axis.

Use conserved energy and angular momentum to characterize torque-free rigid-body motion.

Perform the Legendre transform of a regular Lagrangian with respect to the generalized velocities.

Derive Hamilton’s canonical equations from the Legendre-transformed action.

Construct the Hamiltonian of a particle in Cartesian or curvilinear coordinates.

Interpret a solution of Hamilton’s equations as a trajectory in phase space.

Compute Poisson brackets of functions of canonical coordinates and momenta.

Express time evolution and conservation laws using Poisson brackets.

Use Poisson brackets to show how momentum and angular momentum generate translations and rotations.

Derive Liouville’s theorem for Hamiltonian phase-space flow.

Test whether a transformation is canonical by checking fundamental Poisson brackets.

Derive a canonical transformation from a generating function.

Transform the Hamiltonian under a time-dependent canonical transformation.

Interpret Hamiltonian time evolution as a canonical transformation.

Derive the Hamilton–Jacobi equation from a canonical transformation to constant variables.

Solve the time-independent Hamilton–Jacobi equation for a separable one-dimensional system.

Recover mechanical trajectories from a complete Hamilton–Jacobi solution.

Compute action–angle variables and the oscillation frequency for the harmonic oscillator.

Construct the coordinate-time Lagrangian of a free relativistic particle from the proper-time action.

Compute the canonical momentum and Hamiltonian of a free relativistic particle and recover the relativistic energy–momentum relation.

Derive the relativistic Lorentz-force equation from a particle Lagrangian.

Derive the Euler–Lagrange field equation for a Lagrangian density depending on a field and its first derivatives.

Derive the Klein–Gordon equation from the free scalar-field Lagrangian density.

Apply Noether’s theorem to derive a conserved current for a continuous symmetry of a classical field.

Derive the canonical stress–energy tensor associated with spacetime translations.

Relate Poisson brackets and Hamilton–Jacobi theory to the canonical and semiclassical structures used in quantum mechanics.