Graduate
  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

Graduate → Classical mechanics → Lagrangian and Hamiltonian mechanics ↓


Euler–Lagrange equations


The Euler-Lagrange equations are a fundamental set of equations in classical mechanics, used within the framework of both Lagrangian and Hamiltonian mechanics. They allow us to find the path, or "trajectory", that a physical system takes over time. These equations are derived from the principle of minimum action, which states that the path taken by a system is one for which the action is constant (usually minimal).

Introduction to Lagrangian mechanics

Lagrangian mechanics is a reformulation of classical mechanics, introduced in the 18th century by Joseph-Louis Lagrange. Whereas Newton's laws focus on forces, Lagrangian mechanics redefines mechanics in terms of energy. It uses a function called the Lagrangian, denoted by L, which is defined as the difference between the kinetic energy (T) and potential energy (V) of a system:

L = T - V

The beauty of Lagrangian mechanics is that it simplifies the problem of determining the equations of motion for complex systems, especially when the system has constraints.

Principle of minimum action

The principle of minimum action, also known as Hamilton's principle, is a variational principle that provides the basis for Lagrangian mechanics. According to this principle, the path taken by a system between two states is the one for which the action, S, is maximized (usually minimal). The action is defined as the integral of the Lagrangian over time:

S = ∫ L dt

To find the path that takes the function to the extremum, the function is transformed and its first variation is set to zero:

δS = 0

This brings us to the Euler–Lagrange equations, which are derived by applying the calculus of variations.

Derivation of the Euler–Lagrange equations

Consider a system described by n generalized coordinates qi for i from 1 to n. The Lagrangian L is a function of these generalized coordinates, their time derivatives (velocities) dqi/dt, and time t:

L = L(q1, q2, ..., qn, t)

To derive the Euler–Lagrange equations, consider a small change in the path of the system.

qi(t) → qi(t) + δεi

where δεi is an infinitesimal variation. The action up to first order in δεi becomes as follows:

δS = ∫ (∂L/∂qi) δεi + (∂L/∂(dqi/dt)) δ(dqi/dt) dt

After integrating by parts and assuming that the divergences vanish at the endpoints, we get the Euler–Lagrange equation:

d/dt (∂L/∂(dqi/dt)) - ∂L/∂qi = 0

These are the Euler–Lagrange equations, a set of n second-order differential equations.

Examples of Lagrangian mechanics

Example 1: Simple harmonic oscillator

Consider a simple harmonic oscillator whose mass m is connected to a spring of constant k. The kinetic energy T and potential energy V are given by:

T = 1/2 m (dx/dt)^2
V = 1/2 kx^2

The Lagrangian L is:

L = T - V = 1/2 m (dx/dt)^2 - 1/2 kx^2

Applying the Euler–Lagrange equation:

d/dt (∂L/∂(dx/dt)) = ∂L/∂x

This results in the classic simple harmonic motion equation:

md2x/dt2 + kx = 0

Example 2: Pendulum

Consider a simple pendulum of length l and mass m. The motion is confined to a plane and the angle θ from the vertical is the generalized coordinate. The kinetic energy T and potential energy V are given by:

T = 1/2 m (l^2) (dθ/dt)^2
V = mgl (1 - cos θ)

The Lagrangian L is:

L = 1/2 m (l^2) (dθ/dt)^2 - mgl (1 - cos θ)

Applying the Euler-Lagrange equation we get:

ml^2 d2θ/dt2 + mgl sin θ = 0

Introduction to Hamiltonian mechanics

Hamiltonian mechanics is a further reformulation of classical mechanics, named after Sir William Rowan Hamilton. It provides a powerful framework, especially useful in more advanced mechanics, quantum mechanics, and statistical mechanics. The fundamental quantity in Hamiltonian mechanics is the Hamiltonian, denoted by H, which is often the total energy of the system (the sum of kinetic and potential energies).

From Lagrangian to Hamiltonian

Transforming from the Lagrangian to the Hamiltonian involves a process called the Legendre transformation. The generalized momentum, pi, is defined as:

pi = ∂L/∂(dqi/dt)

The Hamiltonian H is defined as a function of generalized coordinates, momentum, and time:

H(qi, pi, t) = Σ pi (dqi/dt) - L(qi, dqi/dt, t)

Hamilton's equations

Hamilton's equations of motion are derived from the Hamiltonian and are expressed as follows:

d(qi)/dt = ∂H/∂pi
d(pi)/dt = -∂H/∂qi

Example: Simple harmonic oscillator in Hamiltonian mechanics

Returning to the simple harmonic oscillator with Hamiltonian mechanics, the momentum p is given by:

p = m(d(x)/dt)

The Hamiltonian H for a simple harmonic oscillator is:

H = p(d(x)/dt) - (1/2)m(d(x)/dt)2 + 1/2 kx2

This makes it simpler:

H = p2/2m + 1/2 kx2

Applying Hamilton's equations gives the standard equations of motion:

d(x)/dt = p/m
d(p)/dt = -kx

Applications and significance

The Euler–Lagrange equations, and more generally Lagrangian and Hamiltonian mechanics, are fundamental to understanding physical systems. They are particularly useful in dealing with the following:

  • Complex systems: Systems with many degrees of freedom and constraints, such as many-body systems, are simpler in this formulation.
  • Field theories: such as electromagnetism and general relativity, where fields rather than particles are the primary objects of study.
  • Quantum mechanics: where Hamiltonian mechanics provides a bridge to quantum systems through concepts such as operators and eigenstates.

Conclusion

The Euler-Lagrange equations are an essential component of the physicist's toolkit for exploring both classical and modern physics. Using these equations, we can determine the equations of motion for a vast array of systems and understand the principles that govern their dynamic behavior. Lagrangian and Hamiltonian methods provide powerful and elegant approaches to mechanics, highlighting the deep connection between mathematics and physics.


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Euler–Lagrange equations

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