Unit 2: Lagrangian Formalism

1. Generalized Coordinates and Velocities

1.1 Generalized Coordinates

In classical mechanics, describing the motion of a system of N particles using standard Cartesian coordinates (x_i, y_i, z_i) often becomes mathematically cumbersome due to the presence of geometrical constraints. To simplify analysis, we introduce generalized coordinates.

Definition: Generalized coordinates are defined as the minimum number of independent variables required to completely specify the physical state and configuration of a dynamical system at any given instant in time.

If a system consisting of N particles is subjected to k independent constraint equations, the number of degrees of freedom (n) is calculated as:

Degrees of Freedom (n) = 3N - k

The set of generalized coordinates is conventionally denoted by:

q₁, q₂, q₃, ..., q_n

The position vector r_i of the i-th particle can be expressed as a function of the generalized coordinates and explicit time t:

r_i = r_i(q₁, q₂, ..., q_n, t)

1.2 Generalized Velocities

The time derivative of a generalized coordinate is known as a generalized velocity.

Generalized Velocity: q̇_j = dq_j / dt (for j = 1, 2, ..., n)

By differentiating the position vector r_i with respect to time using the chain rule, the velocity v_i of the i-th particle is given by:

v_i = dr_i / dt = ∑_j=1ⁿ (∂r_i / ∂q_j) q̇_j + ∂r_i / ∂t

1.3 Comparison: Cartesian vs. Generalized Coordinates

2. Principle of Virtual Work

2.1 Concept of Virtual Displacement

A virtual displacement, denoted by δr_i, is an infinitesimal, imaginary change in the system's position vector occurring at a fixed instant of time (dt = 0). It must be consistent with the forces and constraints imposed on the system, but does not involve any actual passage of time.

2.2 Principle Statement

Consider a mechanical system in static equilibrium. The total force F_i acting on particle i consists of the applied force F_i^(a) and the force of constraint f_i:

F_i = F_i^(a) + f_i

For equilibrium, the total net force on each particle is zero: F_i = 0. The total virtual work done by these forces during a virtual displacement δr_i is:

δW = ∑_i F_i · δr_i = ∑_i F_i^(a) · δr_i + ∑_i f_i · δr_i = 0

For systems with rigid or smooth constraints (such as ideal constraints like rigid rods, smooth surfaces, or inextensible strings), the constraint forces do zero total virtual work:

∑_i f_i · δr_i = 0

Principle of Virtual Work Statement: The virtual work done by applied, non-constraint forces for a system of connected rigid particles in static equilibrium is zero for any arbitrary virtual displacement compatible with the constraints:

δW = ∑_i F_i^(a) · δr_i = 0

3. D’Alembert’s Principle

3.1 Concept and Formulation

D’Alembert’s principle extends the Principle of Virtual Work from static equilibrium to dynamic systems by introducing the concept of reverse effective force (inertial force).

According to Newton's second law for particle i:

F_i = ṗ_i   or   F_i - ṗ_i = 0

where p_i = m_iv_i is the linear momentum, and ṗ_i is its time derivative (m_ia_i). We can view this as a condition of dynamic equilibrium under effective force (F_i - ṗ_i).

3.2 Mathematical Statement

Taking the dot product of the dynamic equilibrium equation with a virtual displacement δr_i and summing over all particles:

∑_i (F_i - ṗ_i) · δr_i = 0

Substituting F_i = F_i^(a) + f_i:

∑_i (F_i^(a) - ṗ_i) · δr_i + ∑_i f_i · δr_i = 0

Since the virtual work done by constraint forces vanishes for holonomic systems (∑_i f_i · δr_i = 0), we obtain D’Alembert’s Principle:

D’Alembert’s Principle:

∑_i (F_i^(a) - ṗ_i) · δr_i = 0

Key Insight: D’Alembert’s principle successfully eliminates the unknown forces of constraint from the governing dynamic equations of motion.

4. Lagrange’s Equations from D’Alembert’s Principle

To derive Lagrange’s equations, we transform D’Alembert’s principle from Cartesian coordinates (r_i) into independent generalized coordinates (q_j).

Step 1: Transform Virtual Displacement

The virtual displacement δr_i in terms of generalized coordinates is:

δr_i = ∑_j=1ⁿ (∂r_i / ∂q_j) δq_j

Step 2: Transform Applied Work Term

Substituting δr_i into the applied force term gives:

∑_i F_i^(a) · δr_i = ∑_i F_i^(a) · [ ∑_j (∂r_i / ∂q_j) δq_j ] = ∑_j [ ∑_i F_i^(a) · (∂r_i / ∂q_j) ] δq_j

We define the Generalized Force Q_j corresponding to coordinate q_j as:

Q_j = ∑_i F_i^(a) · (∂r_i / ∂q_j)

Thus, the virtual work term becomes: ∑_j Q_j δq_j

Step 3: Transform Inertial Term

The inertial force term is ∑_i ṗ_i · δr_i = ∑_i m_i r̈_i · δr_i. Rewriting using generalized coordinates:

∑_i m_i r̈_i · δr_i = ∑_j [ ∑_i m_i r̈_i · (∂r_i / ∂q_j) ] δq_j

Consider the identity using product rule of differentiation:

m_i r̈_i · (∂r_i / ∂q_j) = (d / dt) [ m_i ṙ_i · (∂r_i / ∂q_j) ] - m_i ṙ_i · (d / dt)(∂r_i / ∂q_j)

Using the kinematic identities:

1. ∂r_i / ∂q_j = ∂ṙ_i / ∂q̇_j (Cancellation of dots property)
2. (d / dt)(∂r_i / ∂q_j) = ∂ṙ_i / ∂q_j (Interchange of derivatives)

Substituting these identities back into the relation:

∑_i m_i r̈_i · (∂r_i / ∂q_j) = (d / dt) [ ∂ / ∂q̇_j ( ∑_i (1/2) m_i v_i² ) ] - ∂ / ∂q_j ( ∑_i (1/2) m_i v_i² )

Recognizing the total kinetic energy T = ∑_i (1/2) m_i v_i², the inertial term simplifies to:

(d / dt)(∂T / ∂q̇_j) - ∂T / ∂q_j

Step 4: Combine Terms into D’Alembert’s Principle

Substituting both the generalized force and kinetic energy terms into D’Alembert's principle:

∑_j=1ⁿ [ (d / dt)(∂T / ∂q̇_j) - ∂T / ∂q_j - Q_j ] δq_j = 0

Since generalized coordinates q_j are completely independent, each term in the brackets must individually equal zero:

(d / dt)(∂T / ∂q̇_j) - ∂T / ∂q_j = Q_j   (for j = 1, 2, ..., n)

Step 5: Conservative Systems and Lagrangian Form

If the forces are conservative, the force can be derived from a scalar potential function V(q_j):

F_i^(a) = -∇_i V   ⇒   Q_j = -∂V / ∂q_j

Since potential energy V depends only on coordinates q_j (and not velocities q̇_j), ∂V / ∂q̇_j = 0. We can rewrite the equation as:

(d / dt) [ ∂(T - V) / ∂q̇_j ] - ∂(T - V) / ∂q_j = 0

Defining the Lagrangian Function L = T - V:

Euler-Lagrange Equations of Motion:

(d / dt)(∂L / ∂q̇_j) - ∂L / ∂q_j = 0   (for j = 1, 2, ..., n)

5. Hamilton’s Principle and Lagrange’s Equations

5.1 Statement of Hamilton’s Principle

Hamilton’s Principle (also known as the Principle of Least Action) provides an integral formulation of classical mechanics.

Hamilton’s Principle: The actual motion of a conservative dynamical system from time t₁ to time t₂ takes a path such that the line integral of the Lagrangian function L (called the Action Integral I) is stationary (has an extremum value) compared to all nearby neighboring paths with identical boundary values at t₁ and t₂.

δI = δ ∫_t1^t₂ L(q_j, q̇_j, t) dt = 0

Boundary conditions require zero variation at endpoints: δq_j(t₁) = 0 and δq_j(t₂) = 0.

5.2 Derivation of Lagrange’s Equations from Hamilton’s Principle

Using calculus of variations, compute the variation of the action integral I:

δI = ∫_t1^t₂ δL(q_j, q̇_j, t) dt = 0

Expand δL using the chain rule:

δL = ∑_j=1ⁿ [ (∂L / ∂q_j) δq_j + (∂L / ∂q̇_j) δq̇_j ]

Substitute δL into the action variation:

δI = ∫_t1^t₂ ∑_j=1ⁿ [ (∂L / ∂q_j) δq_j + (∂L / ∂q̇_j) (d/dt)(δq_j) ] dt = 0

Integrate the second term by parts:

∫_t1^t₂ (∂L / ∂q̇_j) (d/dt)(δq_j) dt = [ (∂L / ∂q̇_j) δq_j ]_t1^t₂ - ∫_t1^t₂ (d / dt)(∂L / ∂q̇_j) δq_j dt

Applying endpoint boundary conditions δq_j(t₁) = δq_j(t₂) = 0, the boundary evaluation term vanishes:

[ (∂L / ∂q̇_j) δq_j ]_t1^t₂ = 0

Combining the remaining integrated terms gives:

δI = ∫_t1^t₂ ∑_j=1ⁿ [ (∂L / ∂q_j) - (d / dt)(∂L / ∂q̇_j) ] δq_j dt = 0

Because variations δq_j are arbitrary and independent within the time interval (t₁, t₂), the integrand must vanish identically for each j:

(d / dt)(∂L / ∂q̇_j) - ∂L / ∂q_j = 0

6. Deduction of Hamilton’s Principle from D’Alembert’s Principle

Hamilton’s Principle can be derived directly from D’Alembert’s Principle by integrating over a finite time interval [t₁, t₂].

Derivation Steps:

1. Start with D’Alembert’s Principle in differential form:

∑_i (m_i r̈_i - F_i^(a)) · δr_i = 0

2. Integrate this relation over time from t₁ to t₂:

∫_t1^t₂ ∑_i m_i r̈_i · δr_i dt - ∫_t1^t₂ ∑_i F_i^(a) · δr_i dt = 0   --- (Equation A)

3. Analyze the first term using integration by parts:

∫_t1^t₂ m_i r̈_i · δr_i dt = [ m_i ṙ_i · δr_i ]_t1^t₂ - ∫_t1^t₂ m_i ṙ_i · (d/dt)(δr_i) dt

Since virtual displacements vanish at endpoint times t₁ and t₂ (δr_i(t₁) = δr_i(t₂) = 0), the boundary term is zero. Using (d/dt)(δr_i) = δṙ_i:

∫_t1^t₂ ∑_i m_i r̈_i · δr_i dt = - ∫_t1^t₂ ∑_i m_i ṙ_i · δṙ_i dt = - ∫_t1^t₂ δ [ ∑_i (1/2) m_i ṙ_i² ] dt = - ∫_t1^t₂ δT dt

4. Analyze the second term for conservative forces:

∫_t1^t₂ ∑_i F_i^(a) · δr_i dt = ∫_t1^t₂ δW dt = - ∫_t1^t₂ δV dt

5. Substitute these back into Equation A:

- ∫_t1^t₂ δT dt - ( - ∫_t1^t₂ δV dt ) = 0   ⇒   ∫_t1^t₂ (δT - δV) dt = 0

Factoring out the variational operator δ:

δ ∫_t1^t₂ (T - V) dt = 0   ⇒   δ ∫_t1^t₂ L dt = 0

This completes the formal deduction of Hamilton’s Principle from D’Alembert’s Principle.

7. Applications of Euler-Lagrange Equations

7.1 One-Dimensional Simple Harmonic Oscillator

Consider a particle of mass m attached to a spring with spring constant k, moving along the x-axis.

• Generalized coordinate: q = x
• Generalized velocity: q̇ = ẋ
• Kinetic Energy: T = (1/2) m ẋ²
• Potential Energy: V = (1/2) k x²

Formulating the Lagrangian:

L = T - V = (1/2) m ẋ² - (1/2) k x²

Partial Derivatives:

∂L / ∂ẋ = m ẋ
(d / dt)(∂L / ∂ẋ) = m ẍ
∂L / ∂x = -k x

Applying Euler-Lagrange Equation:

(d / dt)(∂L / ∂ẋ) - ∂L / ∂x = 0   ⇒   m ẍ - (-k x) = 0   ⇒   m ẍ + k x = 0

Dividing by m yields the standard equation of motion for a SHO:

ẍ + ω² x = 0   (where ω = √(k / m))

7.2 Simple Pendulum

Consider a point mass m suspended by a massless, inextensible string of length l under gravity g.

• Generalized coordinate: q = θ (angular displacement from vertical)
• Position coordinates: x = l sin θ, y = l cos θ (taking fixed support point as origin, downward y)
• Velocity: v² = ẋ² + ẏ² = (l θ̇ cos θ)² + (-l θ̇ sin θ)² = l² θ̇²
• Kinetic Energy: T = (1/2) m l² θ̇²
• Potential Energy: V = -m g l cos θ (referencing origin)

Formulating the Lagrangian:

L = T - V = (1/2) m l² θ̇² + m g l cos θ

Partial Derivatives:

∂L / ∂θ̇ = m l² θ̇
(d / dt)(∂L / ∂θ̇) = m l² θ̈
∂L / ∂θ = -m g l sin θ

Applying Euler-Lagrange Equation:

m l² θ̈ - (-m g l sin θ) = 0   ⇒   m l² θ̈ + m g l sin θ = 0

Simplifying by dividing by m l²:

θ̈ + (g / l) sin θ = 0

Small-angle approximation (sin θ ≈ θ): θ̈ + (g / l) θ = 0, giving angular frequency ω = √(g / l).

7.3 Compound Pendulum

A compound pendulum is any rigid body of mass m oscillating about a horizontal axis passing through a pivot point O under gravity.

• Generalized coordinate: q = θ (angle of deflection of line connecting pivot to center of mass)
• Distance: h = distance from pivot O to Center of Mass (CM)
• Moment of Inertia: I = moment of inertia about pivot axis O
• Kinetic Energy: T = (1/2) I θ̇²
• Potential Energy: V = -m g h cos θ (taking pivot O as zero reference level)

Formulating the Lagrangian:

L = T - V = (1/2) I θ̇² + m g h cos θ

Partial Derivatives:

∂L / ∂θ̇ = I θ̇
(d / dt)(∂L / ∂θ̇) = I θ̈
∂L / ∂θ = -m g h sin θ

Applying Euler-Lagrange Equation:

I θ̈ - (-m g h sin θ) = 0   ⇒   I θ̈ + m g h sin θ = 0

Dividing by I:

θ̈ + (m g h / I) sin θ = 0

For small oscillations (sin θ ≈ θ): θ̈ + (m g h / I) θ = 0.

The time period is T = 2π √(I / (m g h)) = 2π √(L_eq / g), where equivalent length L_eq = I / (m h).

7.4 Free-Falling Body in Uniform Gravity

Consider a body of mass m falling vertically under a uniform gravitational acceleration g.

• Generalized coordinate: q = y (vertical displacement measured positive upwards)
• Kinetic Energy: T = (1/2) m ẏ²
• Potential Energy: V = m g y

Formulating the Lagrangian:

L = T - V = (1/2) m ẏ² - m g y

Partial Derivatives:

∂L / ∂ẏ = m ẏ
(d / dt)(∂L / ∂ẏ) = m ÿ
∂L / ∂y = -m g

Applying Euler-Lagrange Equation:

(d / dt)(∂L / ∂ẏ) - ∂L / ∂y = 0   ⇒   m ÿ - (-m g) = 0   ⇒   m ÿ + m g = 0

Simplifying gives the classical acceleration equation:

ÿ = -g

7.5 Summary of System Lagrangians & Equations of Motion