Variational Principle and Lagrange's Equations

1. Functionals & Extremal Path Problems

In ordinary calculus, we find points $x^*$ that extremize a function $f(x)$. In the calculus of variations, we seek an entire function or path $y(x)$ that extremizes a functional—an integral of the form: where $y'(x) = dy/dx$, and the endpoints are fixed: $y(x_1) = y_1$ and $y(x_2) = y_2$.

2. Variational Derivation of the Euler-Lagrange Equation

Let $y(x)$ be the true path that renders $J$ stationary. Consider a family of varied paths: where $\alpha$ is a continuous parameter, and $\eta(x)$ is an arbitrary differentiable function vanishing at the boundaries: $\eta(x_1) = \eta(x_2) = 0$. The functional becomes a function of $\alpha$: Stationarity requires $\left.\frac{dJ}{d\alpha}\right|_{\alpha=0} = 0$: Integrating the second term by parts: Since $\eta(x_1) = \eta(x_2) = 0$, the boundary term vanishes. Setting $\alpha = 0$: By the Fundamental Lemma of the Calculus of Variations, because $\eta(x)$ is arbitrary, the integrand bracket must vanish everywhere:

1. Definition of the Action Integral

In 1834, Sir William Rowan Hamilton generalized variational mechanics to dynamics. For a conservative dynamical system with configuration described by coordinates $q(t) = (q_1(t), \dots, q_n(t))$, the action integral $S$ is defined as: where $L = T - V$ is the Lagrangian.

2. Statement of Hamilton's Principle

Hamilton's Principle: The actual motion of a holonomic dynamical system from time $t_1$ to $t_2$ follows a trajectory $q(t)$ for which the action integral $S$ is stationary (an extremum, usually a minimum) with respect to arbitrary virtual variations $\delta q_j(t)$ that vanish at the temporal endpoints:

3. Derivation of Lagrange's Equations

Commuting the variation $\delta$ with the time integral: Noting that $\delta \dot{q}_j = \delta \left(\frac{dq_j}{dt}\right) = \frac{d}{dt}(\delta q_j)$ and integrating the second term by parts: The boundary term vanishes identically because $\delta q_j(t_1) = \delta q_j(t_2) = 0$. Therefore: Since the variations $\delta q_j$ are mutually independent, each coefficient must be zero, delivering the Euler-Lagrange equations of motion:

1. Constrained Variations with Undetermined Multipliers

When a system is subjected to $m$ non-holonomic constraints in differential form: virtual displacements (for which $\delta t = 0$) are constrained by: Because the variations $\delta q_j$ are no longer independent, we cannot equate each coefficient in $\delta S = 0$ to zero directly.

2. The Method of Lagrange Multipliers

We introduce $m$ time-dependent undetermined multipliers $\lambda_k(t)$. Multiplying each constraint variation by $\lambda_k(t)$, integrating from $t_1$ to $t_2$, and summing yields: Adding this zero sum to Hamilton's action variation $\delta S = 0$: By appropriately choosing the $m$ multipliers $\lambda_k(t)$, the brackets for $m$ of the coordinates vanish; the remaining $n - m$ variations are independent, requiring their brackets to vanish as well. Thus, for all $j = 1, 2, \dots, n$: The terms $Q_j^{\text{constraint}} = \sum_{k=1}^m \lambda_k a_{kj}$ represent the generalized forces of constraint. Together with the $m$ constraint equations, this provides $n + m$ equations for $n$ coordinates $q_j(t)$ and $m$ multipliers $\lambda_k(t)$.

1. Cyclic Coordinates

If the Lagrangian $L(q, \dot{q}, t)$ does not explicitly depend on a specific coordinate $q_k$, so that: then $q_k$ is termed an ignorable or cyclic coordinate.

2. Immediate Conservation of Conjugate Momentum

Substituting $\frac{\partial L}{\partial q_k} = 0$ into Lagrange's equation:

First Integral of Motion: The generalized momentum conjugate to any cyclic coordinate is strictly conserved throughout the motion.

3. Spatial & Temporal Symmetries

• Spatial Translation Invariance: If the Lagrangian is invariant under spatial translation along direction $\hat{n}$, $\sum_i \vec{F}_i \cdot \hat{n} = 0$, the total linear momentum along $\hat{n}$ is conserved.
• Rotational Invariance: If the Lagrangian is invariant under rotation about axis $\hat{u}$, $\frac{\partial L}{\partial \phi} = 0$, the total angular momentum along $\hat{u}$ is conserved.
• Time Translation Invariance & The Jacobi Integral: If the Lagrangian does not depend explicitly on time ($\frac{\partial L}{\partial t} = 0$), the Jacobi energy integral $h$:
  $$h(q, \dot{q}) = \sum_{j=1}^n \dot{q}_j \frac{\partial L}{\partial \dot{q}_j} - L = \text{constant}$$
  is conserved. When transformations to Cartesian coordinates are scleronomic (time-independent), $h = T + V = E$ (total mechanical energy).

1. Coordinate Parameterization

A planar double pendulum consists of mass $m_1$ connected by a rigid massless rod of length $l_1$ to a fixed pivot, and mass $m_2$ suspended from $m_1$ by a rod of length $l_2$. The generalized coordinates are the deflection angles $\theta_1$ and $\theta_2$ relative to the downward vertical:

2. Kinetic and Potential Energy Formulations

Differentiating positions: The total kinetic energy is: The gravitational potential energy is:

3. Coupled Nonlinear Equations of Motion

Applying the Euler-Lagrange equations yields:

4. Transition to Deterministic Chaos

For small amplitudes ($\theta_1, \theta_2 \ll 1$), these equations linearize into coupled harmonic oscillators yielding two distinct normal modes with real frequencies $\omega_1, \omega_2$. However, at higher energies, the strong nonlinear coupling $\cos(\theta_1 - \theta_2)$ and centrifugal terms induce deterministic chaos:

• Extreme sensitivity to initial conditions (positive Lyapunov exponent $\lambda > 0$).
• Two trajectories separated initially by $|\Delta \vec{x}(0)| \sim 10^{-10}$ diverge exponentially: $|\Delta \vec{x}(t)| \sim |\Delta \vec{x}(0)| e^{\lambda t}$.
• Non-repeating, ergodic phase space orbits while conserving total energy $E = T + V$.