Variational Calculus Part 3: Lagrangian, Hamiltonian, and the Legendre Transform

In the previous part, we derived the Euler-Lagrange equation:

\[\frac{\partial F}{\partial y} - \frac{d}{dx} \left( \frac{\partial F}{\partial y'} \right) = 0\]

This equation is a necessary condition for a function \(y(x)\) to extremize a functional of the form \(J[y] = \int_a^b F(x, y(x), y'(x)) \, dx\).

One of the most celebrated and historically significant applications of variational principles is in classical mechanics. This not only showcases the power of variational calculus but also introduces concepts and mathematical structures, like the Legendre transform, that are fundamental to understanding duality in optimization, including convex conjugation.

In this post, we will:

1. Introduce Lagrangian Mechanics and the Principle of Stationary Action.
2. Show how the Euler-Lagrange equation arises naturally in this context.
3. Discuss the Hamiltonian formulation, derived from the Lagrangian via the Legendre Transform.
4. Highlight the connection between the Legendre transform and convexity, setting the stage for convex analysis.

1. The Principle of Stationary Action and Lagrangian Mechanics

Classical mechanics, as formulated by Newton, describes the motion of objects using forces and accelerations (vectors). However, an alternative and often more powerful perspective was developed by Lagrange and Hamilton, based on the idea that physical systems evolve in a way that optimizes a certain quantity.

Principle. Principle of Stationary Action

A physical system evolves from one configuration to another along a path that makes a quantity called the action stationary (usually a minimum).

Let \(q(t)\) represent the generalized coordinates of a system at time \(t\). These could be positions, angles, or any set of variables that describe the system’s configuration. The time derivative \(\dot{q}(t) = dq/dt\) represents the generalized velocities.

The action, denoted \(S\), is defined as the time integral of a function \(L\), called the Lagrangian:

\[S[q] = \int_{t_1}^{t_2} L(t, q(t), \dot{q}(t)) \, dt\]

Here, the functional is \(S[q]\), the independent variable is time \(t\) (instead of \(x\)), the function being sought is the path \(q(t)\) (instead of \(y(x)\)), and the integrand function is the Lagrangian \(L(t, q, \dot{q})\).

Why does \(L\) depend on \(t, q, \dot{q}\)? This choice is fundamental for describing a vast range of physical systems:

• \(q(t)\): The potential energy of a system often depends on its configuration (e.g., \(V(q)\)).
• \(\dot{q}(t)\): The kinetic energy typically depends on velocities (e.g., \(T \propto \dot{q}^2\)).
• \(t\): Allows for systems where external influences change over time.

The problem is to find the path \(q(t)\) that makes \(S[q]\) stationary. This is precisely the type of problem the Euler-Lagrange equation solves! Substituting \(L\) for \(F\), \(q\) for \(y\), and \(t\) for \(x\), the Euler-Lagrange equation for each generalized coordinate \(q_i\) (if there are multiple) becomes:

\[\frac{\partial L}{\partial q_i} - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) = 0\]

This is the equation of motion in Lagrangian mechanics.

What is the Lagrangian \(L\)?

For many classical mechanical systems, the Lagrangian is found to be the difference between the kinetic energy \(T\) and the potential energy \(V\) of the system:

Definition. Lagrangian for Classical Systems

For a system with kinetic energy \(T(q, \dot{q})\) and potential energy \(V(q, t)\), the Lagrangian is:

\[L(t, q, \dot{q}) = T(q, \dot{q}) - V(q, t)\]

Deriving \(L=T-V\) from Newton’s Laws (1D Example)

Consider a particle of mass \(m\) in one dimension \(q\), subject to a conservative force \(F(q) = -\frac{dV}{dq}\). Newton’s second law is \(m\ddot{q} = F(q)\), or \(m\ddot{q} = -\frac{dV}{dq}\). We can write this as:

\[-\frac{dV}{dq} - \frac{d}{dt}(m\dot{q}) = 0\]

We want to find an \(L(q, \dot{q})\) such that the Euler-Lagrange equation \(\frac{\partial L}{\partial q} - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) = 0\) matches this. Comparing terms:

1. \[\frac{\partial L}{\partial q} = -\frac{dV}{dq}\]
2. \[\frac{\partial L}{\partial \dot{q}} = m\dot{q}\]

Integrating condition (2) with respect to \(\dot{q}\) (treating \(q\) as constant) gives:

\[L = \int m\dot{q} \, d\dot{q} = \frac{1}{2}m\dot{q}^2 + f(q)\]

where \(f(q)\) is an integration “constant” with respect to \(\dot{q}\). We recognize \(\frac{1}{2}m\dot{q}^2\) as the kinetic energy \(T(\dot{q})\). So, \(L = T(\dot{q}) + f(q)\). Now use condition (1):

\[\frac{\partial L}{\partial q} = \frac{\partial}{\partial q}(T(\dot{q}) + f(q)) = \frac{df}{dq}\]

So, \(\frac{df}{dq} = -\frac{dV}{dq}\). Integrating with respect to \(q\) gives \(f(q) = -V(q) + C\). Thus, \(L = T(\dot{q}) - V(q) + C\). The constant \(C\) doesn’t affect the equations of motion, so we can set \(C=0\), yielding \(L = T - V\).

Lagrangian mechanics offers several advantages:

• Scalar Formulation: It works with scalar quantities (energy) rather than vectors (forces).
• Coordinate Independence: The form of the Euler-Lagrange equations is preserved under changes of generalized coordinates.
• Symmetries and Conservation Laws: As we’ll touch upon, symmetries in the Lagrangian directly lead to conserved quantities (Noether’s Theorem).

2. Hamiltonian Mechanics and the Legendre Transform

The Lagrangian formulation uses coordinates \(q\) and velocities \(\dot{q}\). An alternative, often more powerful formulation called Hamiltonian mechanics, uses coordinates \(q\) and generalized momenta \(p\). The transition from the (\(q, \dot{q}\)) description to the (\(q, p\)) description is achieved via a Legendre Transform.

Generalized Momenta

In the Lagrangian framework, the generalized momentum \(p_i\) conjugate to the coordinate \(q_i\) is defined as:

\[p_i = \frac{\partial L}{\partial \dot{q}_i}\]

For a simple particle with \(L = \frac{1}{2}m\dot{x}^2 - V(x)\), the momentum conjugate to \(x\) is \(p_x = \frac{\partial L}{\partial \dot{x}} = m\dot{x}\), which is the familiar linear momentum.

The Legendre Transform

The Legendre transform is a general mathematical procedure for changing the independent variable of a function. If we have a function \(f(x)\), and we define a new variable \(p = \frac{df}{dx}\), the Legendre transform \(f^\ast (p)\) of \(f(x)\) is defined as:

Definition. Legendre Transform

Given a function \(f(x)\), its Legendre transform \(f^\ast (p)\) with respect to \(x\) is:

\[f^\ast(p) = px - f(x)\]

where \(x\) on the right-hand side is implicitly expressed as a function of \(p\) by inverting the relationship \(p = \frac{df}{dx}(x)\). For this inversion to be unique (at least locally), we require \(\frac{d^2f}{dx^2} \neq 0\). Typically, \(f(x)\) is assumed to be strictly convex, meaning \(\frac{d^2f}{dx^2} > 0\), which ensures \(p = f'(x)\) is strictly increasing and thus invertible.

An important property of the Legendre transform is its involutivity (it’s its own inverse, up to a sign if conventions differ) and the symmetric derivative relationship: If \(p = \frac{df}{dx}\), then \(x = \frac{df^\ast }{dp}\).

The Hamiltonian

The Hamiltonian \(H(q, p, t)\) is defined as the Legendre transform of the Lagrangian \(L(q, \dot{q}, t)\) with respect to the generalized velocities \(\dot{q}_i\). The new variables are the generalized momenta \(p_i\).

\[H(q, p, t) = \sum_i p_i \dot{q}_i - L(q, \dot{q}, t)\]

Here, it’s crucial that after performing the transform, the velocities \(\dot{q}_i\) on the right-hand side are expressed as functions of \(q, p, t\) by inverting the set of equations \(p_i = \frac{\partial L}{\partial \dot{q}_i}(q, \dot{q}, t)\). This inversion requires the Hessian matrix \(W_{ij} = \frac{\partial^2 L}{\partial \dot{q}_i \partial \dot{q}_j}\) to be invertible. If \(L\) is strictly convex as a function of the velocities \(\dot{q}\) (i.e., \(W\) is positive definite), this inversion is well-defined.

Hamilton’s Equations of Motion: Using the properties of the Legendre transform and the Euler-Lagrange equations, one can derive Hamilton’s equations of motion:

\[\dot{q}_i = \frac{\partial H}{\partial p_i}\] \[\dot{p}_i = -\frac{\partial H}{\partial q_i}\]

These are a set of first-order differential equations, contrasted with the second-order Euler-Lagrange equations.

What is the Hamiltonian \(H\)? For many common systems where the kinetic energy \(T\) is a quadratic function of velocities and the potential energy \(V\) does not depend on velocities, the Hamiltonian simplifies to the total energy of the system:

\[H = T + V\]

This happens because if \(T = \sum_{j,k} \frac{1}{2} M_{jk}(q) \dot{q}_j \dot{q}_k\) (a quadratic form in \(\dot{q}\)), then Euler’s homogeneous function theorem implies \(\sum_i p_i \dot{q}_i = \sum_i \frac{\partial L}{\partial \dot{q}_i} \dot{q}_i = \sum_i \frac{\partial T}{\partial \dot{q}_i} \dot{q}_i = 2T\). Then \(H = 2T - L = 2T - (T-V) = T+V\).

The Legendre Transform and Convexity

The requirement that \(p = f'(x)\) (or \(p_i = \partial L / \partial \dot{q}_i\)) be invertible is deeply connected to the convexity of the function being transformed.

• If \(f(x)\) is strictly convex, then \(f'(x)\) is strictly increasing, guaranteeing a unique inverse \(x(p)\).
• The Legendre transform of a convex function is also convex. Specifically, if \(W = (\partial^2 L / \partial \dot{q} \partial \dot{q})\) is the Hessian matrix of \(L\) with respect to \(\dot{q}\), then the Hessian matrix of \(H\) with respect to \(p\) is \(W^{-1}\). If \(W\) is positive definite (making \(L\) convex in \(\dot{q}\)), then \(W^{-1}\) is also positive definite (making \(H\) convex in \(p\)).

This connection is not accidental. The Legendre transform is a fundamental example of a duality transformation. In optimization, the Legendre-Fenchel transform (or convex conjugate) generalizes this concept and plays a central role in Lagrangian duality, which is used to solve constrained optimization problems.

Preview: Convex Conjugation The Legendre transform \(f^\ast (p) = \sup_x (px - f(x))\) (using supremum for generality) is known as the convex conjugate or Legendre-Fenchel transform. If \(f\) is convex and closed, then \(f^{\ast \ast }=f\). This concept is pivotal in convex analysis and optimization theory, allowing us to switch between “primal” and “dual” representations of problems. We will explore this in much more detail in the crash course on Convex Analysis.

3. Symmetries and Conservation Laws (Noether’s Theorem)

A profound consequence of the Lagrangian and Hamiltonian formalisms is Noether’s Theorem, which connects continuous symmetries of the Lagrangian to conserved quantities.

• Time Translation Invariance: If \(L\) does not explicitly depend on time (\(\partial L / \partial t = 0\)), then the Hamiltonian \(H\) (often total energy) is conserved (\(dH/dt = 0\)).
• Spatial Translation Invariance: If \(L\) does not depend on a coordinate \(q_k\) (i.e., \(q_k\) is “cyclic” or “ignorable”, so \(\partial L / \partial q_k = 0\)), then the corresponding generalized momentum \(p_k = \partial L / \partial \dot{q}_k\) is conserved (\(dp_k/dt = 0\)).
• Rotational Invariance: If \(L\) is invariant under rotations about an axis, the corresponding angular momentum is conserved.

The fact that physical laws often possess these symmetries (e.g., laws of physics are the same today as yesterday, or here as in a distant galaxy) underpins the fundamental conservation laws of energy, momentum, and angular momentum. These ideas of symmetry and invariants are also powerful in analyzing complex systems, including those in machine learning.

4. What’s Next?

In this post, we’ve seen how variational principles lead to Lagrangian and Hamiltonian mechanics, providing an elegant and powerful framework for describing physical systems. We also introduced the Legendre transform, which not only facilitates the switch from Lagrangian to Hamiltonian views but also serves as a crucial link to the concept of duality and convexity.

This detour through classical mechanics was intended to show:

1. A major historical application and success of variational calculus.
2. The origin of mathematical tools (like the Legendre transform) that are essential in modern optimization.

In the next parts of this crash course on Variational Calculus, we will:

• Apply the Euler-Lagrange equation to solve some classic example problems more directly.
• Discuss generalizations like functionals with higher-order derivatives or multiple independent/dependent variables.
• Briefly touch upon constrained variational problems.

This will further solidify our understanding of variational methods before we eventually move on to more advanced optimization topics like Convex Analysis, where the seeds sown by the Legendre transform will fully blossom.