Glossary

6.1 Legendre transformation and Hamilton's equations

2 min read•july 25, 2024

Legendre transformations are the secret sauce of Hamiltonian mechanics. They let us switch from coordinates and velocities to coordinates and momenta, giving us a new way to look at how things move.

are the heart of this approach. They give us a set of first-order equations that describe how a system changes over time, making it easier to solve complex problems and spot important patterns.

Legendre Transformation and Hamiltonian Mechanics

Legendre transformation in Hamiltonian mechanics

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  • switches between conjugate variables transforming functions of one variable to functions of its derivative
  • Role in Hamiltonian mechanics converts Lagrangian formulation to Hamiltonian formulation switching from generalized coordinates and velocities to coordinates and momenta
  • Key properties preserve information content of the original function allowing for a change of independent variables
  • results from Legendre transform applied to Lagrangian expressed in terms of generalized coordinates and momenta
  • Applications simplify certain mechanical problems (pendulum motion) provide foundation for canonical transformations (action-angle variables)

Derivation of Hamilton's equations

  • Start with Lagrangian L(q,q˙,t)L(q, \dot{q}, t) describing system dynamics
  • Define generalized momentum p=∂L∂q˙p = \frac{\partial L}{\partial \dot{q}} representing system's tendency to change
  • Perform Legendre transformation defining Hamiltonian H(q,p,t)=pq˙−L(q,q˙,t)H(q, p, t) = p\dot{q} - L(q, \dot{q}, t)
  • Derive Hamilton's equations q˙=∂H∂p\dot{q} = \frac{\partial H}{\partial p} and p˙=−∂H∂q\dot{p} = -\frac{\partial H}{\partial q} through partial differentiation
  • Equivalence to Lagrange's equations yields same dynamics as Euler-Lagrange equations ensuring consistency
  • Advantages of Hamiltonian formulation include first-order differential equations and symmetry between position and momentum variables

Solving systems with Hamilton's equations

  • Simple harmonic oscillator example:
    1. Write Lagrangian L=12mx˙2−12kx2L = \frac{1}{2}m\dot{x}^2 - \frac{1}{2}kx^2
    2. Derive Hamiltonian H=p22m+12kx2H = \frac{p^2}{2m} + \frac{1}{2}kx^2
    3. Formulate Hamilton's equations x˙=pm\dot{x} = \frac{p}{m}, p˙=−kx\dot{p} = -kx
    4. Solve resulting first-order differential equations
    5. Compare with solution from Lagrangian approach
  • Advantages of Hamiltonian approach simplify systems with cyclic coordinates (angular momentum conservation) provide natural framework for canonical transformations (generating functions)
  • Comparison with Lagrangian method yields equivalent results choice depends on problem specifics (constraints, symmetries)

Physical meaning of generalized coordinates

  • Generalized coordinates specify system configuration using Cartesian coordinates, angles, or other parameters (pendulum length, spring extension)
  • Generalized momenta conjugate to coordinates represent system's tendency to change without constraints
  • Physical interpretations in linear systems often correspond to mass times velocity in rotational systems may represent angular momentum
  • encompasses all possible values of q and p each point represents unique system state (position and momentum of a particle)
  • Conservation laws emerge from cyclic coordinates leading to conserved momenta total energy conservation when Hamiltonian is time-independent
  • Canonical transformations preserve Hamilton's equations form allow choosing convenient coordinates for specific problems (polar coordinates for central force)

Key Terms to Review (17)

Canonical coordinates: Canonical coordinates are a set of position and momentum variables used in classical mechanics, typically denoted as $(q_i, p_i)$, that form the foundation for Hamiltonian mechanics. These coordinates provide a systematic way to describe the state of a mechanical system and facilitate the formulation of Hamilton's equations, which describe the evolution of a system over time. By transforming from Lagrangian to Hamiltonian formalism through the Legendre transformation, canonical coordinates allow for a clearer understanding of dynamics in both classical and quantum contexts.
Continuity: Continuity refers to a property of functions where small changes in the input result in small changes in the output. In the context of complex analysis and mechanics, continuity is crucial for understanding how functions behave, particularly when evaluating limits, derivatives, and transformations. This property ensures that the function is well-behaved across its domain, allowing for further analysis and application in both theoretical and practical settings.
Convexity: Convexity refers to the property of a function where, for any two points on its graph, the line segment connecting these points lies above or on the graph. This feature is crucial in many areas, especially in optimization and analysis, as it indicates that the function has a unique minimum. In mechanics, convex functions often represent energy landscapes, where the stability and properties of systems can be assessed based on their convexity.
Correspondence Principle: The correspondence principle is a concept that asserts that the predictions of quantum mechanics must converge to those of classical mechanics in the appropriate limit, typically as quantum numbers become large. This principle bridges the gap between the two frameworks, ensuring that quantum theory aligns with classical physics in scenarios where classical mechanics is expected to hold true, thus providing a consistent understanding across different physical regimes.
Extremal Conditions: Extremal conditions refer to the criteria that determine the maximum or minimum values of a function, often used in optimization problems. In the context of mechanics, these conditions are crucial for deriving equations of motion and understanding system behavior by identifying points where energy is minimized or maximized, such as in the Legendre transformation and Hamilton's equations.
Functional Derivative: The functional derivative is a generalization of the derivative concept applied to functionals, which are mappings from a space of functions to real numbers. It represents how a functional changes as the underlying function changes, and it plays a critical role in variational principles, where one seeks to find functions that optimize certain quantities. Understanding functional derivatives is essential for deriving Hamilton's equations from a given Lagrangian framework through the Legendre transformation.
Gradient: The gradient is a vector operator that represents the rate and direction of change of a scalar field. It points in the direction of the greatest rate of increase of the scalar function and has a magnitude equal to the maximum rate of change at that point. This concept is particularly crucial when working with Legendre transformations and Hamilton's equations, where gradients help relate different sets of variables and transitions between different formulations of mechanics.
Hamilton's equations: Hamilton's equations are a set of first-order differential equations that describe the evolution of a physical system in classical mechanics. They provide a reformulation of Newtonian mechanics, emphasizing energy conservation and enabling a clearer understanding of systems with multiple degrees of freedom. These equations are fundamental in connecting classical mechanics to other areas, such as field theory, adiabatic processes, and advanced mathematical techniques like Poisson brackets.
Hamiltonian function: The Hamiltonian function is a fundamental concept in classical mechanics that represents the total energy of a system in terms of its generalized coordinates and momenta. It plays a crucial role in the Legendre transformation, allowing for the transition from Lagrangian mechanics to Hamiltonian mechanics. The Hamiltonian serves as the basis for Hamilton's equations, which describe how a system evolves over time.
Joseph-Louis Lagrange: Joseph-Louis Lagrange was an influential mathematician and physicist known for his contributions to classical mechanics, particularly in the formulation of the Lagrangian framework. His work provided powerful mathematical tools, such as the principle of least action and the method of Lagrange multipliers, which help to analyze systems with constraints and derive equations of motion in a systematic way.
Legendre transformation: The Legendre transformation is a mathematical operation that transforms a function into another function by swapping its dependent and independent variables, particularly useful in physics to change from one set of variables to another. It plays a vital role in mechanics, especially in the formulation of Hamiltonian mechanics where it relates the Lagrangian function to the Hamiltonian function through the momentum variables.
Momentum space: Momentum space is a conceptual framework used in physics to represent the states of a system based on their momentum instead of their position. In this space, each point corresponds to a specific momentum value, allowing for a different perspective on the dynamics of a system, particularly in the context of Hamiltonian mechanics, where it's crucial for formulating Hamilton's equations and analyzing the energy and motion of particles.
Partial Derivatives: Partial derivatives represent the rate of change of a function with respect to one variable while keeping other variables constant. This concept is crucial in understanding how functions behave in multi-variable contexts, especially when dealing with functionals in physics and mechanics, where quantities often depend on several variables simultaneously.
Phase Space: Phase space is a multidimensional space where all possible states of a system are represented, with each state corresponding to a unique point in this space. In classical mechanics, it includes both position and momentum coordinates, allowing for a complete description of the system's dynamics. This concept is crucial as it provides a framework for analyzing the behavior of mechanical systems, connecting various mathematical methods and quantum principles.
Quantization: Quantization refers to the process of constraining an observable to take on discrete values, which is a fundamental concept in both classical and quantum mechanics. This process is crucial for developing mathematical frameworks that describe physical systems, leading to the realization that certain quantities, like energy and momentum, can only exist in specific discrete levels. Understanding quantization allows for the transition from classical descriptions of motion to quantum mechanical representations, highlighting the differences between classical and quantum behavior.
Symplectic structure: A symplectic structure is a geometric framework that arises in the study of Hamiltonian mechanics, characterized by a non-degenerate, skew-symmetric bilinear form on a phase space. It provides the mathematical foundation for the formulation of Hamilton's equations and captures the conservation properties of physical systems, linking classical mechanics to modern geometric concepts.
William Rowan Hamilton: William Rowan Hamilton was an Irish mathematician and physicist known for his groundbreaking work in classical mechanics, particularly through the development of Hamiltonian mechanics. His formulation transformed the way physics understood dynamics by introducing concepts such as the Hamilton-Jacobi equation, which is essential for solving complex mechanical systems and connecting them to action-angle variables.
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