Application: Kinematical and dynamical symmetries - Lec 28 - Frederic Schuller
TLDRThe lecture delves into the intricacies of classical mechanics, focusing on the concepts of kinematical and dynamical symmetries within the framework of the cotangent bundle. The lecturer introduces the audience to the idea of a symplectic manifold and its significance in classical mechanics, highlighting the importance of Darboux coordinates in simplifying the symplectic form. The discussion then transitions to the Hamiltonian vector field, Poisson brackets, and the conservation of observables, leading to the definition of a conserved quantity in a dynamical system. The lecturer further explores the distinction between kinematical and dynamical symmetries, providing examples such as angular momentum and the Runge-Lenz vector, and explaining the conditions for a quantity to be considered a symmetry. The talk is enriched with mathematical derivations and geometric interpretations, offering a comprehensive understanding of the subject matter.
Takeaways
- 📚 The lecture focuses on classical mechanics, specifically kinematical and dynamical symmetries, within the context of the cotangent bundle.
- 🌀 A symplectic manifold is defined by a smooth manifold M with a 2-form Ω that is both closed (exterior derivative is zero) and non-degenerate.
- 📏 Darboux's theorem states that around any point in a symplectic manifold, there exist local coordinates where Ω takes a simple form, with non-zero entries only on the diagonal blocks.
- 🔄 The Hamiltonian vector field associated with an observable is defined geometrically and is crucial for understanding the dynamics of the system.
- ⚖️ Poisson brackets are bilinear maps that reflect the algebraic structure of observables on the symplectic manifold and are derived from the Hamiltonian vector field.
- 🚀 Newton's theorem, or Noether's theorem in this context, implies that if one observable is conserved under the flow of another, then the second is conserved under the flow of the first.
- 🔢 Kinematical symmetries are associated with the kinetic energy part of the Hamiltonian and are independent of the potential energy.
- 🌌 Dynamical symmetries, on the other hand, are related to the potential energy and can vary depending on the specific form of the potential.
- 🔬 The push forward of a Hamiltonian vector field must exist and be a Killing vector field for a symmetry to be considered kinematical, which means it generates isometries on the base manifold.
- 🧮 The Killing tensor condition generalizes the Killing vector condition and is used to find the stopping condition for the expansion of conserved quantities in terms of momentum.
- ⏺ The lecture concludes with an algorithm to systematically find symmetries and conserved quantities in classical mechanics, which can be applied to both kinematical and dynamical cases.
Q & A
What is the main topic of the final lecture?
-The main topic of the final lecture is classical mechanics, specifically focusing on kinematical and dynamical symmetries from the perspective of the cotangent bundle.
What are the two conditions for a symplectic manifold?
-A symplectic manifold is defined by two conditions: the symplectic form must be closed (its exterior derivative is zero), and the symplectic form must be non-degenerate (if Ω(X, Y) = 0 for all vector fields Y, then X must be the zero vector field).
What are Darboux coordinates?
-Darboux coordinates are a special set of coordinates on a symplectic manifold where the symplectic form Ω takes a particularly simple form, with non-zero entries only in a block diagonal structure corresponding to the matrix form of the identity map in phase space.
How does the Hamiltonian vector field relate to observables?
-The Hamiltonian vector field is associated with every observable on the symplectic manifold. It is defined through the equation X_F = Ω^sharp(δF), where δF is the exterior derivative of the observable F, and Ω^sharp is the inverse of the symplectic form Ω.
What is the significance of the Poisson bracket?
-The Poisson bracket is a bilinear map that defines the algebraic structure of observables on the symplectic manifold. It is used to define the time evolution of observables under the Hamiltonian dynamics and satisfies properties such as anti-symmetry and the Jacobi identity.
What is a conserved quantity in classical mechanics?
-A conserved quantity in classical mechanics is an observable for which the Poisson bracket with the Hamiltonian is zero. This implies that the observable does not change in time under the dynamics defined by the Hamiltonian.
What is the difference between kinematical and dynamical symmetries?
-Kinematical symmetries are associated with the geometry of the system and are independent of the potential, such as the symmetries generated by the angular momentum. Dynamical symmetries, on the other hand, are related to the potential and the forces present in the system, such as those arising from the Runge-Lenz vector in the Kepler problem.
What is the Noether theorem in the context of classical mechanics?
-Noether's theorem in classical mechanics states that if one observable is conserved with respect to the flow of another, then the second observable is also conserved with respect to the flow of the first. This is a direct consequence of the conservation of the Poisson bracket.
What is the role of the Hamiltonian in defining the change of an observable over time?
-The Hamiltonian, being a distinguished observable, defines the time evolution of all other observables in the system through the equation dF/dt = {F, H}, where {F, H} is the Poisson bracket between the observable F and the Hamiltonian H.
How does the Killing vector field relate to symmetries in the context of the cotangent bundle?
-A Killing vector field is associated with a kinematical symmetry if the Hamiltonian vector field corresponding to a symmetry observable can be pushed forward to the base manifold as a Killing vector field of the metric on that manifold. This means that the flow generated by the Killing vector field leaves the metric invariant.
What is the generalized Killing condition?
-The generalized Killing condition is a condition that a tensor (called a Killing tensor) must satisfy in order for the expansion of a conserved quantity to terminate. This condition is given by the totally symmetrized derivative of the tensor being zero, which ensures the conservation of the quantity under the dynamics of the system.
Outlines
😀 Introduction to Classical Mechanics and Symmetries
The lecture begins with a review of classical mechanics and introduces the concept of kinematical and dynamical symmetries. The distinction between these symmetries is explored from the perspective of the cotangent bundle. The role of observables as smooth functions on a symplectic manifold is discussed, and the importance of Darboux coordinates in revealing the simplicity of the symplectic form is highlighted.
📚 Detailed Explanation of Hamiltonian Vector Fields
The paragraph delves into the definition of a Hamiltonian vector field, which is associated with every observable on a symplectic manifold, not just the Hamiltonian. It explains how these vector fields are derived from the exterior derivative of an observable function and how they relate to the geometric structure of the manifold. The concept of level sets of an observable and how they interact with the Hamiltonian vector field is also discussed.
🔍 Gradient and Hamiltonian Vector Field Visualization
This section focuses on visualizing the Hamiltonian vector field in the context of an observable's level sets. It describes the process of calculating the gradient of a function and how it gives rise to a covector field, which is then transformed using the symplectic form's inverse. The resulting vector field is orthogonal to the level sets, providing a geometric interpretation of the Hamiltonian dynamics.
🌀 Poisson Brackets and Lie Algebra
The lecture introduces Poisson brackets, which are bilinear maps defined for smooth functions on the symplectic manifold. It discusses the properties of Poisson brackets, including their anti-symmetry and the Jacobi identity, which is essential for proving the algebraic structure of the observables. The connection between Hamiltonian vector fields and Lie algebra is also established.
🚀 Hamiltonian Dynamics and Conserved Quantities
The paragraph explains the concept of a dynamical system in classical mechanics, consisting of a symplectic manifold, a Hamiltonian, and how the change of an observable over time is defined. It also discusses the definition of a conserved quantity, which is an observable with a vanishing Poisson bracket with the Hamiltonian. The importance of the Liouville operator and its role in the formal solution of Hamilton's equations is highlighted.
🔗 Kinematical and Dynamical Symmetries Distinction
The lecture distinguishes between kinematical and dynamical symmetries, with kinematical symmetries referring to the geometry of the system without forces, such as straight-line motion. Dynamical symmetries, on the other hand, are related to the potential and forces acting on the system. The conditions for a symmetry to be kinematical are explored, including the requirement for the push forward of the Hamiltonian vector field to exist and be a Killing vector field.
🌐 Application to Hamiltonian Systems
This section discusses how the concepts of kinematical and dynamical symmetries apply to Hamiltonian systems. It provides examples of each type of symmetry, such as angular momentum for kinematical and the Runge-Lenz vector for dynamical. The importance of the Killing equation and the role of Killing tensors in understanding dynamical symmetries are also covered.
🔬 Algorithm for Finding Symmetries
The paragraph outlines an algorithm for finding symmetries of a Hamiltonian system, based on the assumption that a conserved quantity can be expanded in a series involving the momenta. It details the process of deriving a set of conditions that must be satisfied for each order in the momentum, leading to the generalized Killing condition for conserved quantities.
🎓 Angular Momentum as a Kinematical Symmetry
The lecture uses angular momentum as an example to illustrate the conditions for a quantity to be a conserved quantity and a kinematical symmetry. It shows that the angular momentum components satisfy the stopping condition for the series expansion and are thus conserved. The requirement for the potential to be invariant under rotations for the angular momentum to be conserved is also discussed.
📐 Conditions for Kinematical Symmetry
The final paragraph summarizes the conditions for a quantity to be considered a kinematical symmetry. It emphasizes that only first-order terms in the momentum have the potential to be kinematical symmetries, and any dependence on the momentum higher than the first order indicates a dynamical symmetry. The lecture concludes with a thank you to the audience.
Mindmap
Keywords
💡Classical Mechanics
💡Symplectic Manifold
💡Darboux Coordinates
💡Hamiltonian Vector Field
💡Poisson Bracket
💡Conserved Quantity
💡Kinematical Symmetry
💡Dynamical Symmetry
💡Cotangent Bundle
💡Riemannian Manifold
💡Killing Vector
Highlights
Lecture covers classical mechanics, focusing on kinematical and dynamical symmetries
Introduces concept of a symplectic manifold and its properties
Discusses Darboux's theorem and Darboux coordinates
Defines Hamiltonian vector field and its relation to observables
Explains Poisson bracket, its properties, and its relation to Lie algebra
Introduces conserved quantities and the notion of a Hamiltonian system
Differentiates between kinematical and dynamical symmetries
Explains the Noether theorem and its implications
Derives conditions for a quantity to be a kinematical symmetry
Presents algorithm for finding symmetries based on expansion in momentum
Discusses the generalized Killing condition and its role as a stopping condition
Introduces the concept of a Killing tensor
Provides example of angular momentum as a kinematical symmetry
Explains how to determine if a quantity is a kinematical or dynamical symmetry
Shows that only first order in momentum quantities can be kinematical symmetries
Provides insights into the geometric meaning of kinematical symmetries
Concludes with a summary of key takeaways from the lecture
Transcripts
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