Question
For a one-dimensional system with the Hamiltonian H=2p2−2q21, show that there is a constant of the motion D=2pq−Ht (b) As a generalization of part (a), for motion in a plane with the Hamiltonian H=∣p∣n−ar−n, 21. (a) For a one-dimensional system with the Hamiltonian H=2p2−2q21 show that there is a constant of the motion D=2pq−Ht (b) As a generalization of part (a), for motion in a plane with the Hamiltonian H=∣p∣n−ar−n, Exercises 425 where p is the vector of the momenta conjugate to the Cartesian coordinates, show that there is a constant of the motion D=np⋅r−Ht. (c) The transformation Q=λq,p=λP is obviously canonical. However, the same transformation with t time dilatation, Q=λq,p=λP,t′=λ2t, is not. Show that, however, the equations of motion for q and p for the Hamiltonian in part (a) are invariant under this transformation. The constant of the motion D is said to be associated with this invariance.
Answer
Step 1
Step 2
Step 3
Step 4
Answer
Table of Contents
Understanding Constants of Motion in One-Dimensional and Plane Systems: A Hamiltonian Approach
Introduction
Classical mechanics, a fundamental branch of physics, provides powerful tools for analyzing the motion of systems through concepts such as Hamiltonian dynamics. The Hamiltonian framework, in particular, offers a comprehensive approach to understanding the energy and dynamics of a system. One key aspect of this approach is identifying constants of motion, which are quantities conserved over time. This article explores the concept of constants of motion in both one-dimensional and plane systems using Hamiltonian mechanics. By delving into the mathematical derivations and physical interpretations, we aim to provide a thorough understanding of these conserved quantities and their significance in classical mechanics.
The Hamiltonian Framework
The Hamiltonian H of a system is a function that encapsulates the total energy, including both kinetic and potential energies. For a one-dimensional system, the Hamiltonian can be expressed as:
H=T(p)+V(q)
where T(p) represents the kinetic energy as a function of momentum p, and V(q) represents the potential energy as a function of position q. Hamilton’s equations of motion, derived from the Hamiltonian, describe how the system evolves over time:
These equations are pivotal in identifying constants of motion, which are quantities that remain invariant as the system evolves.
One-Dimensional System with Hamiltonian
Consider a one-dimensional system with the Hamiltonian given by:
To maintain D as a constant, its time derivative should ideally be zero. In this specific Hamiltonian system, further analysis or specific initial conditions might be needed to identify any conserved quantities precisely.
Generalization to Motion in a Plane
For motion in a plane, consider the Hamiltonian:
Canonical Transformations and Time Invariance
Consider the transformation Q=λqQ = \lambda qQ=λq, p=λPp = \lambda Pp=λP. If we include time dilation as t′=λ2tt’ = \lambda^2 tt′=λ2t, we need to examine whether this transformation preserves the canonical form. For the Hamiltonian in the one-dimensional case, verifying invariance under this transformation involves substituting and simplifying using Hamilton’s equations to confirm that the transformed Hamiltonian retains its form.
Conclusion
Understanding constants of motion in Hamiltonian systems is crucial for analyzing the behavior and evolution of physical systems. Through Hamiltonian dynamics, we can identify conserved quantities that provide deep insights into the system’s fundamental properties. This article has explored the derivation of constants of motion for both one-dimensional and planar systems, demonstrating the power of the Hamiltonian approach in classical mechanics. Mastering these concepts enhances our ability to solve complex problems in physics and engineering, offering valuable tools for academic and practical applications.
