Related Experiment Video
Updated: Jul 3, 2026
09:43
Optimized Setup and Protocol for Magnetic Domain Imaging with In Situ Hysteresis Measurement
Published on: November 7, 2017
Symplectic and energy-conserving algorithms for solving magnetic field trajectories
1Department of Physics, Texas A&M University, College Station, Texas 77843, USA.
Summary
New algorithms systematically derive particle trajectory integrators in electric and magnetic fields. These methods offer improved stability and accuracy over traditional symplectic integrators, even for complex field interactions.
Area of Science:
- Computational physics
- Numerical analysis
- Plasma physics
Background:
- Symplectic integrators are commonly used for classical Hamiltonian systems.
- Existing methods for charged particles in magnetic fields offer energy conservation but lack generality.
- Combined electric and magnetic fields present challenges for numerical integration accuracy and stability.
Purpose of the Study:
- To systematically derive exponential-splitting algorithms of any order for particle trajectories in general electric and magnetic fields.
- To analyze the behavior of these algorithms in combined fields, comparing them to symplectic integrators.
- To leverage quantum mechanical operator analysis for developing novel numerical integrators.
Main Methods:
- Applying exponential splitting to the noncanonical evolution operator for charged particles.
- Utilizing the angular momentum operator from quantum mechanics for systematic algorithm derivation.
- Direct evaluation of the magnetic field without a vector potential.
Main Results:
- Algorithms derived from splitting the noncanonical evolution operator conserve energy exactly for magnetic fields.
- In combined fields, these algorithms exhibit symplectic-like behavior with qualitatively correct trajectories and bounded energy errors.
- Operator analysis effectively captures the interplay between electric and magnetic forces, enabling the derivation of complex integrators.
Conclusions:
- Exponential-splitting algorithms offer a systematic and powerful approach for accurate particle trajectory simulations in complex electromagnetic fields.
- These derived integrators provide a superior alternative to traditional symplectic methods, particularly in terms of stability and phase error reduction.
- The use of operator analysis represents a significant advancement in the development of numerical methods for computational physics problems.
Related Concept Videos
01:22Motion Of A Charged Particle In A Magnetic Field
A charged particle experiences a force when moving through a magnetic field. Consider the field to be uniform and the charged particle to move perpendicular to it. If the field is in a vacuum, the magnetic field is the dominant factor determining the motion. Since the magnetic force is perpendicular to the direction of motion, a charged particle follows a curved path. The particle continues to follow this curved path until it forms a complete circle. Another way to look at this is that the...
01:26Orthogonal Trajectories
Orthogonal trajectories describe the geometric relationship between two families of curves that intersect each other at right angles. One illustrative case involves a family of parabolas that open sideways along the x-axis. These curves share a common shape but differ by a scaling parameter, resulting in a set of curves that all pass through the origin and widen at different rates.Determining Orthogonal TrajectoriesTo identify the orthogonal trajectories for these parabolas, the first step...
01:27Magnetic Fields
A moving charge or a current creates a magnetic field in the surrounding space, in addition to its electric field. The magnetic field exerts a force on any other moving charge or current that is present in the field. Like an electric field, the magnetic field is also a vector field. At any position, the direction of the magnetic field is defined as the direction in which the north pole of a compass needle points.
A magnetic field is defined by the force that a charged particle experiences...
A magnetic field is defined by the force that a charged particle experiences...
01:23Magnetic Field due to Moving Charges
A stationary charge creates and interacts with the electric field, while a moving charge creates a magnetic field.
Consider a point charge moving with a constant velocity. Like the electric field, the magnetic field at any point is directly proportional to the magnitude of the charge and inversely proportional to the square of the distance between the source point and the field point. However, unlike the electric field, the magnetic field is always perpendicular to the plane containing the line...
Consider a point charge moving with a constant velocity. Like the electric field, the magnetic field at any point is directly proportional to the magnitude of the charge and inversely proportional to the square of the distance between the source point and the field point. However, unlike the electric field, the magnetic field is always perpendicular to the plane containing the line...
01:24Energy In A Magnetic Field
If a magnetic field is sustained, there must be a current in a closed circuit or loop, implying some energy has been spent in creating the field. If this energy is not dissipated via the circuit's resistance, it is stored in the field.
Take an ideal inductor with zero resistance. Although it's practically impossible, assume that the coil's resistance is so small that it is practically negligible. The loss of the field's energy to dissipate thermal energy (or heat) is thus negligible.
The energy...
Take an ideal inductor with zero resistance. Although it's practically impossible, assume that the coil's resistance is so small that it is practically negligible. The loss of the field's energy to dissipate thermal energy (or heat) is thus negligible.
The energy...
01:26Divergence and Curl of Magnetic Field
The magnetic field due to a volume current distribution given by the Biot–Savart Law can be expressed as follows: