# The Teenager’s Problem:
Efficient Garment Decluttering With Grasp Optimization

Aviv Adler*1, Ayah Ahmad*1, Shengyin Wang2, Wisdom C. Agboh1,2, Edith Llontop1
Tianshuang Qiu1, Jeffrey Ichnowski3, Mehmet Dogar2, Thomas Kollar4, Richard
Cheng4, Ken Goldberg1 ∗Equal Contribution1University of California, Berkeley,
USA.2University of Leeds, UK.3Carnegie Mellon University, USA.4Toyota Research
Institute, USA.

###### Abstract

This paper addresses the “Teenager’s Problem”: efficiently removing scattered
garments from a planar surface. As grasping and transporting individual
garments is highly inefficient, we propose analytical policies to select grasp
locations for multiple garments using an overhead camera. Two classes of
methods are considered: _depth-based_ , which use overhead depth data to find
efficient grasps, and _segment-based_ , which use segmentation on the RGB
overhead image (without requiring any depth data); grasp efficiency is
measured by _Objects per Transport_ , which denotes the average number of
objects removed per trip to the laundry basket. Experiments suggest that both
depth- and segment-based methods easily reduce Objects per Transport (OpT) by
$20\%$; furthermore, these approaches complement each other, with combined
_hybrid_ methods yielding improvements of $34\%$. Finally, a method employing
_consolidation_ (with segmentation) is considered, which manipulates the
garments on the work surface to increase OpT; this yields an improvement of
$67\%$ over the baseline, though at a cost of additional physical actions.

##  I Introduction

We introduce the “Teenager’s Problem”: removing a large number of scattered
garments from a surface (e.g. the floor of a teenager’s room, or a work
surface) in the shortest time. This problem has applications in hotels, retail
dressing rooms, garment manufacturing, and other domains where heaps of
garments must be manipulated efficiently.

In this paper, we first formalize the Teenager’s Problem and then consider
several methods to solve it. Consider Fig. 1 with multiple garments on a work
surface. Given an overhead RGB or RGBD image, what robot pick-and-place
actions would minimize the total time to remove all of the garments? Removing
individual garments, one at a time, would be inefficient. Therefore the robot
should use the deformable nature of garments and pick multiple garments at a
time.

Figure 1: An instance of the Teenager’s Problem in the experimental setup; the
scale automatically records weight data as experiments are run.

Given a scene like the one in Fig. 1, one approach is to ignore the separation
between individual garments and to treat the whole scene as a homogeneous
volume to be removed. This motivates depth-based methods, i.e., methods that
use the depth image to infer grasp points that would then remove as much
volume as possible. We consider two depth-based methods in this paper. The
first method uses height and grasps at the highest point of the scene. The
second method estimates a volume, by integrating the depth data within a grasp
radius, and grasps at the point in the scene that gives the largest estimated
volume.

The depth-based methods are able to identify large heaps and grasp multiple
garments at a time, even when some of these garments are completely buried
under others and therefore not individually visible to the camera. However,
the depth-based methods also miss some good grasps. Particularly, since they
do not detect individual garment positions and boundaries, they miss grasp
points that have lower height/volume but would still pick multiple garments
simultaneously, e.g., points where multiple garment boundaries meet.

Therefore, a second approach to solving the Teenager’s Problem is to
distinguish between individual garments and optimize grasps to pick as many of
these garments as possible. This motivates segment-based methods, i.e.,
methods that use the RGB image to segment the individual garments. We consider
one such method in this paper. This method uses the Segment Anything Model
(SAM) [11] to distinguish the individual garments. Then, given a set of
garments and a candidate grasp point, it predicts the probability that those
garments will be picked by that grasp. The method then uses these predictions
to optimize the grasp to pick the largest number of garments.

While the segment-based method is able to identify grasps that would pick
multiple garments simultaneously, it can also miss some good grasps.
Particularly, since only the top surface of the garment pile is visible to the
camera, garments that are under others are ignored by the segment-based
method.

Therefore, in this paper, we also consider a hybrid method that attempts to
combine the complementary strengths of the two approaches. The hybrid method,
given the current state of the scene, uses either a depth-based method or a
segment-based method, depending on the maximum height available.

Finally, we consider methods that make use of _consolidation_ actions, which
are movements within the workspace to gather the garments into heaps, before
removing them. This improves the efficiency of grasps to transport the objects
to the bin at the cost of the time used in consolidation, which can improve
efficiency in cases where the removal bin is located far from the work
surface.

Experiments suggest that both depth- and segment-based methods significantly
reduce robot trips to the basket by $20\%$; furthermore, these approaches
complement each other, with combined _hybrid_ methods yielding improvements of
$34\%$; finally, OpT can be further increased to $67\%$ by taking additional
_consolidation_ actions within the workspace in order to set up extremely
efficient transport actions.

We make the following contributions:

  * •

A formalization of the Teenager’s Problem.

  * •

Five methods (two depth-based, one segment-based, and two hybrid) to generate
effective multi-garment grasps.

  * •

A method that uses heap consolidation along with the segment-based grasp
generation method to efficiently solve the Teenager’s Problem.

  * •

Physical experiments and data from grasping 2000 garments, that compare the
performance of the various methods above, as well as a random baseline.

##  II Related work

Our work is related to two lines of work: manipulation of deformable objects
and multi-object manipulation.

###  II-A Manipulation of Deformable Objects

Prior work on deformable object manipulation includes folding [3, 8], fabric
smoothing [19, 7, 21], bed-making [20], untangling ropes [22], and singulating
clothes from a heap [27, 23]. Several works aimed to detect specific features,
such as the corners and edges of fabrics, and to identify optimal grasp points
[15, 13, 6]. Other techniques employ deep learning to identify successful
grasps [12, 5]. Some studies have focused on determining optimal grasp points
by considering not only the depth of the cloth but also targeting wrinkles as
highly graspable regions [17, 16, 26]. These prior works focus on manipulating
a single deformable object at a time or singulating a deformable object from
among others. Our work, on the other hand, concentrates on grasping multiple
garments simultaneously.

Figure 2: An example of the segment-based grasp point selection algorithm
(before orientations are chosen). From left to right: _(1)_ The original
overhead RGB image. _(2)_ The cleaned segmentation ${\mathcal{M}}$. _(3)_ The
segments expanded by gripper radius to include ‘nearby’ garment points (all
grasps must be done on garment points as a sanity check); overlapping regions
thus correspond to points that are near multiple segments. _(4)_ The maximal
area (points near a maximal set of segments) with the chosen grasp point shown
in yellow.

###  II-B Multi-object Manipulation

Multi-object grasping can improve decluttering efficiency [1]. It has been
studied, with analytic methods [28], learning-based methods [2, 4], and with
special gripper designs [14]. The focus, however, has remained on rigid
objects. Instead, we consider the problem of grasping multiple deformable
objects at a time, and using such grasps to efficiently clear a surface.

Multi-object manipulation scenarios can encompass cluttered [10] environments,
which can include both deformable and rigid objects [25], however, the goal is
again to singulate the objects to grasp them individually. Prior work on
manipulating multiple rigid objects used methods such as pushing, stacking,
and destacking [9, 1, 18]. In cluttered scenes with multiple rigid objects,
one method for determining how, or where, to grasp is by using image
segmentation [24], detecting and isolating individual objects in the scene. We
also use a segmentation approach but for deformable objects.

##  III The Teenager’s Problem

We formulate the Teenager’s Problem as an abstract framework for studying
garment decluttering, where deformable objects rest on a planar work surface.
Alongside this cluttered work surface, a fixed target bin is provided, and the
goal is to transfer all the garments efficiently from the workspace to the bin
in the shortest possible time. This problem fundamentally translates into
executing a sequence of pick-and-place motions, which fall into two
categories: first, movements from the workspace to the target bin
(“transports”), and second, optional movements within the workspace
(“rearrangements”) to rearrange the garments to simplify future operations.
The importance of this distinction is that the target bin may lie at some
distance from the workspace; when the bin is far, transports become more
expensive relative to rearrangements, which affects the optimal policy. We
assume the availability of a predefined pick-and-place method, from a
designated pick grasp pose to a designated place grasp pose. Consequently, the
central question becomes: for any given scene, which grasp pose should be
chosen for picking, and which for placing? Each pair of grasps, encompassing
the pick and place actions, is denoted as a “move”.

To formalize this problem, let $n$ garments rest on a 2D surface
$[x_{\\_}{\min},x_{\\_}{\max}]\times[y_{\\_}{\min},y_{\\_}{\max}]$, with the
target bin located at the origin $(0,0)$; at any step $k$, garment $i$’s pose
is denoted as $G^{i}(k)$, and the set of all garment poses is denoted
${\mathcal{G}}(k)=\\{G^{1}(k),\dots,G^{n}(k)\\}$ (starting at
${\mathcal{G}}(0)$ at step $0$); a garment which has been removed from the
work surface is also removed from ${\mathcal{G}}(k)$. Let the $k$th _move_ be
defined by a _pick_ tuple
${\bm{p}}_{\\_}{\text{pick}}(k)=(x_{\\_}{\text{pick}}(k),y_{\\_}{\text{pick}}(k),\theta_{\\_}{\text{pick}}(k))$
(representing the grasp) and a _place_ tuple
${\bm{p}}_{\\_}{\text{place}}=(x_{\\_}{\text{place}}(k),y_{\\_}{\text{place}}(k),\theta_{\\_}{\text{place}}(k))$,
where
$(x_{\\_}{\text{pick}}(k),y_{\\_}{\text{pick}}(k))\in[x_{\\_}{\min},x_{\\_}{\max}]\times[y_{\\_}{\min},y_{\\_}{\max}]$,
$(x_{\\_}{\text{place}}(k),y_{\\_}{\text{place}}(k))\in[x_{\\_}{\min},x_{\\_}{\max}]\times[y_{\\_}{\min},y_{\\_}{\max}]\cup(0,0)$,
and
$\theta_{\\_}{\text{pick}}(k),\theta_{\\_}{\text{place}}(k)\in[-\pi/2,\pi/2]$
(since parallel-jaw grippers are symmetric). Moves with
$(x_{\\_}{\text{place}}(k),y_{\\_}{\text{place}}(k))=(0,0)$ represent
transports. Note that transport cost can be adjusted by moving the workspace.

Each move takes time proportional to the Euclidean travel distance, including
the distance from the previous place location to the current pick location,
plus a fixed time $t_{\\_}{\text{grasp}}$ needed to execute the grasp. The
time to execute the $k$th move is:

$\displaystyle T({\bm{p}}_{\\_}{\text{pick}}$
$\displaystyle(k),{\bm{p}}_{\\_}{\text{place}}(k))=$
$\displaystyle\|(x_{\\_}{\text{pick}}(k),y_{\\_}{\text{pick}}(k))-(x_{\\_}{\text{place}}(k-1),y_{\\_}{\text{place}}(k-1))\|_{\\_}2$
$\displaystyle+$
$\displaystyle\|(x_{\\_}{\text{place}}(k),y_{\\_}{\text{place}}(k))-(x_{\\_}{\text{pick}}(k),y_{\\_}{\text{pick}}(k))\|_{\\_}2+t_{\\_}{\text{grasp}}$

Each move then changes the garment poses according to an update function
$f(\cdot)$: given a state ${\mathcal{G}}(k-1)$ and a move
$({\bm{p}}_{\\_}{\text{pick}}(k),{\bm{p}}_{\\_}{\text{place}}(k))$,

$\displaystyle{\mathcal{G}}(k)=f({\mathcal{G}}(k-1),({\bm{p}}_{\\_}{\text{pick}}(k),{\bm{p}}_{\\_}{\text{place}}(k)))$

Then, given starting state ${\mathcal{G}}(0)$ and gripper location
$(x_{\\_}{\text{place}}(0),y_{\\_}{\text{place}}(0),\theta_{\\_}{\text{place}}(0))$
the goal is to find moves
$({\bm{p}}_{\\_}{\text{pick}}(k),{\bm{p}}_{\\_}{\text{place}}(k))$ for
$k=1,\dots,k_{\\_}{\text{finish}}$ to solve:

minimize
$\displaystyle\sum_{\\_}{k=1}^{k_{\\_}{\text{finish}}}T({\bm{p}}_{\\_}{\text{pick}}(k),{\bm{p}}_{\\_}{\text{place}}(k))$
subject to $\displaystyle{\mathcal{G}}(k_{\\_}\text{finish})=\emptyset$

where $k_{\\_}{\text{finish}}$ is also a variable.

However, the above problem formulation does not consider the real-world
complications associated with estimating garment poses and properties.
Garments are deformable objects capable of adopting complex poses and may
obscure one another, leading to partial information about their positions;
furthermore, they behave in complex ways, making accurate state update
predictions very challenging.

Our investigation of the Teenager’s Problem is conducted under the following
conditions:

  * •

The scene is captured using a fixed overhead camera.

  * •

The robot has one parallel-jaw gripper.

The central question remains: When presented with an overhead RGBD image of
garments scattered on a work surface, what should be the next move? For
simplicity, we consider grasps with the gripper in a fully vertical
orientation, just above the work surface, aiming to maximize the number of
garments gripped simultaneously. Each grasp is defined by a tuple
$(x,y,\theta)$, where $(x,y)$ denotes its location on the 2D work surface
(referred to as ${\mathcal{X}}$), and $\theta$ represents the angle of the
gripper jaws.

##  IV Teenager’s Problem Methods

This study explores various strategies for efficiently grasping multiple
garments concurrently, categorized broadly into two types: _depth-based_ and
_segment-based_ approaches.

All the methods described below use a pre-processing of the RGB pixels to
determine the _garment points_ , denoted ${\mathcal{X}}_{\\_}g$. Since we
assume the system knows the color and/or pattern of the background, this is
achieved with color thresholding.

###  IV-A Depth-Based Methods

Depth-based methods use the depth output of the RGBD overhead camera to select
the next grasp. To solve the Teenager’s Problem, these methods are used
repetitively until the workspace is clear: at each step, we capture a new
depth image of the scene, use one of the methods below to generate a new
grasp, and then execute that grasp. We examine two variations:

####  IV-A1 Height

This method selects the highest point in the scene and chooses the orientation
to be the orientation of the major axis of a local principal component
analysis (PCA) around the grasp point.

####  IV-A2 Volume

This method considers the total volume of garments in a disc of radius $R$
around a candidate grasp point $(x,y)$, which is estimated by summing the
heights of all the pixels within that radius, and then selects the point with
the largest total volume. As in the Height method, orientation is selected
using a local PCA.

###  IV-B Segment-Based Methods

The segment-based approach divides the task into _cycles_ ; a segmentation is
generated at the start of each cycle and a sequence of grasps is carried out
based on that segmentation (however, an overhead RGB image is still taken
between each grasp). Each cycle relies on four subroutines:

(1) Segmentation and cleanup; (2) prediction; (3) grasp selection; (4)
execution. Once all the planned moves have been performed the next cycle
begins. The steps of a cycle are detailed below.

####  IV-B1 Segmentation and cleanup

The method uses Meta’s Segment Anything Model (SAM) [11] with the _vit-b_
weights, and prompts the image as a whole. However, the initial segmentation
often contains multiple overlapping segments, gaps, and regions corresponding
to the work surface itself. The method then performs redundant segment
removal, gap filling, and color thresholding to eliminate segments
representing the work surface.

####  IV-B2 Prediction

Given a set of segments ${\mathcal{M}}$ on the workspace and a proposed grasp
point $(x,y,\theta)$, predicting which segments $M\in{\mathcal{M}}$ will be
removed by the grasp is essential. To achieve this, we designed an analytic
predictor. We denote the predictor algorithm as $p$, which takes a grasp
$(x,y,\theta)$ and a segment $M$ and returns a probability

$\displaystyle p((x,y,\theta),M)\in[0,1]\,.$

The predictor models a grasp $(x,y,\theta)$ as affecting an elliptical region
of the workspace, denoted $E(x,y,\theta)$, which is centered at $(x,y)$ with
its major axis oriented along $\theta$; the major and minor axes lengths
($r_{\\_}1,r_{\\_}2$) are scaled to be the length and width of the parallel-
jaw gripper. The probability that a segment $M$ is removed by $(x,y,\theta)$
is then estimated as

$\displaystyle p((x,y,\theta),M)=\frac{\text{area}(E(x,y,\theta)\cap
M)}{\text{area}(E(x,y,\theta)\cap M)+b}$

where $b$ is a normalization constant. This predictor is intended to capture
the following intuition: the gripper directly affects the area under it, and
the more any segment falls in that area, the more likely it is to be removed
by the grasp (but obviously cannot have probability $>1$ of being removed).
Note that this predictor can never be 100% certain that a given segment will
be removed, which is realistic.

####  IV-B3 Grasp selection

With a clean segmentation and predictor, the method faces the task of
determining which grasps to execute from an infinite number of potential
options. To simplify the problem, the method first selects a set of grasp
points (excluding orientations) and then selects an orientation for each
point.

The grasp point selection algorithm does the following (given a radius $r>0$
in pixels, and segmentation ${\mathcal{M}}$):

  * •

For each pixel $(x,y)\in{\mathcal{X}}_{\\_}g$, determine the set
$S(x,y)\subseteq{\mathcal{M}}$ of segments which are within distance $r$ from
$(x,y)$.

  * •

Construct the set ${\mathcal{S}}$ of _maximal sets_ $S(x,y)$, i.e.
${\mathcal{S}}$ consists of every $S(x,y)$ which is not a proper subset of
some other $S(x^{\prime},y^{\prime})$.

  * •

For each maximal set $S_{\\_}i\in{\mathcal{S}}$, choose a (uniformly) random
$(x_{\\_}i,y_{\\_}i)$ such that $S(x_{\\_}i,y_{\\_}i)=S_{\\_}i$.

  * •

Return $\\{(x_{\\_}i,y_{\\_}i)\\}_{\\_}{i=1}^{m}$, where $m$ is the number of
maximal sets.

The rationale behind this algorithm is that a grasp point should be near as
many segments as possible but not too similar to other grasp points and that
for any grasp point that is not in a maximal set, there is another that is
near strictly more segments than it.

Then the algorithm chooses an orientation for each grasp point using a greedy
heuristic. First, it enumerates a list of $\ell$ equally-spaced orientations
in $[-\pi/2,\pi/2]$ and runs the predictor for each orientation on all the
segments; the orientation which is predicted to remove the largest number of
masks (i.e. the sum of the predicted probability of removal over all the
masks) is then chosen for each grasp. For a balance between efficiency and
thoroughness, we used $\ell=6$.

####  IV-B4 Grasp execution

The algorithm then attempts all grasps in sequence, in increasing order of
distance to the bin; this is to prevent, as far as possible, dragging garments
from disturbing the positions of the garments that remain (which may cause
garments to fall off the work surface).

An RGB image is also captured after each transfer to the bin (when the arm is
out of frame), although a new segmentation is _not_ generated (until the next
cycle). Instead, for each planned grasp remaining, the difference between its
current state and the state at the beginning of the cycle (when the
segmentation was generated) is estimated using the squared difference between
the pixel values within a small square neighborhood around the grasp point; if
the difference is too large, the grasp is deemed to be in a different state
from when it was planned and is not performed. This ensures that grasps are
not performed unless the system knows that the local configuration of the
garments is approximately the same as when it was planned.

###  IV-C Hybrid methods

One observation borne out by the experiments was that depth-based methods and
segment-based methods have different strengths – in particular depth-based
methods excel at picking occluded garments (which generally result in taller
piles) while segment-based methods excel at simultaneously picking adjacent
garments. This motivated the idea of considering _hybrid_ methods which make
use of both depth data and segmentation data.

To take advantage of how these methods complement each other, hybrid methods
do the following: given a height threshold, if the tallest pile is taller than
the threshold, execute a (single) grasp as given by the depth-based method; if
all piles are below the threshold, execute one cycle of the segment-based
method.

###  IV-D Segment-based method with consolidation

Another avenue to improving OpT is to first consolidate the garments into
large piles for transport to the bin; this can improve overall efficiency in
cases where the bin is located at some distance from the work surface, making
transports costly relative to manipulations within the workspace. An efficient
primitive for consolidation is the _grasp sequence_ where each pick-and-place
movement picks up where the last one placed; this both saves on robot movement
time (no travel distance to the next pick point) and, ideally, allows the
robot to accumulate more garments as it goes before depositing them in the
bin.

An extension of the segment-based method above to include consolidations is
the following:

  1. 1.

generate the grasp points as in the no-rearrangement segment-based method;

  2. 2.

starting from the furthest grasp point, estimate the _expected area_ of
grasped garments, and execute the next available grasp which does not exceed a
given expected grasped area threshold;

  3. 3.

if no such grasp exists, transport the currently-held garments to the bin.

Going from the furthest grasp point to the closest follows the intuition that
a method using consolidation should consolidate towards the bin since this
will always shorten the distance between the grasped garments and the bin,
even if some are dropped along the way – and this will also tend to compress
them into a smaller space, facilitating later multi-object grasps.

The grasp area threshold corresponds to the intuition that the gripper has a
limit to the amount of fabric it can hold and thus trying to accumulate more
than that limit in one grasp is counterproductive.

###  IV-E Baseline

Finally, as a baseline, we use the _random_ method, which uniformly randomly
selects a garment point $(x,y)\in{\mathcal{X}}_{\\_}g$ with a uniformly random
orientation $\theta\in[-\pi/2,\pi/2]$, accounting for the gripper’s symmetry.

TABLE I: OpT (Objects per Transport) by grasping method | Depth-Based Methods |  | Hybrid Methods |
---|---|---|---|---
Random | Volume | Height | Segment | Volume/Segment | Height/Segment | Segment with Consolidation
$1.48\pm 0.11$ | $1.76\pm 0.13$ | $1.77\pm 0.20$ | $1.77\pm 0.13$ | $1.87\pm 0.13$ | $1.98\pm 0.24$ | $2.47\pm 0.2$

##  V Experiments

We tested all algorithms on the test set of 10 garments (see Fig. 3) with 25
sample runs. Each sample run begins with a randomized scene containing all 10
test set garments, and ends when the workspace is cleared of garments.

Figure 3: The test set of 10 garments, representing a variety of different
sizes, weights, textures, colors, patterns, flexibility, and garment classes.

###  V-A Data collection pipeline

To run the experiments, we used a semi-autonomous data collection pipeline, in
which experimental scene reset, randomization and data recording are done
automatically, with the experimenter only needing to correct problems when
they arise (for instance, if a garment falls off the work surface, the
experimenter must return it for the next sample). The system uses the recorded
weight data to automatically notify the experimenter when such a problem
occurs, to minimize the amount of human attention necessary for data
collection.

The scene is automatically reset in the following way:

  1. 1.

The robot grasps the bin and empties it over the work surface to deposit the
garments, then places the bin back to its original position.

  2. 2.

The robot executes a sequence of random pick-and-place actions on the surface
to shuffle the garments; in our experiments, 10 such moves were performed for
every scene reset.

Then the experiment is performed with the selected method, recording at each
step the overhead RGBD output, grasp location and orientation, and weight of
the garments in the bin. The experiment is paused for 3 seconds after every
transport to allow the scale’s output to settle.

###  V-B Metrics evaluated

Our metric for evaluating the methods is _Objects per Transport_ (OpT), which
denotes the average number of objects taken during each transport and measures
the general effectiveness of the performed grasps. We use OpT (as opposed to
Picks Per Hour (PPH)) as our metric as OpT directly measures grasp quality and
PPH depends heavily on implementation details (particularly concerning
computation time) which reduces its reliability as a metric in this setting.

For each algorithm tested, OpT was evaluated on all 25 sample runs, which were
then averaged to yield the final result and $95\%$ confidence bounds.

##  VI Results

The results of the experiments are given in Table I, show that both depth-
based and segment-based methods yield clear improvements (approximately $20\%$
additional OpT) over the baseline. Furthermore, these approaches complement
each other: hybridizing them yielded $26\%$ and $34\%$ more OpT (for
volume/segment and height/segment respectively) as compared to the baseline.
However, it is important to note that only the segment-based method achieves
this without the use of depth information, which may not be available on all
systems.

It should be noted that while grasp quality is the focus and OpT is the most
meaningful metric for this work, the ultimate goal remains improving pick
efficiency as measured by PPH. The depth-based methods, which do not perform
significant computations to find grasps, improve PPH from $477$ for the
baseline to $526$ and $525$ for max-volume and max-height grasps respectively.
While the segmentation method registers a slight decrease in PPH (to $453$),
optimizing the methods’ speed may increase the PPH up to a comparable $523$
(determined by subtracting computation time in the experiments) with no need
for depth data.

Finally, at the cost of both computational overhead and additional physical
actions, the segmentation with consolidation method improved OpT by $67\%$
over the baseline.

###  VI-A Comparison of Methods

What are the inherent advantages and disadvantages of the two approaches
outlined above?

  * •

Depth-based methods require both RGB and depth images to compute grasps. In
contrast, segment-based methods solely rely on RGB images, rendering them
suitable for systems lacking depth cameras.

  * •

Segment-based methods often demand more computational resources, as they
involve neural network-driven segmentation and subsequent cleanup. To ensure
efficient computation without compromising speed, it may be necessary to
deploy GPUs or opt for segmentation methods optimized for CPU processing.

  * •

A notable challenge faced by segment-based methods is their limited ability to
detect grasps that remove occluded garments. In contrast, depth-based methods
more often grasp over occluded garments due to their utilization of depth
information, which provides an enhanced perception of garment depth.

  * •

Conversely, segment-based methods explicitly choose grasps to simultaneously
capture garments situated closely together, whereas depth-based methods cannot
determine which points are in proximity to multiple visible garments.

##  VII Conclusion

In this work, we tackle the challenging problem of robotic garment
decluttering, by formalizing the Teenager’s Problem and developing both depth-
and segment-based methods to solve it. We used recent advances in image
segmentation [11] to explore an approach that uses it to distinguish garments
in the image and find grasps that are likely to capture as many as possible.

###  VII-A Dirty Laundry and Future Work

However, this work has certain limitations and leaves a number of areas open
for improvement:

  * •

All the proposed methods rely on accurately separating the garments from the
work surface using the RGB image, which is done here via color thresholding.
While this was reliable in our experimental setup, a different system may be
needed in practice.

  * •

While all the methods considered here grasp at a fixed height above the work
surface with a perfectly vertical gripper, the most efficient grasp may not
share those characteristics. Additional improvements might be obtained by
optimizing the grasp height or angle.

  * •

While the $67\%$ OpT increase from the segmentation with consolidation method
is large, an average of roughly $4.0$ rearrangement actions were performed per
transport saved over the baseline. Nevertheless, such methods may increase
efficiency in cases where transports are relatively costly, e.g. when the
target bin is far from the workspace.

Extensions such as sorting of clothes (for instance, separating clothing by
type or color) are a natural fit for the techniques discussed here, especially
the segment-based approach. Additionally, although we present only the
analytic grasp predictor, the segment-based method described in Section IV-B
is designed to be compatible with any predictor. Thus, a possible direction
for future work is to improve the predictor.

## References

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# High-order Field Theory and Weak Euler-Lagrange-Barut Equation for Classical
Relativistic Particle-Field Systems

Peifeng Fan  corresponding author<EMAIL_ADDRESS>Key Laboratory of
Optoelectronic Devices and Systems, College of Physics and Optoelectronic
Engineering, Shenzhen University, Shenzhen 518060, China  Advanced Energy
Research Center, Shenzhen University, Shenzhen 518060, China  Qiang Chen
corresponding author<EMAIL_ADDRESS>National Supercomputing Center in
Zhengzhou, Zhengzhou University, Zhengzhou, Henan 450001, China  Jianyuan Xiao
corresponding author<EMAIL_ADDRESS>School of nuclear science and
technology, University of Science and Technology of China, Hefei, Anhui
230026, China

###### Abstract

It is widely accepted that conservation laws, especially energy-momentum
conservation, have fundamental importance for both classical and quantum
systems in physics. A widely used method to derive the conservation laws is
based on Noether’s theorem. However, for classical relativistic particle-field
systems, this process is still impeded. Different from the quantum situation,
the obstruction emerged when we regard the particle’s field as a classical
world line. The difficulties come from two aspects. One is the mass-shell
constraint and the other comes from the heterogeneous-manifolds that particles
and fields reside on. This study develops a general geometric (manifestly
covariant) field theory for classical relativistic particle-field systems. In
considering the mass-shell constraint, the Euler-Lagrange-Barut (ELB) equation
as a geometric version of the Euler-Lagrange (EL) equation is applied to
determine the world lines of the relativistic particles. As a differential
equation in the standard field theory, the infinitesimal criterion of the
symmetry condition is converted into an integro-differential equation. To
overcome the second difficulty, we develop a weak ELB equation on the 4D
space-time. The weak version of the ELB equation will play an essential role
in establishing the connections between symmetries and local conservation
laws. Using field theory together with the weak ELB equation developed here,
the conservation laws can be systematically derived from the symmetries that
the systems admit.

high-order field theory, weak Euler-Lagrange-Barut equation, infinitesimal
criterion of symmetric condition, Noether’s theorem, geometric conservation
laws

##  I introduction

Classical relativistic particle-field systems, where many particles interact
with the self-generated and high background fields, are often encountered in
astrophysics (Ma and Wang, 1995; Wei and Wu, 2021; Beklemishev and Tessarotto,
2004), accelerator physics (Xie and Du, 2007) and gyrokinetic systems in
plasma physics (Littlejohn, 1984; Beklemishev and Tessarotto, 2004, 1999). For
these systems, one of the significant topics is focused on the derivation of
energy and momentum conservation laws (Boghosian, 1987; Similon, 1985; Brizard
and Chan, 1999).

As a widely accepted fundamental principle, conservation laws can be derive by
the corresponding symmetry that the Lagrangian (or action) of the system
admit. This is the so-called Noether’s theorem (Noether, 1918). This method
has been widely used in deriving the energy-momentum conservation in quantum
field systems by the space-time translation symmetry (Peskin and Schroeder,
2018). However, for classical particle-field systems, the derivation process
is still elusive. For example, for electromagnetic system coupled with
relativistic particles, the energy-momentum conservation was first derived by
Landau and Lifshitz (Landau and Lifshitz, 1971), and just reformulated into a
geometric form recently from the space-time translation symmetry (Fan  _et
al._ , 2018).

Different from the quantum systems, the dynamics of the particles and the
fields reside on heterogeneous manifolds. The fields, e.g., the
electromagnetic fields, are defined on the 4D space-time domain, while the
particles as world lines in Minkowski space are only defined on 1D parametric
space (e.g, time-axis), which make the standard Euler-Lagrange (EL) breakdown
(Qin  _et al._ , 2014). This difficulty has been overcame recently by
transformed the EL equation into a weak form (Qin  _et al._ , 2014; Fan  _et
al._ , 2019, 2020), and the corresponding method has been applied in non-
relativistic systems, such as the Vlasov-Possion (VP) system, Vlasov-Darwin
(VD) system and gyrokinetic system in plasma physics.

However, for relativistic situation with the geometric setting, this theory
was only reformulated for the Maxwell’s system (Fan  _et al._ , 2018). In
promoting the method to geometric formalism, the proper-time parameter rather
than time parameter is often used, which yields the mass-shell constraint. In
considering this constraint, the equation of motion, derived by the Hamilton’s
principle, will be the Euler-Lagrange-Barut (ELB) equation instead of the
standard EL equation (Barut, 1964; Brehme, 1971). Besides, the using of
proper-time parameter also makes the Lagrangian density a functional instead a
function of particle’s world line. Due to the existing difficulties, for
general classical relativistic particle-field systems, the weak form equation
of particle’s motion and infinitesimal criterion of symmetry condition for is
still unclear.

In this study, we will extended the theory constructed in Ref. (Fan  _et al._
, 2018) to general high-order field systems. High-order electromagnetic field
theories always appear in the study of gyrokinetic systems for magnetized
plasma (Qin  _et al._ , 2007) and the Podolsky system (Podolsky, 1942; Bopp,
1940) for the radiation reaction of classical charged particles. The weak ELB
equation for Maxwell’s system is also extended to general situations. The
infinitesimal criterion of symmetry condition is also reformulated in
considering the geometric setting. Moreover, because the Lagrangian density is
a functional, the criterion we derived here is then a integro-differential
equation. This is quite different from the standard field theory, in which the
Lagrangian density is a function, and the infinitesimal criterion is
consequently a differential equation (Olver, 1993). Using the the general weak
ELB equation and the infinitesimal criterion developed here, the conservation
laws can be systematically derived from the underlying symmetries. As a simple
but nontrivial high-order case, we will calculate the energy-moment
conservation laws for Podolsky system coupled with relativistic charged
particles.

The rest of this paper is organized as follows. In Sec. II, we will introduce
the action of a general classical relativistic particle-field systems and the
weak ELB equation is established as well. In Sec. III, the geometric
infinitesimal criterion of the symmetry is developed as required by the mass-
shell constraint. Using the geometric weak Euler-Lagrange equation and
geometric infinitesimal criterion, the conservation laws are obtained in Sec.
IV. In the last section, we will derive the energy-momentum conservation laws
for high-order electromagnetic systems coupled with charged particles.

##  II classical relativistic particle-field systems and geometric weak Euler-
Lagrange-Barut equation

We start from the geometric action of the particle-field system and extend the
theory developed in Ref. (Fan  _et al._ , 2018, 2019) to the general
relativistic situation. The geometric action of a classical relativistic
particle-field system is generally written as

$\mathcal{A}=\sum_{a}\int_{a_{1}}^{a_{2}}L_{a}\left(\tau_{a},\bm{\chi}_{a}\left(\tau_{a}\right),\dot{\bm{\chi}}_{a}\left(\tau_{a}\right),\mathrm{pr}^{(n)}\bm{\psi}\left(\bm{\chi}_{a}\left(\tau_{a}\right)\right)\right)d\tau_{a}+\int_{\Omega}\mathcal{L}_{F}\left(\bm{\chi},\mathrm{pr}^{(n)}\bm{\psi}\left(\bm{\chi}\right)\right)d^{4}\bm{\chi},$
(1)

where the subscript $a$ labels particles, $\bm{\chi}$ is the space-time
position, $\bm{\psi}\left(\bm{\chi}\right)$ is a vector (or any tensor) field
defined on space-time. Here,
$\mathrm{pr}^{(n)}\bm{\psi}\left(\bm{\chi}\right)$ is the prolongation of the
field $\bm{\psi}\left(\bm{\chi}\right)$, which contains
$\bm{\psi}\left(\bm{\chi}\right)$ and its derivatives up to the $n$th order,
i.e.,

$\mathrm{pr}^{(n)}\bm{\psi}\left(\bm{\chi}\right)=\left(\bm{\psi}\left(\bm{\chi}\right),\partial_{\mu_{1}}\bm{\psi}\left(\bm{\chi}\right),\cdots,\partial_{\mu_{1}}\partial_{\mu_{2}}\cdots\partial_{\mu_{n}}\bm{\psi}\left(\bm{\chi}\right)\right),$
(2)

where $\partial_{\mu_{i}}$ $\left(i=1,2,\cdots,n\right)$ represents derivative
with respect to space time coordinates. In Eq. (1),
$\bm{\chi}_{a}\left(\tau_{a}\right)$ is the world line of the $a$th particle
and $\tau_{a}$ is the proper time parameter.
$\dot{\bm{\chi}}_{a}\left(\tau_{a}\right)\equiv
d\bm{\chi}_{a}\left(\tau_{a}\right)/d\tau_{a}$ is the 4-velocity of the $a$th
particle which satisfy the mass-shell constraint

$\dot{\chi}_{a}^{\mu}\dot{\chi}_{a\mu}=c^{2},$ (3)

where the Lorentzian metric
$\eta_{\mu\nu}=\mathrm{diag}\left\\{1,-1,-1,-1\right\\}$ is applied to define
$\dot{\chi}_{a\mu}$, i.e.,
$\dot{\chi}_{a\mu}=\eta_{\mu\nu}\dot{\chi}_{a}^{\nu}$. Due to the subsidiary
condition (3), the variation of the action $\mathcal{A}$ with respect to
$\delta\bm{\chi}_{a}$ won’t yields the standard Euler-Lagrange equation but
the equation as follows

$E_{\bm{\chi}_{a}\mu}\left(L_{a}\right)=0,$ (4)

where the linear operator $E_{\bm{\chi}_{a}\mu}$ is defined by

$E_{\bm{\chi}_{a}\mu}\coloneqq\frac{\partial}{\partial\chi_{a}^{\mu}}-\frac{d}{d\tau_{a}}\left[\frac{\partial}{\partial\dot{\chi}_{a}^{\mu}}+\frac{1}{c^{2}}\left(\mathrm{id}-\dot{\chi}_{a}^{\nu}\frac{\partial}{\partial\dot{\chi}_{a}^{\nu}}\right)\right],$
(5)

which is different from the standard Euler operator by the existence of the
last two terms. To be faithful to history, equation
$\left(\ref{eq:ELB}\right)$ was first obtained by Barut using the Lagrange
multiplier method (Barut, 1964). Therefore, we will refer to the linear
operator $E_{\bm{\chi}_{a}\mu}$ as the Euler-Lagrange-Barut (ELB) operator and
call equation (4) the ELB equation.

The integral of the $a$th particle’s Lagrangian $L_{a}$ in Eq. (1) is along an
arbitrary time-like world line (denoted by $l_{a}$) which connected two fixed
world points $a_{1}$ and $a_{2}$ at the space-time $\mathbb{R}^{4}$, while the
integral of the field’s Lagrangian density $\mathcal{L}_{F}$ is over the the
space-time domain $\Omega$. Hence, the integral of the Lagrangian density
$\mathcal{L}_{F}$ for the field $\bm{\psi}$ is over space-time, and the
integral of Lagrangian $L_{a}$ for the $a$th particle is over proper time
only. As a consequence, the action in the form (1) is not easily applicable
for Noether’s procedure of deriving local conservation laws.

To deal with this problem, we multiply the first part on the right-hand side
of Eq. (1) by the following identity

$\int\delta_{a}d^{4}\bm{\chi}=1,$ (6)

where
$\delta_{a}\equiv\delta\left(\bm{\chi}-\bm{\chi}_{a}\left(\tau_{a}\right)\right)$
is the Dirac’s delta function. The action $\mathcal{A}$ in Eq. (1) is then
transformed into one integral over space-time,

$\displaystyle\mathcal{A}=\int_{\Omega}\mathcal{L}\left(\bm{\chi},\left[\bm{\chi}_{a}\right],\mathrm{pr}^{(n)}\bm{\psi}\left(\bm{\chi}\right)\right)d^{4}\bm{\chi},$
(7)

where

$\displaystyle\mathcal{L}=\sum_{a}\mathcal{L}_{a}+\mathcal{L}_{F}\left(\bm{\chi},\mathrm{pr}^{(n)}\bm{\psi}\left(\bm{\chi}\right)\right),$
(8)
$\displaystyle\mathcal{L}_{a}=\int_{a_{1}}^{a_{2}}\ell_{a}d\tau_{a},\;\ell_{a}=\ell_{a}\left(\tau_{a},\bm{\chi},\mathrm{pr}^{(1)}\bm{\chi}_{a}\left(\tau_{a}\right),\mathrm{pr}^{(n)}\bm{\psi}\left(\bm{\chi}\right)\right)=L_{a}\delta_{a}.$
(9)

Different from the non-geometric situation, the Lagrangian density
$\mathcal{L}$ is not a function but a functional of the particle’s world line.
To differ from the other local variables (such as $\bm{\chi}$ and
$\bm{\psi}\left(\bm{\chi}\right)$), we enclose $\bm{\chi}_{a}$ by square
brackets.

We now calculate how the action (7) varies in response to the variations of
$\delta\bm{\chi}_{a}$ and $\delta\bm{\psi}$,

$\delta\mathcal{A}=\sum_{a}\int_{a_{1}}^{a_{2}}\left[\int_{\Omega}E_{\bm{\chi}_{a}\mu}\left(\ell_{a}\right)d^{4}\bm{\chi}\right]\delta\chi_{a}^{\mu}d\tau_{a}+\int_{\Omega}E_{\bm{\psi}}\left(\mathcal{L}\right)\cdot\delta\bm{\psi}d^{4}\bm{\chi},$
(10)

where

$E_{\bm{\psi}}=\frac{\partial}{\partial\bm{\psi}}+\sum_{j=1}^{n}\left(-1\right)^{j}D_{\mu_{1}}\cdots
D_{\mu_{j}}\frac{\partial}{\partial\left(\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}\bm{\psi}\right)}$
(11)

is the Euler operator for the field $\bm{\psi}$. Here, $\mathrm{id}$ in Eq.
(5) is the identity operator. Using Hamilton’s principle, we immediately
obtain the equation of motion for particles and fields

$\displaystyle\int_{\Omega}E_{\bm{\chi}_{a}\mu}\left(\ell_{a}\right)d^{4}\bm{\chi}=0,$
(12) $\displaystyle E_{\bm{\psi}}\left(\mathcal{L}\right)=0.$ (13)

Similar to the submanifold EL equation given in Refs. (Qin  _et al._ , 2014;
Fan  _et al._ , 2019), equation (12) is called the submanifold ELB equation.
Using the linear property of ELB operator, we can easily prove that the
submanifold ELB equation (12) is equivalent to ELB equation (4).

To apply equation (12) to Noether’s method, we need the explicit expression of
$E_{\bm{\chi}_{a}\mu}\left(\ell_{a}\right)$. For electromagnetic system, the
expression was given in the previous work (Fan  _et al._ , 2018). We now
derive a general expression of $E_{\bm{\chi}_{a}\mu}\left(\ell_{a}\right)$. We
first transform the first and the last three terms of Eq. (12) into following
forms

$\displaystyle\frac{\partial\ell_{a}}{\partial\chi_{a}^{\mu}}=\frac{\partial}{\partial\chi_{a}^{\mu}}\left(L_{a}\delta_{a}\right)=\frac{\partial\delta_{a}}{\partial\chi_{a}^{\mu}}L_{a}+\frac{\partial
L_{a}}{\partial\chi_{a}^{\mu}}\delta_{a}=-\frac{\partial\delta_{a}}{\partial\chi^{\mu}}L_{a}+\frac{\partial
L_{a}}{\partial\chi_{a}^{\mu}}\delta_{a}$
$\displaystyle=-\frac{D}{D\chi^{\mu}}\left(L_{a}\delta_{a}\right)+\frac{\partial
L_{a}}{\partial\chi_{a}^{\mu}}\delta_{a}=\frac{D}{D\chi^{\nu}}\left(-L_{a}\delta_{a}\eta_{\>\mu}^{\nu}\right)+\frac{\partial
L_{a}}{\partial\chi_{a}^{\mu}}\delta_{a}.$ (14)
$\displaystyle-\frac{d}{d\tau_{a}}\left\\{\left[\frac{\partial
L_{a}}{\partial\dot{\chi}_{a}^{\mu}}+\frac{1}{c^{2}}\dot{\chi}_{a\mu}\left(L_{a}-\dot{\chi}_{a}^{\sigma}\frac{\partial
L_{a}}{\partial\dot{\chi}_{a}^{\sigma}}\right)\right]\delta_{a}\right\\}$
$\displaystyle=-\frac{d\delta_{a}}{d\tau_{a}}\left[\frac{\partial
L_{a}}{\partial\dot{\chi}_{a}^{\mu}}+\frac{1}{c^{2}}\dot{\chi}_{a\mu}\left(L_{a}-\dot{\chi}_{a}^{\sigma}\frac{\partial
L_{a}}{\partial\dot{\chi}_{a}^{\sigma}}\right)\right]-\frac{d}{d\tau_{a}}\left[\frac{\partial
L_{a}}{\partial\dot{\chi}_{a}^{\mu}}+\frac{1}{c^{2}}\dot{\chi}_{a\mu}\left(L_{a}-\dot{\chi}_{a}^{\sigma}\frac{\partial
L_{a}}{\partial\dot{\chi}_{a}^{\sigma}}\right)\right]\delta_{a}.$ (15)

The first term on the right-hand side of Eq. (15) can be rewritten as

$\displaystyle-\frac{d\delta_{a}}{d\tau_{a}}\left[\frac{\partial
L_{a}}{\partial\dot{\chi}_{a}^{\mu}}+\frac{1}{c^{2}}\dot{\chi}_{a\mu}\left(L_{a}-\dot{\chi}_{a}^{\sigma}\frac{\partial
L_{a}}{\partial\dot{\chi}_{a}^{\sigma}}\right)\right]=-\dot{\chi}_{a}^{\nu}\frac{\partial\delta_{a}}{\partial\chi_{a}^{\nu}}\left[\frac{\partial
L_{a}}{\partial\dot{\chi}_{a}^{\mu}}+\frac{1}{c^{2}}\dot{\chi}_{a\mu}\left(L_{a}-\dot{\chi}_{a}^{\sigma}\frac{\partial
L_{a}}{\partial\dot{\chi}_{a}^{\sigma}}\right)\right]$
$\displaystyle=\frac{D}{D\chi^{\nu}}\left\\{\dot{\chi}_{a}^{\nu}\left[\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\mu}}+\frac{1}{c^{2}}\dot{\chi}_{a\mu}\left(\ell_{a}-\dot{\chi}_{a}^{\sigma}\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\sigma}}\right)\right]\right\\}.$
(16)

Substituting Eqs. (14)-(16) into $E_{\bm{\chi}_{a}\mu}\left(\ell_{a}\right)$,
we have

$E_{\bm{\chi}_{a}\mu}\left(\ell_{a}\right)=\frac{D}{D\chi^{\nu}}\left\\{-\ell_{a}\eta_{\>\mu}^{\nu}+\dot{\chi}_{a}^{\nu}\left[\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\mu}}+\frac{1}{c^{2}}\dot{\chi}_{a\mu}\left(\ell_{a}-\dot{\chi}_{a}^{\sigma}\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\sigma}}\right)\right]\right\\},$
(17)

where we used the ELB equation (4). We will refer to Eq. (17) as the weak ELB
equation, which as a differential equation is equivalent to the submanifold
ELB equation (12). Just like the weak EL equation used in non-relativistic
particle-field systems, the weak ELB equation is also elemental in deriving
local conservation laws for relativistic situation.

##  III geometric infinitesimal criterion of symmetry condition

We now turn to discuss the symmetries of the relativistic particle-field
systems. A symmetry of the action $\mathcal{A}$ is a group of transformation

$\displaystyle
g_{\epsilon}\cdot\left(\tau_{a},\bm{\chi},\bm{\chi}_{a}\left(\tau_{a}\right),\bm{\psi}\left(\bm{\chi}\right)\right)\equiv\left(Z_{a\epsilon}\left(\tau_{a},\bm{\chi},\bm{\chi}_{a},\bm{\psi}\right),\bm{\Xi}_{\epsilon}\left(\tau_{a},\bm{\chi},\bm{\chi}_{a},\bm{\psi}\right),\right.$
$\displaystyle\left.\vphantom{\bm{\Phi}_{\epsilon}}\bm{\Theta}_{a\epsilon}\left(\tau_{a},\bm{\chi},\bm{\chi}_{a},\bm{\psi}\right),\bm{\Phi}_{\epsilon}\left(\tau_{a},\bm{\chi},\bm{\chi}_{a},\bm{\psi}\right)\right)=\left(\tilde{\tau}_{a},\tilde{\bm{\chi}},\tilde{\bm{\chi}}_{a},\tilde{\bm{\psi}}\left(\tilde{\bm{\chi}}\right)\right)$
(18)

such that

$\displaystyle\int_{\Omega}\mathcal{L}\left(\bm{\chi},\left[\bm{\chi}_{a}\right],\mathrm{pr}^{(n)}\bm{\psi}\left(\bm{\chi}\right)\right)d^{4}\bm{\chi}=\int_{\tilde{\Omega}}\mathcal{L}\left(\tilde{\tau}_{a},\tilde{\bm{\chi}},\left[\tilde{\bm{\chi}}_{a}\right],\tilde{\bm{\psi}}\left(\tilde{\bm{\chi}}\right)\right)d^{4}\tilde{\bm{\chi}},$
(19)

and

$\dot{\chi}_{a}^{\mu}\dot{\chi}_{a\mu}=\dot{\tilde{\chi}}_{a}^{\mu}\dot{\tilde{\chi}}_{a\mu}=c^{2},$
(20)

where
$g_{\epsilon}=\left(Z_{a\epsilon},\bm{\Xi}_{\epsilon},\bm{\Theta}_{a\epsilon},\bm{\Phi}_{\epsilon}\right)$
constitutes a continuous group of transformations parameterized by $\epsilon$.
Equations (19) and (20) are called the symmetry conditions. Different from the
standard field systems, the second condition (20) is needed now due to the
mass-shell constraint.

To derive the local conservation laws, an infinitesimal version of the
symmetry condition is required. For this purpose, we take the derivative of
Eqs. (19) and (20) with respect to $\epsilon$ at $\epsilon=0$,

$\frac{d}{d\epsilon}\bigg{|}_{0}\int_{\tilde{\Omega}}\left[\sum_{a}\int_{\tilde{a}_{1}}^{\tilde{a}_{2}}\ell_{a}\left(\tilde{\tau}_{a},\tilde{\bm{\chi}},\mathrm{pr}^{(1)}\tilde{\bm{\chi}}_{a}\left(\tilde{\tau}_{a}\right),\mathrm{pr}^{(n)}\tilde{\bm{\psi}}\left(\tilde{\bm{\chi}}\right)\right)d\tilde{\tau}_{a}+\mathcal{L}_{F}\left(\tilde{\bm{\chi}},\mathrm{pr}^{(n)}\tilde{\bm{\psi}}\left(\tilde{\bm{\chi}}\right)\right)\right]d^{4}\tilde{\bm{\chi}}=0.$
(21)
$\frac{d}{d\epsilon}\bigg{|}_{0}\left[\dot{\tilde{\chi}}_{a}^{\mu}\dot{\tilde{\chi}}_{a\mu}\right]=0$
(22)

Substituting Eqs. (8) and (9) into Eq. (21), we have

$\displaystyle\int_{\Omega}\frac{d}{d\epsilon}\bigg{|}_{0}\left\\{\left[\sum_{a}\int_{a_{1}}^{a_{2}}\ell_{a}\left(\tilde{\tau}_{a},\tilde{\bm{\chi}},\mathrm{pr}^{(1)}\tilde{\bm{\chi}}_{a}\left(\tilde{\tau}_{a}\right),\mathrm{pr}^{(n)}\tilde{\bm{\psi}}\left(\tilde{\bm{\chi}}\right)\right)j_{a}\left(\epsilon\right)d\tau_{a}\right.\right.$
$\displaystyle\left.\vphantom{\int_{\tilde{a}_{1}}^{\tilde{a}_{2}}}\left.\vphantom{\int_{\tilde{a}_{1}}^{\tilde{a}_{2}}}+\mathcal{L}_{F}\left(\tilde{\bm{\chi}},\mathrm{pr}^{(n)}\tilde{\bm{\psi}}\left(\tilde{\bm{\chi}}\right)\right)\right]\mathrm{det}\bm{J}\left(\epsilon\right)\right\\}d^{4}\bm{\chi}=0,$
(23)

where

$j_{a}\left(\epsilon\right)\equiv\frac{d\tilde{\tau}_{a}}{d\tau_{a}},\;\bm{J}\left(\epsilon\right)\equiv\frac{\partial\tilde{\bm{\chi}}}{\partial\bm{\chi}},$
(24)

Because equation (23) survive for any small integral domains, the integrand
must be zero, i.e.,

$\displaystyle\frac{d}{d\epsilon}\bigg{|}_{0}\left[\sum_{a}\int_{a_{1}}^{a_{2}}\ell_{a}\left(\tilde{\tau}_{a},\tilde{\bm{\chi}},\mathrm{pr}^{(1)}\tilde{\bm{\chi}}_{a}\left(\tilde{\tau}_{a}\right),\mathrm{pr}^{(n)}\tilde{\bm{\psi}}\left(\tilde{\bm{\chi}}\right)\right)j_{a}\left(\epsilon\right)d\tau_{a}\right.$
$\displaystyle\left.\vphantom{\int_{\tilde{a}_{1}}^{\tilde{a}_{2}}}+\mathcal{L}_{F}\left(\tilde{\bm{\chi}},\mathrm{pr}^{(n)}\tilde{\bm{\psi}}\left(\tilde{\bm{\chi}}\right)\right)\right]\mathrm{det}\bm{J}\left(\epsilon\right)=0.$
(25)

Equation (25) can be finally transformed into

$\sum_{a}\int_{a_{1}}^{a_{2}}\left[\mathrm{pr}^{\left(n\right)}\bm{v}\left(\ell_{a}\right)+\ell_{a}\dot{\zeta}_{a}\right]d\tau_{a}+\mathrm{pr}^{\left(n\right)}\bm{v}\left(\mathcal{L}_{F}\right)+\mathcal{L}D_{\mu}\xi^{\mu}=0,$
(26)

where

$\bm{v}=\frac{d}{d\epsilon}\bigg{|}_{0}g_{\epsilon}\cdot\left(\tau_{a},\bm{\chi},\bm{\chi}_{a},\bm{\psi}\right)=\sum_{a}\zeta_{a}\frac{\partial}{\partial\tau_{a}}+\bm{\xi}\cdot\frac{\partial}{\partial\bm{\chi}}+\sum_{a}\bm{\theta}_{a}\cdot\frac{\partial}{\partial\bm{\chi}_{a}}+\bm{\phi}\cdot\frac{\partial}{\partial\bm{\psi}}$
(27)

is the infinitesimal generator of the group transformation, and where

$\begin{cases}\zeta_{a}=\zeta_{a}\left(\tau_{a},\bm{\chi},\bm{\chi}_{a},\bm{\psi}\right)=\frac{d}{d\epsilon}\bigg{|}_{0}Z_{a\epsilon}\left(\tau_{a},\bm{\chi},\bm{\chi}_{a},\bm{\psi}\right),\\\
\bm{\xi}=\bm{\xi}\left(\tau_{a},\bm{\chi},\bm{\chi}_{a},\bm{\psi}\right)=\frac{d}{d\epsilon}\bigg{|}_{0}\bm{\Xi}_{\epsilon}\left(\tau_{a},\bm{\chi},\bm{\chi}_{a},\bm{\psi}\right),\\\
\bm{\theta}_{a}=\bm{\theta}_{a}\left(\tau_{a},\bm{\chi},\bm{\chi}_{a},\bm{\psi}\right)=\frac{d}{d\epsilon}\bigg{|}_{0}\bm{\Theta}_{a\epsilon}\left(\tau_{a},\bm{\chi},\bm{\chi}_{a},\bm{\psi}\right),\\\
\bm{\phi}=\bm{\phi}\left(\tau_{a},\bm{\chi},\bm{\chi}_{a},\bm{\psi}\right)=\frac{d}{d\epsilon}\bigg{|}_{0}\bm{\Phi}_{\epsilon}\left(\tau_{a},\bm{\chi},\bm{\chi}_{a},\bm{\psi}\right).\end{cases}$
(28)

Here, $\mathrm{pr}^{\left(n\right)}\bm{v}$ is the prolongation of $\bm{v}$
defined by

$\displaystyle\mathrm{pr}^{\left(n\right)}\bm{v}=\frac{d}{d\epsilon}\bigg{|}_{0}\left(\tilde{\tau}_{a},\tilde{\bm{\chi}},\mathrm{pr}^{\left(1\right)}\tilde{\bm{\chi}}_{a},\mathrm{pr}^{\left(n\right)}\tilde{\bm{\psi}}\left(\tilde{\bm{\chi}}\right)\right)$
$\displaystyle=\bm{v}+\sum_{a}\theta_{a1}^{\mu}\frac{\partial}{\partial\dot{\chi}_{a}^{\mu}}+\sum_{k=1}^{n}\phi_{\mu_{1}\dots\mu_{k}}^{\alpha}\frac{\partial}{\partial\left(D_{\mu_{1}}\cdots
D_{\mu_{k}}\psi^{\alpha}\right)},$ (29)

where

$\displaystyle\begin{cases}\theta_{a1}^{\mu}=\zeta_{a}\ddot{\chi}_{a}^{\mu}+\dot{q}_{a}^{\mu},\\\
\phi_{\mu_{1}\dots\mu_{k}}^{\alpha}=\xi^{\mu}D_{\mu_{1}}\cdots
D_{\mu_{k}}\left(D_{\mu}\psi^{\alpha}\right)+D_{\mu_{1}}\cdots
D_{\mu_{k}}Q^{\alpha},\end{cases}$ (30)

and

$\begin{cases}\bm{q}_{a}\equiv\bm{\theta}_{a}-\zeta_{a}\dot{\bm{\chi}}_{a},\\\
Q^{\alpha}\equiv\phi^{\alpha}-\xi^{\mu}D_{\mu}\psi^{\alpha},\end{cases}$ (31)

are characteristics of the infinitesimal generator $\bm{v}$. Equation (26) is
the infinitesimal criterion of the symmetry condition (19). Detailed
derivation process of the prolongation formula (29) can be found in Ref.
(Olver, 1993). Different from the situation in the previous references, an
integral along the particle’s world line appeared in the infinitesimal
criterion (see Eq. (26)). This originally comes from the fact that the
Lagrangian density $\mathcal{L}$ is a functional rather than a function of
particle’s world lines. Owing to this integral, the infinitesimal criterion
(III) is not a differential equation but a integro-differential equation.

Similarly, using Eq. (22), we can obtained another infinitesimal criterion as

$\dot{q}_{a}^{\mu}\dot{\chi}_{a\mu}=0.$ (32)

In deriving Eq. (32), we used the following equations

$\displaystyle\frac{d}{d\epsilon}\bigg{|}_{0}\dot{\tilde{\chi}}^{\mu}=\theta_{a1}^{\mu}=\zeta_{a}\ddot{\chi}_{a}^{\mu}+\dot{q}_{a}^{\mu},$
(33) $\displaystyle\ddot{\chi}_{a}^{\mu}\dot{\chi}_{a\mu}\equiv 0.$ (34)

##  IV conservation laws

Due to the particularities of the infinitesimal criterion shown in Sec. III,
especially the existence of the additional infinitesimal criterion (32), the
derivation process of local conservation law and the and the final results are
quite different from the standard situation without constraints. We next show
how the infinitesimal criterion (26) and (32) determine a conservation law.

We first transform Eq. (26) into another equivalent form. The first term in
Eq. (26) can be rewritten by components as

$\displaystyle\mathrm{pr}^{\left(n\right)}\bm{v}\left(\ell_{a}\right)+\ell_{a}\dot{\zeta}_{a}=\zeta_{a}\frac{\partial\ell_{a}}{\partial\tau_{a}}+\theta_{a}^{\mu}\frac{\partial\ell_{a}}{\partial\chi_{a}^{\mu}}+\theta_{a1}^{\mu}\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\mu}}+\ell_{a}\dot{\zeta}_{a}$
$\displaystyle+\xi^{\mu}\frac{\partial\ell_{a}}{\partial\chi^{\mu}}+\phi^{\alpha}\frac{\partial\ell_{a}}{\partial\psi^{\alpha}}+\sum_{k=1}^{n}\phi_{\mu_{1}\dots\mu_{k}}^{\alpha}\frac{\partial\ell_{a}}{\partial\left(D_{\mu_{1}}\cdots
D_{\mu_{k}}\psi^{\alpha}\right)}.$ (35)

From the first four terms of right-hand side of Eq. (35), we have

$\displaystyle\zeta_{a}\frac{\partial\ell_{a}}{\partial\tau_{a}}+\theta_{a}^{\mu}\frac{\partial\ell_{a}}{\partial\chi_{a}^{\mu}}+\left(\zeta_{a}\ddot{\chi}_{a}^{\mu}+\dot{q}_{a}^{\mu}\right)\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\mu}}+\ell_{a}\dot{\zeta}_{a}$
$\displaystyle=\zeta_{a}\left(\frac{\partial\ell_{a}}{\partial\tau_{a}}+\dot{\chi}_{a}^{\mu}\frac{\partial\ell_{a}}{\partial\chi_{a}^{\mu}}+\ddot{\chi}_{a}^{\mu}\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\mu}}\right)+q_{a}^{\mu}\frac{\partial\ell_{a}}{\partial\chi_{a}^{\mu}}+\dot{q}_{a}^{\mu}\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\mu}}+\ell_{a}\dot{\zeta}_{a}$
$\displaystyle=\zeta_{a}\frac{d\ell_{a}}{d\tau_{a}}+\ell_{a}\dot{\zeta}_{a}+\frac{d}{d\tau_{a}}\left(q_{a}^{\mu}\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\mu}}\right)+q_{a}^{\mu}\left(\frac{\partial\ell_{a}}{\partial\chi_{a}^{\mu}}-\frac{d}{d\tau_{a}}\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\mu}}\right)$
$\displaystyle=\frac{d}{d\tau_{a}}\left(\ell_{a}\zeta_{a}\right)+\frac{d}{d\tau_{a}}\left[q_{a}^{\mu}\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\mu}}\right]+\frac{q_{a}^{\mu}}{c^{2}}\frac{d}{d\tau_{a}}\left[\left(\ell_{a}-\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\nu}}\dot{\chi}_{a}^{\nu}\right)\dot{\chi}_{a\mu}\right]+q_{a}^{\mu}\bm{E}_{\bm{\chi}_{a}\mu}\left(\ell_{a}\right)$
$\displaystyle=\frac{d}{d\tau_{a}}\left[\ell_{a}\zeta_{a}+q_{a}^{\mu}\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\mu}}+\frac{q_{a}^{\mu}}{c^{2}}\left(\ell_{a}-\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\nu}}\dot{\chi}_{a}^{\nu}\right)\dot{\chi}_{a\mu}\right]-\frac{1}{c^{2}}\left(\ell_{a}-\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\nu}}\dot{\chi}_{a}^{\nu}\right)\dot{q}_{a}^{\mu}\dot{\chi}_{a\mu}+q_{a}^{\mu}\bm{E}_{\bm{\chi}_{a}\mu}\left(\ell_{a}\right)$
$\displaystyle=\frac{d}{d\tau_{a}}\left[\ell_{a}\zeta_{a}+q_{a}^{\mu}\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\mu}}+\frac{q_{a}^{\mu}}{c^{2}}\left(\ell_{a}-\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\nu}}\dot{\chi}_{a}^{\nu}\right)\dot{\chi}_{a\mu}\right]+q_{a}^{\mu}\bm{E}_{\bm{\chi}_{a}\mu}\left(\ell_{a}\right),$
(36)

where we used the second infinitesimal criterion (32) in the last step. The
last three terms of Eq. (35) can be transformed into

$\displaystyle\xi^{\mu}\frac{\partial\ell_{a}}{\partial\chi^{\mu}}+\phi^{\alpha}\frac{\partial\ell_{a}}{\partial\psi^{\alpha}}+\sum_{k=1}^{n}\phi_{\mu_{1}\dots\mu_{k}}^{\alpha}\frac{\partial\ell_{a}}{\partial\left(D_{\mu_{1}}\cdots
D_{\mu_{k}}\psi^{\alpha}\right)}$
$\displaystyle=\xi^{\mu}D_{\mu}\ell_{a}+D_{\mu}\mathbb{P}_{a}^{\mu}+\bm{E}_{\bm{\psi}}\left(\ell_{a}\right)\cdot\bm{Q}$
(37)

by the standard derivation process (see Ref. (Olver, 1993)), where

$\mathbb{P}_{a}^{\mu}=\sum_{i=1}^{n}\sum_{j=1}^{i}\left(-1\right)^{j+1}D_{\mu_{j+1}}\cdots
D_{\mu_{k}}Q^{\alpha}\left[D_{\mu_{1}}\cdots
D_{\mu_{j-1}}\frac{\partial\ell_{a}}{\partial\left(D_{\mu_{1}}\cdots
D_{\mu_{j-1}}D_{\mu}D_{\mu_{j+1}}\cdots
D_{\mu_{i}}\psi^{\alpha}\right)}\right].$ (38)

Similarly, the last two terms in Eq. (35), (26) can be written as

$\mathrm{pr}^{\left(n\right)}\bm{v}\left(\mathcal{L}_{F}\right)+\mathcal{L}D_{\mu}\xi^{\mu}=\xi^{\mu}D_{\mu}\mathcal{L}_{F}+\mathcal{L}D_{\mu}\xi^{\mu}+D_{\mu}\mathbb{P}_{F}^{\mu}+\bm{E}_{\bm{\psi}}\left(\mathcal{L}_{F}\right)\cdot\bm{Q},$
(39)

where

$\mathbb{P}_{F}^{\mu}=\sum_{i=1}^{n}\sum_{j=1}^{i}\left(-1\right)^{j+1}D_{\mu_{j+1}}\cdots
D_{\mu_{k}}Q^{\alpha}\left[D_{\mu_{1}}\cdots
D_{\mu_{j-1}}\frac{\partial\mathcal{L}_{F}}{\partial\left(D_{\mu_{1}}\cdots
D_{\mu_{j-1}}D_{\mu}D_{\mu_{j+1}}\cdots
D_{\mu_{i}}\psi^{\alpha}\right)}\right].$ (40)

Combing Eqs. (35), (36), (37) and (39), equation (26) is then transformed into

$\displaystyle\sum_{a}\int_{a_{1}}^{a_{2}}\left[q_{a}^{\mu}\bm{E}_{\bm{\chi}_{a}\mu}\left(\ell_{a}\right)+\xi^{\mu}D_{\mu}\ell_{a}+D_{\mu}P_{a}^{\mu}+\bm{E}_{\bm{\psi}}\left(\ell_{a}\right)\cdot\bm{Q}\right]d\tau_{a}$
$\displaystyle+\xi^{\mu}D_{\mu}\mathcal{L}_{F}+D_{\mu}\mathbb{P}_{F}^{\mu}+\bm{E}_{\bm{\psi}}\left(\mathcal{L}_{F}\right)\cdot\bm{Q}+\mathcal{L}D_{\mu}\xi^{\mu}=0,$
(41)

where we used

$\displaystyle\sum_{a}\int_{a_{1}}^{a_{2}}\frac{d}{d\tau_{a}}\left[\ell_{a}\zeta_{a}+q_{a}^{\mu}\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\mu}}+\frac{q_{a}^{\mu}}{c^{2}}\dot{\chi}_{a\mu}\left(\ell_{a}-\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\nu}}\dot{\chi}_{a}^{\nu}\right)\right]d\tau_{a}$
$\displaystyle=\left[\ell_{a}\zeta_{a}+q_{a}^{\mu}\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\mu}}+\frac{q_{a}^{\mu}}{c^{2}}\dot{\chi}_{a\mu}\left(\ell_{a}-\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\nu}}\dot{\chi}_{a}^{\nu}\right)\right]_{a_{1}}^{a_{2}}=0.$
(42)

Suppose $\bm{\xi}$ and $\bm{Q}$ are independent of $\tau_{a}$, equation (41)
becomes

$\displaystyle
D_{\mu}\left[\mathcal{L}\xi^{\mu}+\mathbb{Q}^{\mu}\right]+\sum_{a}\int_{a_{1}}^{a_{2}}\left[q_{a}^{\mu}\bm{E}_{\bm{\chi}_{a}\mu}\left(\ell_{a}\right)\right]d\tau_{a}+\bm{E}_{\bm{\psi}}\left(\mathcal{L}\right)\cdot\bm{Q}=0,$
(43)

where

$\mathbb{Q}^{\mu}=\sum_{i=1}^{n}\sum_{j=1}^{i}\left(-1\right)^{j+1}D_{\mu_{j+1}}\cdots
D_{\mu_{k}}Q^{\alpha}\left[D_{\mu_{1}}\cdots
D_{\mu_{j-1}}\frac{\partial\mathcal{L}}{\partial\left(D_{\mu_{1}}\cdots
D_{\mu_{j-1}}D_{\mu}D_{\mu_{j+1}}\cdots
D_{\mu_{i}}\psi^{\alpha}\right)}\right].$ (44)

Using the EL equation (13) of the field $\bm{\psi}$, the last term in the Eq.
(43) vanishes. However, due to the the weak ELB equation (17), the second term
(43) is not zero. If the characteristic $\bm{q}_{a}$ are independent of
$\bm{\chi}$ and $\bm{\psi}$, this term can be written as a divergence form,
i.e.,

$\sum_{a}\int_{a_{1}}^{a_{2}}\left[q_{a}^{\mu}\bm{E}_{\bm{\chi}_{a}\mu}\left(\ell_{a}\right)\right]d\tau_{a}=D_{\nu}\mathbb{S}^{\nu}$
(45)

where

$\mathbb{S}^{\nu}=\sum_{a}\int_{a_{1}}^{a_{2}}d\tau_{a}\left\\{q_{a}^{\mu}\left[-\ell_{a}\eta_{\>\mu}^{\nu}+\dot{\chi}_{a}^{\nu}\left[\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\mu}}+\frac{1}{c^{2}}\dot{\chi}_{a\mu}\left(\ell_{a}-\dot{\chi}_{a}^{\sigma}\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\sigma}}\right)\right]\right]\right\\}$
(46)

Substituting Eq. (45) into Eq. (43), we finally arrive at the geometric
conservation law

$D_{\mu}\left[\mathcal{L}\xi^{\mu}+\mathbb{Q}^{\mu}+\mathbb{S}^{\mu}\right]=0.$
(47)

##  V gauge-symmetric energy-momentum conservation laws for high-order
electromagnetic systems coupled with charged particles

We now apply the general theory to high-order electromagnetic system coupled
with charged particles. The Lagrangian $L_{a}$ for this system can be
generally written as a gauge symmetric form as

$\displaystyle
L_{a}=L_{a}\left(\chi_{a}^{\mu},A_{\mu},F_{\mu\nu},DF_{\mu\nu},\cdots,D^{\left(n\right)}F_{\mu\nu}\right),$
(48)
$\displaystyle\mathcal{L}_{F}=\mathcal{L}_{F}\left(\chi^{\mu},F_{\mu\nu},DF_{\mu\nu},\cdots,D^{\left(n\right)}F_{\mu\nu}\right).$
(49)

where $D$ is the space-time derivative operator, i.e.,
$D=\left(\left(1/c\right)\partial/\partial t,\bm{\nabla}\right)$, $A_{\mu}$ as
the field $\bm{\psi}$ is the 4-potential and $F_{\mu\nu}$ is the Faraday
tensor defined by

$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}.$ (50)

Using Eq. (13), we can obtain the equation of motion for electromagnetic
field,

$E_{A}^{\mu}\left(\mathcal{L}\right)=0,$ (51)

where $E_{A}^{\mu}$ is the Euler operator for $A_{\mu}.$ To obtain gauge-
symmetric energy-momentum conservation laws, we transform (51) into

$\partial_{\mu}\mathcal{D}^{\mu\nu}=\frac{4\pi}{c}J_{f}^{\nu}$ (52)

using the derivatives with respect to $F_{\mu\nu}$ (see Ref. (Fan  _et al._ ,
2021)), where

$\mathcal{D}^{\mu\nu}=-8\pi
E_{F}^{\left[\mu\nu\right]}\left(\mathcal{L}\right)$ (53)

is regarded as the electric displacement tensor. Here, $E_{F}^{\mu\nu}$ is the
Euler operator for the Faraday tensor, i.e.,

$E_{F}^{\mu\nu}\coloneqq\frac{\partial}{\partial
F_{\mu\nu}}+\sum_{i=1}^{n}\left(-1\right)^{i}D_{\mu_{1}}\cdots
D_{\mu_{i}}\frac{\partial}{\partial\partial_{\mu_{1}}\cdots\partial_{\mu_{i}}F_{\mu\nu}},$
(54)

$J_{f}^{\nu}$ is the free 4-current which is defined by

$J_{f}^{\nu}\coloneqq-c\frac{\partial\mathcal{L}}{\partial
A_{\nu}}=-\int_{a_{1}}^{a_{2}}\frac{\partial\ell_{a}}{\partial
A_{\nu}}ds_{a}.$ (55)

Here, $ds_{a}=cd\tau_{a}$ is the line element in the Minkowski space. The
superscript $[\mu\thinspace\nu]$ in Eq. (53) denotes the anti-symmetrization
with respect of $\mu$ and $\nu$. Suppose Eq. (52) is gauge symmetric and the
Lagrangian density obey the minimal coupling which can be written as

$\mathcal{L}=\sum_{a}\int_{a_{1}}^{a_{2}}\left[-\frac{q_{a}}{c}A_{\mu}\dot{\chi}_{a}^{\mu}\right]\delta_{a}d\tau_{a}+\mathrm{GSP}\left(\mathcal{L}\right)=-\frac{1}{c}A_{\mu}J^{\mu}+\mathrm{GSP}\left(\mathcal{L}\right),$
(56)

where $q_{a}$ is the charge of the $a$th particle and
“$\mathrm{GSP}\left(\mathcal{L}\right)$” denotes the gauge-symmetric parts of
the Lagrangian density. To get the gauge-symmetric energy-momentum
conservation laws, the prolongation formula is also need reconstructed by
using the derivatives with respect to $F_{\mu\nu},$ that is

$\displaystyle\text{pr}^{\left(n\right)}\bm{v}=\bm{v}+\sum_{a}\theta_{a1}^{\mu}\frac{\partial}{\partial\dot{\chi}_{a}^{\mu}}+\left[G_{\sigma\rho}+\xi^{\sigma}D_{\sigma}F_{\sigma\rho}\right]\frac{\partial}{\partial
F_{\sigma\rho}}$ $\displaystyle+\sum_{i=1}^{n}\left[D_{\mu_{1}}\cdots
D_{\mu_{i}}G_{\sigma\rho}+\xi^{\sigma}D_{\sigma}D_{\mu_{1}}\cdots
D_{\mu_{i}}F_{\sigma\rho}\right]\frac{\partial}{\partial\left(\partial_{\mu_{1}}\cdots\partial_{\mu_{i}}F_{\sigma\rho}\right)},$
(57)

where

$G_{\sigma\rho}=\partial_{\sigma}Q_{\rho}-\partial_{\rho}Q_{\sigma}=2\partial_{[\sigma}Q_{\rho]}.$
(58)

Finally, the conservation law (47) now read

$D_{\mu}\left\\{\mathcal{L}\xi^{\mu}-\frac{1}{4\pi}\mathcal{D}^{\mu\sigma}Q_{\sigma}+\mathbb{P}^{\mu}+\mathbb{S}^{\mu}\right\\}=0,$
(59)

where

$\mathbb{P}^{\mu}=\sum_{i=1}^{n}\sum_{j=1}^{i}\left(-1\right)^{j+1}\left(D_{\mu_{j+1}}\cdots
D_{\mu_{i}}G_{\sigma\rho}\right)\left[D_{\mu_{1}}\cdots
D_{\mu_{j-1}}\frac{\partial\mathcal{L}}{\partial\left(\partial_{\mu_{1}}\cdots\partial_{\mu_{j-1}}\partial_{\mu}\partial_{\mu_{j+1}}\cdots\partial_{\mu_{i}}F_{\sigma\rho}\right)}\right].$
(60)

We now turn to discuss the space-time translation symmetry and energy-momentum
conservation law. Suppose the action of the system is invariant under the
space-time translation,

$g_{\epsilon}\cdot\left(\tau_{a},\chi^{\mu},\chi_{a}^{\mu},A^{\mu}\right)=\left(\tilde{\tau}_{a},\tilde{\chi}^{\mu},\tilde{\chi}_{a}^{\mu},\tilde{A}^{\mu}\right)=\left(\tau_{a},\chi^{\mu}+\epsilon\chi_{0}^{\mu},\chi_{a}^{\mu}+\epsilon\chi_{0}^{\mu},A^{\mu}\right).$
(61)

The corresponding infinitesimal generator of the group transformation (61) is

$\bm{v}=\chi_{0}^{\mu}\frac{\partial}{\partial\chi^{\mu}}+\sum_{a}\chi_{0}^{\mu}\frac{\partial}{\partial\chi_{a}^{\mu}},$
(62)

where $\zeta_{a}=\phi^{\alpha}=0,\;\xi^{\mu}=\theta_{a}^{\mu}=\chi_{0}^{\mu}.$
The characteristics $\bm{Q}$, $\bm{q}_{a}$ and the term $G_{\sigma\rho}$ are
calculated as

$\displaystyle q_{a}^{\mu}\equiv\chi_{0}^{\mu},$ (63) $\displaystyle
Q^{\alpha}\equiv-\chi_{0}^{\nu}\partial_{\nu}A^{\alpha},$ (64) $\displaystyle
G_{\sigma\rho}=-\chi_{0}^{\nu}\partial_{\nu}F_{\sigma\rho}.$ (65)

It is clear that the second infinitesimal criterion (32) is satisfied by
$\dot{q}_{a}^{\mu}\equiv 0$. Substituting Eqs. (63)-(65) into Eq. (59), we
obtain the canonical energy-momentum conservation law,

$\displaystyle D_{\mu}T_{N}^{\mu\nu}=0,$ (66) $\displaystyle
T_{N}^{\mu\nu}=\mathcal{L}\eta^{\mu\nu}+\frac{1}{4\pi}\mathcal{D}^{\mu\sigma}\partial^{\nu}A_{\sigma}-\Sigma^{\mu\nu}+\Pi^{\mu\nu},$
(67)

where

$\displaystyle\Sigma^{\mu\nu}=\sum_{i=1}^{n}\sum_{j=1}^{i}\left(-1\right)^{j+1}\left(D_{\mu_{j+1}}\cdots
D_{\mu_{i}}\partial^{\nu}F_{\sigma\rho}\right)\left[D_{\mu_{1}}\cdots
D_{\mu_{j-1}}\frac{\partial\mathcal{L}}{\partial\left(\partial_{\mu_{1}}\cdots\partial_{\mu_{j-1}}\partial_{\mu}\partial_{\mu_{j+1}}\cdots\partial_{\mu_{i}}F_{\sigma\rho}\right)}\right],$
(68)
$\displaystyle\Pi^{\mu\nu}=\sum_{a}\int_{a_{1}}^{a_{2}}d\tau_{a}\left\\{-\ell_{a}\eta^{\mu\nu}+\dot{\chi}_{a}^{\mu}\left[\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a\nu}}+\frac{1}{c^{2}}\dot{\chi}_{a}^{\nu}\left(\ell_{a}-\dot{\chi}_{a}^{\sigma}\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\sigma}}\right)\right]\right\\}.$
(69)

Here, $T_{N}^{\mu\nu}$ is the canonical energy-momentum tensor which is gauge
dependent. We next “improve” $T_{N}^{\mu\nu}$ to a gauge-symmetric form. We
add the following identity

$D_{\mu}\left(D_{\sigma}\mathcal{F}^{\sigma\mu\nu}\right)=0,\quad\mathcal{F}^{\sigma\mu\nu}\equiv\frac{1}{4\pi}\mathcal{D}^{\sigma\mu}A^{\nu}$
(70)

to Eq. (66) to get the explicitly gauge invariant conservation law

$D_{\mu}T_{\text{GS}}^{\mu\nu}=0,$ (71)

where

$T_{\text{GS}}^{\mu\nu}=\mathcal{L}\eta^{\mu\nu}+\frac{1}{c}J_{f}^{\mu}A^{\nu}+\Pi^{\mu\nu}+\frac{1}{4\pi}\mathcal{D}^{\mu\sigma}F_{\;\sigma}^{\nu}-\Sigma^{\mu\nu}$
(72)

is the improved energy-momentum tensor, where we used Eq. (52). We next prove
that $T_{\text{GS}}^{\mu\nu}$ is gauge invariant. It is sufficient to show
that the first three terms in the right-hand side of Eq. (72) is gauge
invariant. Substituting Eqs. (56) and (69) into Eq. (72), these terms are

$\displaystyle\mathcal{L}\eta^{\mu\nu}+\frac{1}{c}J_{f}^{\mu}A^{\nu}+\Pi^{\mu\nu}$
$\displaystyle=\mathcal{L}\eta^{\mu\nu}+\frac{1}{c}J_{f}^{\mu}A^{\nu}+\sum_{a}\int_{a_{1}}^{a_{2}}d\tau_{a}\left\\{-\ell_{a}\eta^{\mu\nu}+\dot{\chi}_{a}^{\mu}\left[\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a\nu}}+\frac{1}{c^{2}}\dot{\chi}_{a}^{\nu}\left(\ell_{a}-\dot{\chi}_{a}^{\sigma}\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\sigma}}\right)\right]\right\\}$
$\displaystyle=\mathcal{L}\eta^{\mu\nu}-\left(\sum_{a}\int_{a_{1}}^{a_{2}}d\ell_{a}\tau_{a}\right)\eta^{\mu\nu}+\frac{1}{c}J_{f}^{\mu}A^{\nu}$
$\displaystyle+\sum_{a}\int_{a_{1}}^{a_{2}}d\tau_{a}\left\\{-\frac{q_{a}}{c}\dot{\chi}_{a}^{\mu}A^{\nu}+\dot{\chi}_{a}^{\mu}\text{GSP}\left(\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a\nu}}\right)+\frac{1}{c^{2}}\dot{\chi}_{a}^{\mu}\dot{\chi}_{a}^{\nu}\left[\ell_{a}-\frac{q_{a}}{c}\dot{\chi}_{a}^{\sigma}A_{\sigma}-\dot{\chi}_{a}^{\sigma}\text{GSP}\left(\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\sigma}}\right)\right]\right\\}$
$\displaystyle=\mathcal{L}_{F}\eta^{\mu\nu}+\frac{1}{c}J_{f}^{\mu}A^{\nu}-\frac{1}{c}\left(\sum_{a}\int_{a_{1}}^{a_{2}}q_{a}\dot{\chi}_{a}^{\mu}d\tau_{a}\right)A^{\nu}$
$\displaystyle+\frac{1}{c^{2}}\sum_{a}\int_{a_{1}}^{a_{2}}d\tau_{a}\left\\{\dot{\chi}_{a}^{\mu}\text{GSP}\left(\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a\nu}}\right)+\dot{\chi}_{a}^{\mu}\dot{\chi}_{a}^{\nu}\left[\text{GSP}\left(\ell_{a}\right)-\dot{\chi}_{a}^{\sigma}\text{GSP}\left(\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\sigma}}\right)\right]\right\\}$
$\displaystyle=\mathcal{L}_{F}\eta^{\mu\nu}+\sum_{a}\int_{a_{1}}^{a_{2}}d\tau_{a}\left\\{\dot{\chi}_{a}^{\mu}\left[\text{GSP}\left(\ell_{a}\right)-\dot{\chi}_{a}^{\sigma}\text{GSP}\left(\frac{\partial\ell_{a}}{\partial\dot{\chi}_{a}^{\sigma}}\right)\right]\right\\},$
(73)

where Eq. (55) is used in the last step. Equation (73) confirms that
$T_{\text{inv}}^{\mu\nu}$ is gauge invariant.

Lastly, we discuss the Podolsky system (Podolsky, 1942) coupled with charged
particles, where the Lagrangian density of particles and fields are
respectively written as

$\displaystyle\ell_{\text{P}a}=-\left(m_{a}c^{2}+\frac{q_{a}}{c}A_{\mu}\dot{\chi}_{a}^{\mu}\right)\delta_{a}$
(74)
$\displaystyle\mathcal{L}_{\text{PF}}=-\frac{1}{16\pi}F_{\sigma\rho}F^{\sigma\rho}-\frac{a^{2}}{8\pi}\partial_{\sigma}F^{\sigma\lambda}\partial^{\rho}F_{\rho\lambda}.$
(75)

Substituting Eq. (74) into Eq. (69) to obtain $\Pi_{\text{P}}^{\mu\nu}$

$\Pi_{\text{P}}^{\mu\nu}=\sum_{a}\int_{a_{1}}^{a_{2}}d\tau_{a}\left\\{\left[m_{a}c^{2}+\frac{q_{a}}{c}A_{\sigma}\dot{\chi}_{a}^{\sigma}\right]\delta_{a}\eta^{\mu\nu}-\left[m_{a}\dot{\chi}_{a}^{\mu}\dot{\chi}_{a}^{\nu}+\frac{q_{a}}{c}\dot{\chi}_{a}^{\mu}A^{\nu}\right]\delta_{a}\right\\}.$
(76)

$\Sigma^{\mu\nu}$ and $\mathcal{D}^{\sigma\mu}$ in Eq. (72) has been shown in
Ref. (Fan  _et al._ , 2021), we rewritten here as

$\displaystyle\Sigma_{\text{P}}^{\mu\nu}$
$\displaystyle=-\frac{a^{2}}{4\pi}\left(\partial^{\nu}F_{\;\rho}^{\mu}\right)\left(\partial_{\sigma}F^{\sigma\rho}\right),$
(77) $\displaystyle\mathcal{D}_{\text{P }}^{\sigma\mu}$
$\displaystyle=-F^{\mu\sigma}+a^{2}\left(\partial^{\mu}\partial_{\lambda}F^{\lambda\sigma}-\partial^{\sigma}\partial_{\lambda}F^{\lambda\mu}\right).$
(78)

Using Eqs. 74)-(78), the gauge invariant energy-momentum tensor is now become

$\displaystyle-
T_{\text{inv}}^{\mu\nu}=\sum_{a}\int_{a_{1}}^{a_{2}}\left(m_{a}\dot{\chi}_{a}^{\mu}\dot{\chi}_{a}^{\nu}\delta_{a}\right)d\tau_{a}+\frac{1}{4\pi}\left[-F^{\mu\sigma}F_{\nu\sigma}+\frac{1}{4}\left(F_{\sigma\rho}F^{\sigma\rho}\right)\eta^{\mu\nu}\right]$
$\displaystyle+\frac{a^{2}}{8\pi}\left(\partial_{\sigma}F^{\sigma\lambda}\partial^{\rho}F_{\rho\lambda}\right)\eta^{\mu\nu}+\frac{a^{2}}{4\pi}F_{\;\sigma}^{\nu}\left(\partial^{\mu}\partial_{\rho}F^{\rho\sigma}\right)-\frac{a^{2}}{4\pi}F_{\;\sigma}^{\nu}\left(\partial^{\sigma}\partial_{\rho}F^{\rho\mu}\right)-\frac{a^{2}}{4\pi}\left(\partial^{\nu}F_{\;\rho}^{\mu}\right)\left(\partial_{\sigma}F^{\sigma\rho}\right).$
(79)

##  VI conclusion and discussion

In this work, we developed a general geometric (or manifestly covariant) field
theory for classical relativistic particle-field systems and established the
connections between general symmetries and local conservation laws for the
systems. To achieve this goal, we overcame two difficulties.

The first difficulty associated with the mass-shell constraint (see Eq. (3)).
As a result, the standard Euler-Lagrange (EL) equation is reconstructed by the
Euler-Lagrange-Barut (ELB) equation (see Eq. (17)). Besides, the use of proper
time parameter makes the Lagrangian density (8) a function of the field
$\bm{\psi}$ and also a functional of the particle’s world line, which directly
lead to the standard infinitesimal criterion a integro-differential equation
rather than a differential one (see Eq. (26)). Furthermore, to satisfy the
mass-shell condition, an extra criterion (32) is derived.

The second difficulty comes from the heterogeneous-manifolds that the
particles and fields reside on. The fields are defined on the 4D space-time,
while each particle’s word line is defined only on the 1D parameter space. As
a consequence, the standard Noether’s procedure for deriving local
conservation laws from symmetries is not applicable without modification. To
overcome this difficulty, we developed a weak version of the Euler-Lagrange-
Barut (ELB) equation for particles on the 4D space-time, which is rewritten
here as

$E_{\bm{\chi}_{a}\mu}\left(\ell_{a}\right)=D_{\nu}H_{\thinspace\mu}^{\nu},$
(80)

where the definition of $H_{\thinspace\mu}^{\nu}$ can be easily read from Eq.
(17). This non-vanishing term $H_{\thinspace\mu}^{\nu}$ is emerged in the
transformed infinitesimal criterion (43) as

$\sum_{a}\int_{a_{1}}^{a_{2}}\left[q_{a}^{\mu}D_{\nu}H_{\thinspace\mu}^{\nu}\right]d\tau_{a}.$
(81)

If the characteristic $q_{a}^{\mu}$ is independent of space-time position
$\bm{\chi}$ and the field $\bm{\psi}$, the derivative operator $D_{\nu}$ can
be moved out from the integral, i.e.,
$\sum_{a}\int_{a_{1}}^{a_{2}}\left[q_{a}^{\mu}H_{\thinspace\mu}^{\nu}\right]d\tau_{a}=D_{\nu}\mathcal{J^{\nu}},$the
new current
$\mathcal{J^{\nu}}\equiv\sum_{a}\int_{a_{1}}^{a_{2}}\left[q_{a}^{\mu}H_{\thinspace\mu}^{\nu}\right]d\tau_{a}$
is then induced.

Combing the weak ELB equation and infinitesimal criterion of the symmetry
conditions, the general geometric conservation laws is systematically derived.
Using general theory constructed here, we obtain the energy-momentum
conservation laws for high-order relativistic electromagnetic systems by the
space-time translation symmetry.

Interestingly, when the characteristic $q_{a}^{\mu}$ of the transformation
(18) is related with space-time position $\bm{\chi}$ or the field $\bm{\psi}$,
the derivative operator $D_{\nu}$ cannot be moved out from the integral. As
such, equation (81) cannot be transformed into a divergence form, and a
conservation law cannot be given even the system admitting a continuous
symmetry. We have not find a suitable symmetry which won’t lead to a
conservation law. Here, we recommend it as an open question and it would be an
exciting research project to shed more light on that question.

###### Acknowledgements.

P. Fan was supported by Shenzhen Clean Energy Research Institute and National
Natural Science Foundation of China (NSFC-12005141). Q. Chen was supported by
the National Natural Science Foundation of China (NSFC-11805273). J. Xiao was
supported by the National MC Energy R&D Program (2018YFE0304100), National Key
Research and Development Program (2016YFA0400600, 2016YFA0400601 and
2016YFA0400602), and the National Natural Science Foundation of China
(NSFC-11905220).

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