1) Noether's theorem states that for every continuous symmetry of the Lagrangian of a physical system, there exists a corresponding conservation law. For example, if the Lagrangian is symmetric under rotations, angular momentum is conserved.
2) The theorem can be used to derive conservation laws from the symmetries of a physical system. It provides a relationship between symmetry and conservation laws.
3) Noether's theorem expresses conservation laws in terms of quantities called "Noether charges." A Noether charge is the conserved physical quantity associated with a particular symmetry of the Lagrangian.
Introduction to Classical Mechanics:
UNIT-I : Elementary survey of Classical Mechanics: Newtonian mechanics for single particle and system of particles, Types of the forces and the single particle system examples, Limitation of Newton’s program, conservation laws viz Linear momentum, Angular Momentum & Total Energy, work-energy theorem; open systems (with variable mass). Principle of Virtual work, D’Alembert’s principle’ applications.
UNIT-II : Constraints; Definition, Types, cause & effects, Need, Justification for realizing constraints on the system
LECTURE 1 PHY5521 Classical Mechanics Honour to Masters LevelDavidTinarwo1
Classical mechanics, a well-organized introductory lecture. This is easy to follow, and a must-go-through lecture. UNIT-I : Elementary survey of Classical Mechanics: Newtonian mechanics for single particle and system of particles, Types of the forces and the single particle system examples, Limitation of Newton’s program, conservation laws viz Linear momentum, Angular Momentum & Total Energy, work-energy theorem; open systems (with variable mass). Principle of Virtual work, D’Alembert’s principle’ applications.
UNIT-II : Constraints; Definition, Types, cause & effects, Need, Justification for realizing constraints on the system, Difficulties introduced by imposing constraints on the system, Examples of constraints, Introduction of generalized coordinates justification. Lagrange’s equations; Linear generalized potentials, Generalized coordinates and momenta & energy; Gauge function for Lagrangian and its gauge invariance, Applications to constrained systems and generalized forces.
Theory of Vibrations: Introduction to the theory of vibrations in multi-degree-of-freedom systems, Normal modes and modal analysis, Nonlinear oscillations and chaos theory.
Canonical Transformations: Properties and classification of canonical transformations, Action-angle variables and their applications in integrable systems, Canonical perturbation theory and perturbation methods.
Poisson's and Lagrange's Brackets: Definitions and properties of Poisson's brackets, Relationship between Poisson's brackets and Hamilton's equations, Lagrange's brackets and their applications in dynamics. UNIT-III : Cyclic coordinates, Integrals of the motion, Concepts of symmetry, homogeneity and isotropy, Invariance under Galilean transformations Hamilton’s equation of motion: Legendre’s dual transformation, Principle of least action; derivation of equations of motion; variation and end points; Hamilton’s principle and characteristic functions; Hamilton-Jacobi equation.
UNIT-IV : Central force fields: Definition and properties, Two-body central force problem, gravitational and electrostatic potentials in central force fields, closure and stability of circular orbits; general analysis of orbits; Kepler’s laws and equation, Classification of orbits, orbital dynamics and celestial mechanics, differential equation of orbit, Virial Theorem.
UNIT-V : Canonical transformation; generating functions; Properties; group property; examples; infinitesimal generators; Poisson bracket; Poisson theorems; angular momentum PBs; Transition from discrete to continuous system, small oscillations (longitudinal oscillations in elastic rod); normal modes and coordinates.
On the Unification of Physic and the Elimination of Unbound Quantitiesijrap
This paper supports Descartes' idea of a constant quantity of motion, modernized by Leibniz. Unlike Leibniz, the paper emphasizes that the idea is not realized by forms of energy, but by energy itself. It remains constant regardless of the form, type, or speed of motion, even that of light. Through force, energy is only transformed. Here it is proved that force is its derivative. It exists even at rest, representing the object's minimal energy state. With speed, we achieve its multiplication up to the maximum energy state, from which a maximum force is derived from the object. From this point, corresponding to Planck's Length, we find the value of the force wherever we want. Achieving this removes the differences between various natural forces. The new idea eliminates infinite magnitudes. The process allows the laws to transition from simple to complex forms and vice versa, through differentiation-integration. For this paper, this means achieving the Unification Theory.
Physics Project On Physical World, Units and MeasurementSamiran Ghosh
This PowerPoint is Physical World, Units and Measurement. This is basically the first chapter of 11th class/grade. This power point explains the basic or fundamental physics with some information about SI units and fundamental forces.
Introduction to Classical Mechanics:
UNIT-I : Elementary survey of Classical Mechanics: Newtonian mechanics for single particle and system of particles, Types of the forces and the single particle system examples, Limitation of Newton’s program, conservation laws viz Linear momentum, Angular Momentum & Total Energy, work-energy theorem; open systems (with variable mass). Principle of Virtual work, D’Alembert’s principle’ applications.
UNIT-II : Constraints; Definition, Types, cause & effects, Need, Justification for realizing constraints on the system
LECTURE 1 PHY5521 Classical Mechanics Honour to Masters LevelDavidTinarwo1
Classical mechanics, a well-organized introductory lecture. This is easy to follow, and a must-go-through lecture. UNIT-I : Elementary survey of Classical Mechanics: Newtonian mechanics for single particle and system of particles, Types of the forces and the single particle system examples, Limitation of Newton’s program, conservation laws viz Linear momentum, Angular Momentum & Total Energy, work-energy theorem; open systems (with variable mass). Principle of Virtual work, D’Alembert’s principle’ applications.
UNIT-II : Constraints; Definition, Types, cause & effects, Need, Justification for realizing constraints on the system, Difficulties introduced by imposing constraints on the system, Examples of constraints, Introduction of generalized coordinates justification. Lagrange’s equations; Linear generalized potentials, Generalized coordinates and momenta & energy; Gauge function for Lagrangian and its gauge invariance, Applications to constrained systems and generalized forces.
Theory of Vibrations: Introduction to the theory of vibrations in multi-degree-of-freedom systems, Normal modes and modal analysis, Nonlinear oscillations and chaos theory.
Canonical Transformations: Properties and classification of canonical transformations, Action-angle variables and their applications in integrable systems, Canonical perturbation theory and perturbation methods.
Poisson's and Lagrange's Brackets: Definitions and properties of Poisson's brackets, Relationship between Poisson's brackets and Hamilton's equations, Lagrange's brackets and their applications in dynamics. UNIT-III : Cyclic coordinates, Integrals of the motion, Concepts of symmetry, homogeneity and isotropy, Invariance under Galilean transformations Hamilton’s equation of motion: Legendre’s dual transformation, Principle of least action; derivation of equations of motion; variation and end points; Hamilton’s principle and characteristic functions; Hamilton-Jacobi equation.
UNIT-IV : Central force fields: Definition and properties, Two-body central force problem, gravitational and electrostatic potentials in central force fields, closure and stability of circular orbits; general analysis of orbits; Kepler’s laws and equation, Classification of orbits, orbital dynamics and celestial mechanics, differential equation of orbit, Virial Theorem.
UNIT-V : Canonical transformation; generating functions; Properties; group property; examples; infinitesimal generators; Poisson bracket; Poisson theorems; angular momentum PBs; Transition from discrete to continuous system, small oscillations (longitudinal oscillations in elastic rod); normal modes and coordinates.
On the Unification of Physic and the Elimination of Unbound Quantitiesijrap
This paper supports Descartes' idea of a constant quantity of motion, modernized by Leibniz. Unlike Leibniz, the paper emphasizes that the idea is not realized by forms of energy, but by energy itself. It remains constant regardless of the form, type, or speed of motion, even that of light. Through force, energy is only transformed. Here it is proved that force is its derivative. It exists even at rest, representing the object's minimal energy state. With speed, we achieve its multiplication up to the maximum energy state, from which a maximum force is derived from the object. From this point, corresponding to Planck's Length, we find the value of the force wherever we want. Achieving this removes the differences between various natural forces. The new idea eliminates infinite magnitudes. The process allows the laws to transition from simple to complex forms and vice versa, through differentiation-integration. For this paper, this means achieving the Unification Theory.
Physics Project On Physical World, Units and MeasurementSamiran Ghosh
This PowerPoint is Physical World, Units and Measurement. This is basically the first chapter of 11th class/grade. This power point explains the basic or fundamental physics with some information about SI units and fundamental forces.
The Phase Theory towards the Unification of the Forces of Nature the Heart Be...IOSR Journals
A new theory has been presented, for the first time, called the "Phase Theory", which is the natural evolution of the physical thought and is considered the one beyond the super string theory. This theory solves the unsolved problems of the mysterious of matter, antimatter and interactions and makes a wide step towards the unification of the forces of nature. In this theory, the vibrating string of different frequency modes which determines the different types of elementary particles is replaced by a three dimensional infinitesimal pulsating (black)holes with the same frequency. Different types of elementary particles are determined by different phase angles associated with the same frequency. This allows the force of interactions to take place among elementary particles, without the need to invoke the notion of the force carrier particles, as the (stable) force of interactions can never take place between elementary particles at different frequencies. Besides the strong mathematical proofs given in this paper to prove its truthfulness, an experimental prediction has been given to confirm the theory presented in the form of the relation between the electron radius and quarks radii. The paper shows that quarks are direct consequence of this theory, and solves "the flavor problem" in QCD, and gives the clue to answer the questions of "Why are there so many flavors? The paper also derives the equation of the big bang theory which describes the singularity of the moment of creation of the universe.
In this paper, the underlying principles about the theory of relativity are briefly introduced and reviewed. The mathematical prerequisite needed for the understanding of general relativity and of Einstein field equations are discussed. Concepts such as the principle of least action will be included and its explanation using the Lagrange equations will be given. Where possible, the mathematical details and rigorous analysis of the subject has been given in order to ensure a more precise and thorough understanding of the theory of relativity. A brief mathematical analysis of how to derive the Einstein’s field’s equations from the Einstein-Hilbert action and the Schwarzschild solution was also given.
Schrodinger wave equation and its application
a very good animated presentation.
Bs level.
semester 6th.
how to make a very good appreciable presentation.
From last four decades of research it is well-established that all electrophysiological signals are nonlinear, irregular and aperiodic. Since those signals are used in everyday clinical practice as diagnostic tools (EMG, ECG, EEG), a huge progress in using it in making diagnostic more precise and
The Phase Theory towards the Unification of the Forces of Nature the Heart Be...IOSR Journals
A new theory has been presented, for the first time, called the "Phase Theory", which is the natural evolution of the physical thought and is considered the one beyond the super string theory. This theory solves the unsolved problems of the mysterious of matter, antimatter and interactions and makes a wide step towards the unification of the forces of nature. In this theory, the vibrating string of different frequency modes which determines the different types of elementary particles is replaced by a three dimensional infinitesimal pulsating (black)holes with the same frequency. Different types of elementary particles are determined by different phase angles associated with the same frequency. This allows the force of interactions to take place among elementary particles, without the need to invoke the notion of the force carrier particles, as the (stable) force of interactions can never take place between elementary particles at different frequencies. Besides the strong mathematical proofs given in this paper to prove its truthfulness, an experimental prediction has been given to confirm the theory presented in the form of the relation between the electron radius and quarks radii. The paper shows that quarks are direct consequence of this theory, and solves "the flavor problem" in QCD, and gives the clue to answer the questions of "Why are there so many flavors? The paper also derives the equation of the big bang theory which describes the singularity of the moment of creation of the universe.
In this paper, the underlying principles about the theory of relativity are briefly introduced and reviewed. The mathematical prerequisite needed for the understanding of general relativity and of Einstein field equations are discussed. Concepts such as the principle of least action will be included and its explanation using the Lagrange equations will be given. Where possible, the mathematical details and rigorous analysis of the subject has been given in order to ensure a more precise and thorough understanding of the theory of relativity. A brief mathematical analysis of how to derive the Einstein’s field’s equations from the Einstein-Hilbert action and the Schwarzschild solution was also given.
Schrodinger wave equation and its application
a very good animated presentation.
Bs level.
semester 6th.
how to make a very good appreciable presentation.
From last four decades of research it is well-established that all electrophysiological signals are nonlinear, irregular and aperiodic. Since those signals are used in everyday clinical practice as diagnostic tools (EMG, ECG, EEG), a huge progress in using it in making diagnostic more precise and
Similar to Noethers theoram theoritical approach (20)
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
1. BIRLA INSTITUTE OF
TECHNOLOGY
Submission of assignment of the subject
Advance Electrodynamics [SAP3007] on the topic
Noether’s Theorem and Its Consequences
To -
Dr Saurabh Lahiri
Department of physics
30th
September 2021
By –
Shivam Parasar IPH/10033/17
Aditya Narayan Singh IPH/10034/17
2. INTRODUCTION
Noether’s Theorem
If the physical system behaves the same no matter how it is oriented in space, its Lagrangian
is symmetric under continuous rotation. According to this symmetry, Noord's theorem shows
that the angular momentum of the system is conserved. This is the result of the law. The
physical system itself does not have to be symmetric. Zigzag asteroids that fall into space
maintain angular momentum even if they are asymmetric. Basically, the laws of motion are
symmetric.
To give another example, if a physical process shows the same result regardless of location or
time, its Lagrangian quantity is symmetric under a continuous transformation of space and
time: according to Noether's theorem, these symmetries explain the law of conservation of
momentum. The linearity and energy of this system.
Noether's theorem is important because it provides information about conservation law and is
a practical calculus tool. This allows researchers to determine the conserved quantity
(invariant) from the symmetry of the observed physical system. Instead, researchers can use
the given invariants to examine the entire virtual Lagrangian category to describe the physical
system.
Noether's theorem can be expressed as follows.
If the system has continuous symmetry, there is a corresponding number of values stored in
time.
More accurate version:
Conserved currents correspond to all differentiable symmetries produced by local interactions.
The term "symmetry" in the previous statement more accurately refers to the form of
covariance adopted by the laws of physics for a series of one-dimensional Lie group of
3. transformations that meet certain technical standards. The conservation law of physical
quantities is generally expressed as a continuity equation.
The formal proof of the theorem uses invariants to derive the equation for the current associated
with the conserved physical quantity.
The conservation law says that in the mathematical description of the evolution of a system,
the value of X remains constant throughout its entire motion - it is irreversible.
Mathematically, the rate of change of X (time derivative) is zero.
𝑑𝑥
𝑑𝑡
= 𝑥̇ = 0
It is said that this amount is preserved. It is often said that they are an integral part of the
movement (it is not necessary to include the movement itself, but only the evolution of time).
For example, if the energy of a system is conserved, that energy is always irreversible, which
can impose constraints on the behavior of the system and help solve the system. These constant
movements not only provide insight into the nature of the system, but are also useful
computational tools. For example, you can change the approximate solution by finding the
closest position that satisfies the corresponding conservation law.
From the late eighteenth to the early nineteenth century, physicists developed more systematic
methods for discovering inventions. In 1788, great progress was made in the development of
Lagrangian mechanics associated with the principle of least action. With this approach, the
state of the system can be described in any number of generalized coordinates q. As is
customary in Newtonian mechanics, there is no need to represent the laws of motion in
Cartesian coordinate systems. The function is defined as the integral I multiplied by a function
known as the Lagrangian L.
𝐼 = ∫ 𝐿(𝑞, 𝑞̇, 𝑡)𝑑𝑡
Here, the coordinates of the point at q represent the rate of change of q.
𝑞̇ =
𝑑𝑞
𝑑𝑡
Hamilton's principle states that a physical path q(t) (the path actually used by the system) is a
path in which small changes in this path do not change I, at least until the first order. This
principle gives rise to the Euler-Lagrange equation.
𝑑
𝑑𝑡
(
𝜕𝐿
𝜕𝑞̇
) =
𝜕𝐿
𝜕𝑞
Thus, if one of the coordinates, for example qk, does not appear in the Lagrangian, then the
right side of the equation is zero, and the left side needs it.
𝑑
𝑑𝑡
(
𝜕𝐿
𝜕𝑞̇𝑘
) =
𝑑𝑝̇𝑘
𝑑𝑡
= 0
4. Momentum defined as
𝑝𝑘 =
𝜕𝐿
𝜕𝑞𝑘
̇
speed is maintained means momentum is conserved (on the physical path).
Therefore, the absence of a negligibly small coordinate qk in the Lagrangian means that the
Lagrangian is not affected by changes or transformations of qk. Lagrangian are invariant and
are said to have symmetry under such transformations. This is the original idea, generalized
by Noether's theorem. We can directly make a statement that Noether’s theorem gives the
relation ship between symmetry and conservation laws. It is very essential to know that by
which transformation we get what physical quantity conservation. And this theorem can help
up. Now we would dig little deep to know about symmetries.
Symmetry
Physical symmetry is generalized to be invariant or immutable with respect to all kinds of
transformations, such as arbitrary coordinate transformations. This concept has become one of
the most powerful tools in theoretical physics, and virtually all laws of nature follow from
symmetry. Indeed, this role should be ascribed to Nobel laureate P. V. Anderson in his widely
read 1972 paper "There Is More Other": "It would not be an exaggeration to say that physics is
the study of symmetry." said. Noether's theorem (in a very simplified form states that for all
continuous mathematical symmetries there are corresponding conserved quantities such as
energy and momentum, and in Noether's native language there are conserved growth currents).
Wigner also believes that the symmetry of the laws of physics determines the properties of
particles found in nature.
Important symmetries in physics include continuous space-time symmetry and discrete
symmetry. The internal symmetry of particles, super symmetry of the theory of physics.
Continuous symmetry is an intuitive idea that fits with the idea of some symmetry as motion
rather than discrete symmetry. Reflection symmetry, which is irreversible from state to state
with one type of inversion. However, discrete symmetry can always be rethought as a subset
of higher dimensional continuous symmetry. The reflectance of a 2D object in 3D space can
be obtained by continuously rotating the object 180 degrees in a non-parallel plane.
Discrete symmetry is symmetry that represents discontinuous changes in the system. For
example, a square has discrete rotation symmetry because only rotation through an angle that
is a multiple of a straight line retains the original square shape. Discrete symmetry can include
a kind of "permutation". These swaps are commonly referred to as reflections or swaps. In
mathematics and theoretical physics, discrete symmetry is a symmetry about the transformation
of discrete groups. A group of topologies with a discrete topology, the elements of which form
a finite or countable set.
In supersymmetric theory, the equation of force and the equation of matter are the same. In
theoretical and mathematical physics, theories with this property are called supersymmetry
5. theory. There are dozens of supersymmetric theories. Supersymmetry is the space-time
symmetry between the two main classes of particles. Bosons have integer spins and obey
boson-Einstein statistics. Fermions have half-integer spins and obey the Fermi - Dirac statistics.
In supersymmetry, every particle of one class has a particle attached to another class, called its
super partner, whose spins differ by half of the integers. For example, when an electron is
present in supersymmetric theory, there is a particle called an "electron" (super partner
electron) that is the bosonic partner of the electron. In the simplest supersymmetry theory,
which has a completely "monolithic" supersymmetry, each pair of supersymmetric particles
has the same mass and internal quantum number, with the exception of spin. More sophisticated
supersymmetry theories can spontaneously break symmetry, and supersymmetric particles
have different masses.
Supersymmetry includes quantum mechanics, statistical mechanics, quantum field theory,
condensed matter physics, nuclear physics, optics, stochastic mechanics, particle physics,
astronomical physics, quantum gravity, string theory, cosmology, etc. It is used in different
ways in different areas of physics. Supersymmetry is applicable not only to physics, such as
finance.
Continuous Symmetries and Conservation Law
A conservation law states a particular measurable property of an isolated physical system does
not change as the system evolves over time. If we talk over exact conservation laws then we
have conservation of mass, linear momentum, angular momentum, electric charge, etc. And
again, if we talk about approximate conservation then we have conservation law for mass
parity, lepton number, baryon number, hypercharge. Also, one other conservation (local
conservation) states that the amount of the conserved quantity at a point or within a volume
can only be changed by the amount of quantity that flows in or out of the volume.
More definition of Noether’s theorem:
there is one to one correspondence between each one of them and the differentiable symmetry
of nature.
Derivation:
Let A be invariant under a set of transformation.
𝑞(𝑡) → 𝑞′(𝑡) ⇒ 𝑓(𝑞(𝑡), 𝑞′(𝑡))
{
𝑓(𝑞(𝑡), 𝑞′(𝑡))𝑖𝑠 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑞(𝑡)
𝑞(𝑡)𝑖𝑠 𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡, 𝑞 𝑖𝑛𝑑𝑖𝑐𝑎𝑡𝑒𝑠 𝑚𝑜𝑡𝑖𝑜𝑛
Perform symmetry again and again then it is again symmetry and in general, called
symmetric group. So
𝛿𝑠𝑞(𝑡) = 𝑞′(𝑡) − 𝑞(𝑡) → 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙 𝑖𝑛 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑎𝑛𝑑 𝑓𝑖𝑛𝑎𝑙 𝑠𝑦𝑠𝑡𝑒𝑚
𝛿𝑠 → 𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦 𝑣𝑎𝑟𝑖𝑎𝑖𝑜𝑛
Equation 1
6. 𝛿𝑠𝑞(𝑡) = ∈ ∆(𝑞(𝑡) , 𝑞′(𝑡), 𝑡) → 𝐺𝑒𝑛𝑒𝑟𝑎𝑙 𝑓𝑜𝑟𝑚
We must note that symmetry variations are non-zero at the ends unlike the euler-lagrange
equation.
using chain rule differential and integration by parts we have,
𝛿𝑠𝐴 = ∫ 𝑑𝑡 [
𝜕𝐿
𝜕𝑞(𝑡)
− 𝜕𝑡
𝜕𝐿
𝜕𝑞′(𝑡)
] 𝛿𝑠𝑞(𝑡) +
𝜕𝐿
𝜕𝑞′(𝑡)
𝛿𝑠𝑞(𝑡)
𝑡𝑏
𝑡𝑎
|𝑡𝑎
𝑡𝑏
For the path q(t) that satisfy Euler-lagrange equation the first term gets vanish as it is equal to
zero.
So, we are left with
𝛿𝑠𝐴 = ∈
𝜕𝐿
𝜕𝑞′
∆(𝑞, 𝑞′
, 𝑡)|𝑡𝑏
𝑡𝑎
Sins according to symmetry group assumption 𝛿𝑠𝐴 can vanish anytime
𝑄(𝑡) =
𝜕𝐿
𝜕𝑞′ ∆(𝑞, 𝑞′
, 𝑡)
Q(t) Is independent of time t so we can write it
Q(t) = Q
so, it is a conserved quantity, also called a constant of motion and
𝜕𝐿
𝜕𝑞′
∆(𝑞, 𝑞′
, 𝑡) is Noether’s
charge.
If we generalize equation 5 for which the action is directly not invariant but its symmetry
variation is equal to an arbitrary boundary condition. So,
𝛿𝑠𝐴 = ∈∧ (𝑞, 𝑞̇, 𝑡)|𝑡𝑎
𝑡𝑏
Using equation 5
𝑄(𝑡) =
𝜕𝐿
𝜕𝑞′
∆(𝑞, 𝑞′
, 𝑡) − ∧ (𝑞, 𝑞′
, 𝑡)
{
𝑄(𝑡) → 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑜𝑓 𝑚𝑜𝑡𝑖𝑜𝑛
𝜕𝐿
𝜕𝑞′
∆(𝑞, 𝑞′
, 𝑡) → 𝑁𝑜𝑒𝑡ℎ𝑒𝑟′
𝑠 𝐶ℎ𝑎𝑟𝑔𝑒
Now if we consider the action considering Lagrangian to symmetry variation can be stated as
follows,
𝛿𝑠𝐿̇ = 𝐿(𝑞 + 𝛿𝑠𝑞 , 𝑞̇ + 𝛿𝑠𝑞̇) − 𝐿̇(𝑞 − 𝑞̇)
𝛿𝑠𝐿̇ = [
𝜕𝐿
𝜕𝑞(𝑡)
− 𝜕𝑡
𝜕𝐿
𝜕𝑞̇(𝑡)
] 𝛿𝑠𝑞̇(𝑡) +
𝑑
𝑑𝑡
[
𝜕𝐿
𝜕𝑞(𝑡)
𝛿𝑠𝑞(𝑡)]
Equation 2
Equation 3
Equation 4
Equation 5
Equation 6
Equation 7
Equation 8
Equation 9
7. Again, the Euler-lagrange term vanishes. So, the assumption of invariance of action in
equation 7 is equivalent to assuming that the symmetry variation of Lagrangian is the total
time derivative of some function ∧ (𝑞, 𝑞̇, 𝑡).
𝛿𝑠𝐿(𝑞, 𝑞̇, 𝑡) = ∈
𝑑
𝑑𝑡
∧ (𝑞, 𝑞̇, 𝑡)
So, using equation five again we can write equation 10 as,
∈
𝑑
𝑑𝑡
[
𝑑𝐿
𝑑𝑞̇
∆(𝑞, 𝑞̇, 𝑡) −∧ (𝑞, 𝑞̇, 𝑡)] = 0
And hence we again recovered Noether’s Charge.
The existence of a conserved quantity for every continuous symmetry is the content of
Noether’s theorem.
Displacement and Energy Conservation
Consider the case that Lagrangian does not explicitly depend on the time
therefore, we can write
t’ = t - E (time translation equation)
Also, the above statement can be written as
𝐿(𝑞, 𝑞̇, 𝑡) = 𝐿(𝑞, 𝑞̇)
Sue for the same part say q(t) we can write
𝑞̇(𝑡′) = 𝑞(𝑡)
𝑞̇(𝑡′) 𝑤𝑖𝑡ℎ 𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑒𝑑 𝑡𝑖𝑚𝑒 𝑖𝑠 𝑠𝑎𝑚𝑒 𝑎𝑠 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑡𝑖𝑚𝑒 𝑡, 𝑞(𝑡)
Using symmetry variation described in this section we say,
𝛿𝑠𝑞(𝑡) = 𝑞′(𝑡) − 𝑞(𝑡) = 𝑞(𝑡′
+ 𝑡) − 𝑞(𝑡)
= 𝑞(𝑡′)− ∈ 𝑞̇(𝑡′) − 𝑞(𝑡) = ∈ 𝑞̇(𝑡)
So, symmetry variation of Lagrangian is
𝛿𝑠𝐿 = 𝐿(𝑞′(𝑡), 𝑞̇′(𝑡)) − 𝐿(𝑞(𝑡) − 𝑞̇(𝑡)) =
𝜕𝐿
𝜕𝑞
𝛿𝑠𝑞(𝑡) +
𝜕𝐿
𝜕𝑞̇
𝛿𝑠𝑞̇(𝑡)
Inserting equation 3 in equation 4 the value of 𝛿𝑠𝑞(𝑡)
Equation 10
Equation 11
Equation 1
Equation 2
Equation 3
Equation 4
8. 𝛿𝑠𝐿 = ∈ (
𝜕𝐿
𝜕𝑞̇
𝑞̇ +
𝜕𝐿
𝜕𝑞̇
𝑞̈) = ∈
𝑑𝐿
𝑑𝑡
this equation is of the form like equation 10 of the previous section with 𝛬 = 𝐿 as we know
that time translation is symmetry transformation. so 𝛬 in equation 10 of the previous section
concide with Lagrangian.
Also, we have Noether’s charge equation as
𝑄 =
𝜕𝐿
𝜕𝑞̇
𝑞̇ − 𝐿(𝑞, 𝑞̇) → 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑜𝑓 𝑚𝑜𝑡𝑖𝑜𝑛
Let the time-dependent symmetry variation is
𝛿𝑠
𝑡
𝑞(𝑡) = ∈ (𝑡)𝑞̇(𝑡)
So corresponding Lagrangian charge is given as
𝛿𝑠
𝑡
𝐿 =
𝜕𝐿
𝜕𝑞
∈ 𝑞̇ +
𝜕𝐿
𝜕𝑞̇
(∈ 𝑞+∈ 𝑞̈)
̇ =
𝜕𝐿𝑡
𝜕𝑡̇
∈ +
𝜕𝐿𝑡
𝜕𝑡̇
∈̇
Where,
𝜕𝐿𝑡
𝜕𝑡̇
=
𝜕𝐿
𝜕𝑞̇
𝑞̇
𝜕𝐿𝑡
𝜕𝑡̇
=
𝜕𝐿
𝜕𝑞
𝑞̇ +
𝜕𝐿
𝜕𝑞̇
∈ 𝑞̈ =
𝑑𝐿
𝑑𝑡
The Noether’s charge coincide with hamiltonian of system
(Means time translation fulfil the
symmetry condition)
Equation 5
Equation 6